Optimal. Leaf size=581 \[ \frac{\tan (e+f x) \sec ^2(e+f x)^{5/6} F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a f (d \sec (e+f x))^{5/3}}+\frac{3 b}{5 f \left (a^2+b^2\right ) (d \sec (e+f x))^{5/3}}+\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}+\frac{\sqrt{3} b^{8/3} \sec ^2(e+f x)^{5/6} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac{\sqrt{3} b^{8/3} \sec ^2(e+f x)^{5/6} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right )}{2 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}} \]
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Rubi [A] time = 0.819225, antiderivative size = 581, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3512, 757, 429, 444, 51, 63, 210, 634, 618, 204, 628, 208} \[ \frac{\tan (e+f x) \sec ^2(e+f x)^{5/6} F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a f (d \sec (e+f x))^{5/3}}+\frac{3 b}{5 f \left (a^2+b^2\right ) (d \sec (e+f x))^{5/3}}+\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}+\frac{\sqrt{3} b^{8/3} \sec ^2(e+f x)^{5/6} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac{\sqrt{3} b^{8/3} \sec ^2(e+f x)^{5/6} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right )}{2 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 757
Rule 429
Rule 444
Rule 51
Rule 63
Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))} \, dx &=\frac{\sec ^2(e+f x)^{5/6} \operatorname{Subst}\left (\int \frac{1}{(a+x) \left (1+\frac{x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}\\ &=\frac{\sec ^2(e+f x)^{5/6} \operatorname{Subst}\left (\int \left (\frac{a}{\left (a^2-x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{11/6}}+\frac{x}{\left (-a^2+x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{11/6}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}\\ &=\frac{\sec ^2(e+f x)^{5/6} \operatorname{Subst}\left (\int \frac{x}{\left (-a^2+x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}+\frac{\left (a \sec ^2(e+f x)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}\\ &=\frac{F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}+\frac{\sec ^2(e+f x)^{5/6} \operatorname{Subst}\left (\int \frac{1}{\left (-a^2+x\right ) \left (1+\frac{x}{b^2}\right )^{11/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{2 b f (d \sec (e+f x))^{5/3}}\\ &=\frac{3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac{F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}+\frac{\left (b \sec ^2(e+f x)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-a^2+x\right ) \left (1+\frac{x}{b^2}\right )^{5/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{2 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}\\ &=\frac{3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac{F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}+\frac{\left (3 b^3 \sec ^2(e+f x)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{-a^2-b^2+b^2 x^6} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}\\ &=\frac{3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac{F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}-\frac{\left (b^3 \sec ^2(e+f x)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{\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 )}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac{\left (b^3 \sec ^2(e+f x)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{\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 )}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac{\left (b^3 \sec ^2(e+f x)^{5/6}\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 )}{\left (a^2+b^2\right )^{5/3} f (d \sec (e+f x))^{5/3}}\\ &=\frac{3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac{F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}+\frac{\left (b^{8/3} \sec ^2(e+f x)^{5/6}\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 )}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac{\left (b^{8/3} \sec ^2(e+f x)^{5/6}\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 )}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac{\left (3 b^3 \sec ^2(e+f x)^{5/6}\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 )^{5/3} f (d \sec (e+f x))^{5/3}}-\frac{\left (3 b^3 \sec ^2(e+f x)^{5/6}\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 )^{5/3} f (d \sec (e+f x))^{5/3}}\\ &=\frac{3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac{b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac{F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}-\frac{\left (3 b^{8/3} \sec ^2(e+f x)^{5/6}\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 )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac{\left (3 b^{8/3} \sec ^2(e+f x)^{5/6}\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 )^{11/6} f (d \sec (e+f x))^{5/3}}\\ &=\frac{3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac{\sqrt{3} b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{2 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac{\sqrt{3} b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{2 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac{b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac{F_1\left (\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}\\ \end{align*}
Mathematica [B] time = 31.7372, size = 6862, normalized size = 11.81 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.169, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\tan \left ( fx+e \right ) } \left ( d\sec \left ( fx+e \right ) \right ) ^{-{\frac{5}{3}}}}\, 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{5}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}}\,{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}{\left (d \sec{\left (e + f x \right )}\right )^{\frac{5}{3}} \left (a + b \tan{\left (e + f x \right )}\right )}\, 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{5}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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