Optimal. Leaf size=1141 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.65882, antiderivative size = 1141, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 14, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.824, Rules used = {6742, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 260} \[ \frac{8 c d^3 \log (d+e x) e^7}{\left (c d^4+a e^4\right )^3}-\frac{2 c d^3 \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^3}-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac{\sqrt{c} d \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e^5}{\sqrt{a} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} \left (\sqrt{c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt{a} e^2 \left (7 c d^4-a e^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} \left (\sqrt{c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt{a} e^2 \left (7 c d^4-a e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (7 c d^4-a e^4\right )\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (7 c d^4-a e^4\right )\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt{c} d \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e}{2 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac{c \left (4 a d^3 e^3+x \left (\left (c d^4-3 a e^4\right ) d^2-2 e \left (c d^4-a e^4\right ) x d+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (c x^4+a\right )}-\frac{\sqrt [4]{c} \left (3 \sqrt{c} \left (c d^4-3 a e^4\right ) d^2+\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} \left (3 \sqrt{c} \left (c d^4-3 a e^4\right ) d^2+\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} \left (3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} \left (3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 1854
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 1248
Rule 635
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx &=\int \left (\frac{e^8}{\left (c d^4+a e^4\right )^2 (d+e x)^2}+\frac{8 c d^3 e^8}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2-4 c d^3 e^3 x^3\right )}{\left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2-8 c d^3 e^3 x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac{\left (c e^4\right ) \int \frac{d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2-8 c d^3 e^3 x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}+\frac{c \int \frac{d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2-4 c d^3 e^3 x^3}{\left (a+c x^4\right )^2} \, dx}{\left (c d^4+a e^4\right )^2}\\ &=-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac{\left (c e^4\right ) \int \left (\frac{x \left (-2 d e \left (3 c d^4-a e^4\right )-8 c d^3 e^3 x^2\right )}{a+c x^4}+\frac{d^2 \left (5 c d^4-3 a e^4\right )+e^2 \left (7 c d^4-a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^3}-\frac{c \int \frac{-3 d^2 \left (c d^4-3 a e^4\right )+4 d e \left (c d^4-a e^4\right ) x-e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac{\left (c e^4\right ) \int \frac{x \left (-2 d e \left (3 c d^4-a e^4\right )-8 c d^3 e^3 x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}+\frac{\left (c e^4\right ) \int \frac{d^2 \left (5 c d^4-3 a e^4\right )+e^2 \left (7 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}-\frac{c \int \left (\frac{4 d e \left (c d^4-a e^4\right ) x}{a+c x^4}+\frac{-3 d^2 \left (c d^4-3 a e^4\right )-e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac{\left (c e^4\right ) \operatorname{Subst}\left (\int \frac{-2 d e \left (3 c d^4-a e^4\right )-8 c d^3 e^3 x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}-\frac{c \int \frac{-3 d^2 \left (c d^4-3 a e^4\right )-e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^2}-\frac{\left (c d e \left (c d^4-a e^4\right )\right ) \int \frac{x}{a+c x^4} \, dx}{a \left (c d^4+a e^4\right )^2}-\frac{\left (e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}+\frac{\left (e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}\\ &=-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac{\left (4 c^2 d^3 e^7\right ) \operatorname{Subst}\left (\int \frac{x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^3}-\frac{\left (c d e^5 \left (3 c d^4-a e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^3}-\frac{\left (c d e \left (c d^4-a e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^4+a e^4\right )^2}-\frac{\left (3 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^2}+\frac{\left (3 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^2}+\frac{\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac{\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac{\left (e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}+\frac{\left (e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}\\ &=-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}-\frac{\sqrt{c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^4+a e^4\right )^3}-\frac{\sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac{8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac{2 c d^3 e^7 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^3}+\frac{\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}+\frac{\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}+\frac{\left (3 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )^2}+\frac{\left (3 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )^2}+\frac{\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac{\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}\\ &=-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}-\frac{\sqrt{c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^4+a e^4\right )^3}-\frac{\sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac{8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac{2 c d^3 e^7 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^3}+\frac{\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}-\frac{\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}-\frac{\sqrt{c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^4+a e^4\right )^3}-\frac{\sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac{8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac{2 c d^3 e^7 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^3}\\ \end{align*}
Mathematica [A] time = 0.983025, size = 807, normalized size = 0.71 \[ \frac{256 c d^3 \log (d+e x) e^7-64 c d^3 \log \left (c x^4+a\right ) e^7-\frac{32 \left (c d^4+a e^4\right ) e^7}{d+e x}+\frac{8 c \left (c d^4+a e^4\right ) \left (c x \left (d^2-2 e x d+3 e^2 x^2\right ) d^4+a e^3 \left (4 d^3-3 e x d^2+2 e^2 x^2 d-e^3 x^3\right )\right )}{a \left (c x^4+a\right )}+\frac{2 \sqrt [4]{c} \left (-3 \sqrt{2} c^{5/2} d^{10}+8 \sqrt [4]{a} c^{9/4} e d^9-3 \sqrt{2} \sqrt{a} c^2 e^2 d^8-14 \sqrt{2} a c^{3/2} e^4 d^6+48 a^{5/4} c^{5/4} e^5 d^5-30 \sqrt{2} a^{3/2} c e^6 d^4+21 \sqrt{2} a^2 \sqrt{c} e^8 d^2-24 a^{9/4} \sqrt [4]{c} e^9 d+5 \sqrt{2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{2 \sqrt [4]{c} \left (3 \sqrt{2} c^{5/2} d^{10}+8 \sqrt [4]{a} c^{9/4} e d^9+3 \sqrt{2} \sqrt{a} c^2 e^2 d^8+14 \sqrt{2} a c^{3/2} e^4 d^6+48 a^{5/4} c^{5/4} e^5 d^5+30 \sqrt{2} a^{3/2} c e^6 d^4-21 \sqrt{2} a^2 \sqrt{c} e^8 d^2-24 a^{9/4} \sqrt [4]{c} e^9 d-5 \sqrt{2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{\sqrt{2} \sqrt [4]{c} \left (3 c^{5/2} d^{10}-3 \sqrt{a} c^2 e^2 d^8+14 a c^{3/2} e^4 d^6-30 a^{3/2} c e^6 d^4-21 a^2 \sqrt{c} e^8 d^2+5 a^{5/2} e^{10}\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{7/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (3 c^{5/2} d^{10}-3 \sqrt{a} c^2 e^2 d^8+14 a c^{3/2} e^4 d^6-30 a^{3/2} c e^6 d^4-21 a^2 \sqrt{c} e^8 d^2+5 a^{5/2} e^{10}\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{7/4}}}{32 \left (c d^4+a e^4\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 1636, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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