Optimal. Leaf size=715 \[ \frac{b^2 \tan ^3(e+f x) \sqrt [6]{\sec ^2(e+f x)} F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}+\frac{\tan (e+f x) \sqrt [6]{\sec ^2(e+f x)} F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b}{f \left (a^2+b^2\right )^2 \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt{3} f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt{3} f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{f \left (a^2+b^2\right ) \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )} \]
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Rubi [A] time = 0.99437, antiderivative size = 715, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {3512, 757, 429, 444, 51, 63, 296, 634, 618, 204, 628, 208, 510} \[ \frac{b^2 \tan ^3(e+f x) \sqrt [6]{\sec ^2(e+f x)} F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}+\frac{\tan (e+f x) \sqrt [6]{\sec ^2(e+f x)} F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b}{f \left (a^2+b^2\right )^2 \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt{3} f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt{3} f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 f \left (a^2+b^2\right )^{13/6} \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{f \left (a^2+b^2\right ) \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 757
Rule 429
Rule 444
Rule 51
Rule 63
Rule 296
Rule 634
Rule 618
Rule 204
Rule 628
Rule 208
Rule 510
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))^2} \, dx &=\frac{\sqrt [6]{\sec ^2(e+f x)} \operatorname{Subst}\left (\int \frac{1}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )^{7/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{\sqrt [6]{\sec ^2(e+f x)} \operatorname{Subst}\left (\int \left (\frac{a^2}{\left (a^2-x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{7/6}}-\frac{2 a x}{\left (a^2-x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{7/6}}+\frac{x^2}{\left (-a^2+x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{7/6}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{\sqrt [6]{\sec ^2(e+f x)} \operatorname{Subst}\left (\int \frac{x^2}{\left (-a^2+x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{7/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (2 a \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a^2-x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{7/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (a^2 \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{7/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (a \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right )^2 \left (1+\frac{x}{b^2}\right )^{7/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac{\left (7 a \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right ) \left (1+\frac{x}{b^2}\right )^{7/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{6 b \left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{7 a b}{\left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac{\left (7 a b \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right ) \sqrt [6]{1+\frac{x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{6 \left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{7 a b}{\left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac{\left (7 a b^3 \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^4}{a^2+b^2-b^2 x^6} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{7 a b}{\left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac{\left (7 a b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{1}{2} \sqrt [6]{a^2+b^2}-\frac{\sqrt [3]{b} x}{2}}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (7 a b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{1}{2} \sqrt [6]{a^2+b^2}+\frac{\sqrt [3]{b} x}{2}}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (7 a b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a^2+b^2}-b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{7 a b}{\left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{3 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac{\left (7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{b} \sqrt [6]{a^2+b^2}+2 b^{2/3} x}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{12 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{b} \sqrt [6]{a^2+b^2}+2 b^{2/3} x}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{12 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (7 a b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (7 a b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{7 a b}{\left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{3 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b^{4/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [6]{\sec ^2(e+f x)}}{12 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [6]{\sec ^2(e+f x)}}{12 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac{\left (7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (7 a b^{4/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{7 a b}{\left (a^2+b^2\right )^2 f \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt{3}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{2 \sqrt{3} \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b^{4/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt{3}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{2 \sqrt{3} \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{3 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{7 a b^{4/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [6]{\sec ^2(e+f x)}}{12 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [6]{\sec ^2(e+f x)}}{12 \left (a^2+b^2\right )^{13/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a^2 f \sqrt [3]{d \sec (e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [3]{d \sec (e+f x)}}-\frac{a b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)} \left (a^2-b^2 \tan ^2(e+f x)\right )}\\ \end{align*}
Mathematica [C] time = 51.3887, size = 9626, normalized size = 13.46 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.278, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}}{\frac{1}{\sqrt [3]{d\sec \left ( fx+e \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{d \sec{\left (e + f x \right )}} \left (a + b \tan{\left (e + f x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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