How is the induced drag calculated for a wing with elliptical planform?
Induced drag is caused by the downward deflection of the air streaming around the wing. The resulting aerodynamic force is tilted backwards by half the deflection angle, and the air flows off the wing with an added vertical speed component, producing downwash. Increasing the downwash angle means increasing both lift and the backward tilt, so the induced drag goes up with the square of the lift produced.

If you want to minimize induced drag for a given lift, this quadratic dependence means the optimum is reached when the downwash angle is constant over span.


  How is the induced drag calculated for a wing with elliptical planform?


The elliptical, untwisted wing has the same angle of attack and the same lift coefficient over span, and produces the desired constant downwash angle.  To simplify things, let's assume the wing is just acting on the air with the density $\rho$ flowing with the speed $v$ through a circle with a diameter equal to the span $b$ of the wing. If we just look at this stream tube, the mass flow is $$\frac{dm}{dt} = \frac{b^2}{4}\cdot\pi\cdot\rho\cdot v$$

Lift $L$ is then the impulse change which is caused by the wing. With the downward air speed $v_z$ imparted by the wing, lift is: $$L = \frac{b^2}{4}\cdot\pi\cdot\rho\cdot v\cdot v_z = S\cdot c_L\cdot\frac{v^2}{2}\cdot\rho$$

$S$ is the wing area and $c_L$ the overall lift coefficient. If we now solve for the vertical air speed, we get $$v_z = \frac{S\cdot c_L\cdot\frac{v^2}{2}\cdot\rho}{\frac{b^2}{4}\cdot\pi\cdot\rho\cdot v} = \frac{2\cdot c_L\cdot v}{\pi\cdot AR}$$
with $AR = \frac{b^2}{S}$ the aspect ratio of the wing. Now we can divide the vertical speed by the air speed to calculate the angle by which the air has been deflected by the wing. Let's call it $\alpha_w$: $$\alpha_w = arctan\left(\frac{v_z}{v}\right) = arctan \left(\frac{2\cdot c_L}{\pi\cdot AR}\right)$$

The deflection happens gradually along the wing chord, so the mean local flow angle along the chord is just $\alpha_w / 2$. Lift acts perpendicularly to this local flow, thus is tilted backwards by $\alpha_w / 2$. In coefficients, lift is $c_L$, and the backwards component is $\alpha_w / 2 \cdot c_L$. Let's call this component $c_{Di}$: $$c_{Di} = arctan \left(\frac{c_L}{\pi\cdot AR}\right)\cdot c_L$$

For small $\alpha_w$s the arcus tangens can be neglected, and we get this familiar looking equation for the backwards-pointing component of the reaction force: $$c_{Di} = \frac{c_L^2}{\pi\cdot AR}$$

If the circulation over span has an elliptic distribution, the local change in circulation times the local amount of circulation is constant, and the induced drag $c_{Di}$ is at it's minimum. If this would be different, a higher local $v_z$ causes a quadratic increase in local induced drag, so the whole wing will create it's lift less efficiently.


  Is this wing shape the most efficient ?


Only if you ask an aerodynamicist would the answer be yes. An elliptic wing will give you the best ratio of lift to drag, which clearly is one way to express efficiency.

In reality, the wing has to lift itself plus a payload, but only lifting the payload should be considered when formulating efficiency. Therefore, pure lift/drag optimization is too narrow. What should count is the best ratio of lift minus wing weight relative to drag. R. T. Jones wrote a NACA Technical Note back in 1950 in which he looked at this problem analytically. Wing weight goes up when much lift is created near the tips, because this lift will cause a disproportional root bending moment, and the wing spar which has to carry this bending moment is a significant part of the wing structure. Therefore, reducing lift at the tips and adding more lift at the root will create a lighter wing for a modest drag increase, resulting in an overall optimum for an almost triangular lift distribution. When compared to an elliptical wing planform, the total wing span of such an optimized wing is bigger for the same overall drag, but this wing will weigh less.



Spanwise loading comparison for wings of the same fixed lift, from NACA Technical Note 2249. 

But this is too easy. Scaling laws must be considered in addition. You know that elephants have much more massive legs relative to their body size than antelopes (or even ants, for an even more drastic comparison), since body mass scales with the cube of linear dimension while structural strength scales only with the square of linear dimension. This means that wing spar weight will be proportionally higher for larger aircraft.

As a consequence, insects have more elliptic wings than albatrosses, and model aircraft have optimum wings which are much more elliptic than the optimum wing of an airliner. The optimum shifts from an elliptic load distribution at very small scales to an almost triangular distribution at large scales.