Optimal. Leaf size=99 \[ -\frac{\tan ^3(c+d x)}{a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{13 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{11 \tan (c+d x) \sec (c+d x)}{8 a^3 d}+\frac{x}{a^3} \]
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Rubi [A] time = 0.204299, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3888, 3886, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ -\frac{\tan ^3(c+d x)}{a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{13 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{11 \tan (c+d x) \sec (c+d x)}{8 a^3 d}+\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3886
Rule 3473
Rule 8
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rule 3768
Rubi steps
\begin{align*} \int \frac{\tan ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\frac{\int (-a+a \sec (c+d x))^3 \tan ^2(c+d x) \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \tan ^2(c+d x)+3 a^3 \sec (c+d x) \tan ^2(c+d x)-3 a^3 \sec ^2(c+d x) \tan ^2(c+d x)+a^3 \sec ^3(c+d x) \tan ^2(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \tan ^2(c+d x) \, dx}{a^3}+\frac{\int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{a^3}+\frac{3 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{a^3}-\frac{3 \int \sec ^2(c+d x) \tan ^2(c+d x) \, dx}{a^3}\\ &=-\frac{\tan (c+d x)}{a^3 d}+\frac{3 \sec (c+d x) \tan (c+d x)}{2 a^3 d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{4 a^3 d}-\frac{\int \sec ^3(c+d x) \, dx}{4 a^3}+\frac{\int 1 \, dx}{a^3}-\frac{3 \int \sec (c+d x) \, dx}{2 a^3}-\frac{3 \operatorname{Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{a^3 d}\\ &=\frac{x}{a^3}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{11 \sec (c+d x) \tan (c+d x)}{8 a^3 d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{4 a^3 d}-\frac{\tan ^3(c+d x)}{a^3 d}-\frac{\int \sec (c+d x) \, dx}{8 a^3}\\ &=\frac{x}{a^3}-\frac{13 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{11 \sec (c+d x) \tan (c+d x)}{8 a^3 d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{4 a^3 d}-\frac{\tan ^3(c+d x)}{a^3 d}\\ \end{align*}
Mathematica [B] time = 0.729249, size = 230, normalized size = 2.32 \[ \frac{\sec ^4(c+d x) \left (38 \sin (c+d x)-32 \sin (2 (c+d x))+22 \sin (3 (c+d x))+39 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 \cos (2 (c+d x)) \left (13 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-13 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 d x\right )+\cos (4 (c+d x)) \left (13 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-13 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 d x\right )-39 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+24 d x\right )}{64 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.088, size = 228, normalized size = 2.3 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-{\frac{1}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-4}}+{\frac{3}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{27}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{21}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{13}{8\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-4}}+{\frac{3}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}+{\frac{27}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{21}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{13}{8\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.64603, size = 347, normalized size = 3.51 \begin{align*} \frac{\frac{2 \,{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{13 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{21 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{3} - \frac{4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac{16 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac{13 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac{13 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19605, size = 265, normalized size = 2.68 \begin{align*} \frac{16 \, d x \cos \left (d x + c\right )^{4} - 13 \, \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) + 13 \, \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (11 \, \cos \left (d x + c\right )^{2} - 8 \, \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{16 \, a^{3} d \cos \left (d x + c\right )^{4}} \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 [A] time = 13.369, size = 166, normalized size = 1.68 \begin{align*} \frac{\frac{8 \,{\left (d x + c\right )}}{a^{3}} - \frac{13 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac{13 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{2 \,{\left (21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 13 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{4} a^{3}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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