# Simplifying Elliptic Integrals

According to wolfram alpha, the following equation holds.
$$\int_0^1 \frac{ku^2}{\sqrt{(1 - u^2) (1 - k u^2)}}=K(k)-E(k)$$
where $$K$$ and $$E$$ are complete elliptic integrals of the first and second kind, respectively.
It looked so easy that I tried to prove it myself, but without success.
Can someone please explain this to me?

\begin{align*} I &= \int_0^1 \frac{ku^2}{\sqrt{(1 - u^2) (1 - k u^2)}}\,du \\ &= \int_0^1 \frac{1}{\sqrt{(1 - u^2) (1 - k u^2)}}\,du-\int_0^1 \frac{1-ku^2}{\sqrt{(1 - u^2) (1 - k u^2)}}\,du \\ &= \int_0^1 \frac{du}{\sqrt{(1 - u^2) (1 - k u^2)}}\,du-\int_0^1 \frac{\sqrt{1-ku^2}}{\sqrt{1 - u^2}}\,du \\ &= K(k)-E(k) \end{align*}
per Wolfram|Alpha's convention for $$K$$ and $$E$$.
{"ft_lang_label":"__label__en","ft_lang_prob":0.7129868,"math_prob":0.9999924,"size":726,"snap":"2023-40-2023-50","text_gpt3_token_len":299,"char_repetition_ratio":0.18144044,"word_repetition_ratio":0.24271844,"special_character_ratio":0.45592287,"punctuation_ratio":0.08695652,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999968,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-29T11:36:13Z\",\"WARC-Record-ID\":\"<urn:uuid:032bbe73-7e54-4b66-9a5a-8b1f69d25bf2>\",\"Content-Length\":\"139598\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1e9cfac6-7945-430b-9e9f-7363fbd0ad52>\",\"WARC-Concurrent-To\":\"<urn:uuid:37e9b9e6-a50e-47e2-9175-47579a3ec0b1>\",\"WARC-IP-Address\":\"104.18.43.226\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/4693466/simplifying-elliptic-integrals\",\"WARC-Payload-Digest\":\"sha1:RTZ62VYNSV7G7FWJ4PRXJF6DQ6TG7UHF\",\"WARC-Block-Digest\":\"sha1:ZSMWO526QOQGU4HWHNPDO2L3TMHGQUIH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100081.47_warc_CC-MAIN-20231129105306-20231129135306-00704.warc.gz\"}"}