Optimal. Leaf size=579 \[ \frac{\tan (e+f x) \sqrt [6]{\sec ^2(e+f x)} F_1\left (\frac{1}{2};1,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a f \sqrt [3]{d \sec (e+f x)}}+\frac{3 b}{f \left (a^2+b^2\right ) \sqrt [3]{d \sec (e+f x)}}+\frac{b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}-\frac{b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}-\frac{\sqrt{3} b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}+\frac{\sqrt{3} b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right )}{2 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}-\frac{b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}} \]
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Rubi [A] time = 0.850136, antiderivative size = 579, 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, 296, 634, 618, 204, 628, 208} \[ \frac{\tan (e+f x) \sqrt [6]{\sec ^2(e+f x)} F_1\left (\frac{1}{2};1,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a f \sqrt [3]{d \sec (e+f x)}}+\frac{3 b}{f \left (a^2+b^2\right ) \sqrt [3]{d \sec (e+f x)}}+\frac{b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}-\frac{b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}-\frac{\sqrt{3} b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}+\frac{\sqrt{3} b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right )}{2 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}-\frac{b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}} \]
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
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx &=\frac{\sqrt [6]{\sec ^2(e+f x)} \operatorname{Subst}\left (\int \frac{1}{(a+x) \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}{\left (a^2-x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{7/6}}+\frac{x}{\left (-a^2+x^2\right ) \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}{\left (-a^2+x^2\right ) \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 \sqrt [6]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x^2\right ) \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};1,\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 f \sqrt [3]{d \sec (e+f x)}}+\frac{\sqrt [6]{\sec ^2(e+f x)} \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 )}{2 b f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};1,\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 f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (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 )}{2 \left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};1,\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 f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (3 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 ) f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};1,\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 f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (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 )}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (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 )}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (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 )}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}-\frac{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)}}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};1,\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 f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (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 )}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (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 )}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (3 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 ) f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (3 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 ) f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}-\frac{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)}}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{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)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{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)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};1,\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 f \sqrt [3]{d \sec (e+f x)}}+\frac{\left (3 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 )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{\left (3 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 )^{7/6} f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}-\frac{\sqrt{3} 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 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{\sqrt{3} 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 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{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)}}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{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)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac{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)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac{F_1\left (\frac{1}{2};1,\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 f \sqrt [3]{d \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 21.3437, size = 285, normalized size = 0.49 \[ -\frac{60 d F_1\left (\frac{7}{3};\frac{7}{6},\frac{7}{6};\frac{10}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right ) (a \cos (e+f x)+b \sin (e+f x))}{7 b f (d \sec (e+f x))^{4/3} \left (7 (a+i b) F_1\left (\frac{10}{3};\frac{7}{6},\frac{13}{6};\frac{13}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right )+7 (a-i b) F_1\left (\frac{10}{3};\frac{13}{6},\frac{7}{6};\frac{13}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right )+20 (a+b \tan (e+f x)) F_1\left (\frac{7}{3};\frac{7}{6},\frac{7}{6};\frac{10}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.167, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\tan \left ( fx+e \right ) }{\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 )}}\,{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 )}\, 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 )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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