Optimal. Leaf size=237 \[ -\frac{3 \tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{125 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}+\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a^3 d}-\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a^3 d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a^3 d}+\frac{\tan ^5(c+d x) \sec (c+d x)}{2 a^3 d}-\frac{5 \tan ^3(c+d x) \sec (c+d x)}{8 a^3 d}+\frac{115 \tan (c+d x) \sec (c+d x)}{128 a^3 d}+\frac{x}{a^3} \]
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Rubi [A] time = 0.362998, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ -\frac{3 \tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{125 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}+\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a^3 d}-\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a^3 d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a^3 d}+\frac{\tan ^5(c+d x) \sec (c+d x)}{2 a^3 d}-\frac{5 \tan ^3(c+d x) \sec (c+d x)}{8 a^3 d}+\frac{115 \tan (c+d x) \sec (c+d x)}{128 a^3 d}+\frac{x}{a^3} \]
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 ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\frac{\int (-a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \tan ^6(c+d x)+3 a^3 \sec (c+d x) \tan ^6(c+d x)-3 a^3 \sec ^2(c+d x) \tan ^6(c+d x)+a^3 \sec ^3(c+d x) \tan ^6(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \tan ^6(c+d x) \, dx}{a^3}+\frac{\int \sec ^3(c+d x) \tan ^6(c+d x) \, dx}{a^3}+\frac{3 \int \sec (c+d x) \tan ^6(c+d x) \, dx}{a^3}-\frac{3 \int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a^3}\\ &=-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\sec (c+d x) \tan ^5(c+d x)}{2 a^3 d}+\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a^3 d}-\frac{5 \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx}{8 a^3}+\frac{\int \tan ^4(c+d x) \, dx}{a^3}-\frac{5 \int \sec (c+d x) \tan ^4(c+d x) \, dx}{2 a^3}-\frac{3 \operatorname{Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a^3 d}\\ &=\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{5 \sec (c+d x) \tan ^3(c+d x)}{8 a^3 d}-\frac{5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\sec (c+d x) \tan ^5(c+d x)}{2 a^3 d}+\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a^3 d}-\frac{3 \tan ^7(c+d x)}{7 a^3 d}+\frac{5 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{16 a^3}-\frac{\int \tan ^2(c+d x) \, dx}{a^3}+\frac{15 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{8 a^3}\\ &=-\frac{\tan (c+d x)}{a^3 d}+\frac{15 \sec (c+d x) \tan (c+d x)}{16 a^3 d}+\frac{5 \sec ^3(c+d x) \tan (c+d x)}{64 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{5 \sec (c+d x) \tan ^3(c+d x)}{8 a^3 d}-\frac{5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\sec (c+d x) \tan ^5(c+d x)}{2 a^3 d}+\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a^3 d}-\frac{3 \tan ^7(c+d x)}{7 a^3 d}-\frac{5 \int \sec ^3(c+d x) \, dx}{64 a^3}-\frac{15 \int \sec (c+d x) \, dx}{16 a^3}+\frac{\int 1 \, dx}{a^3}\\ &=\frac{x}{a^3}-\frac{15 \tanh ^{-1}(\sin (c+d x))}{16 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{115 \sec (c+d x) \tan (c+d x)}{128 a^3 d}+\frac{5 \sec ^3(c+d x) \tan (c+d x)}{64 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{5 \sec (c+d x) \tan ^3(c+d x)}{8 a^3 d}-\frac{5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\sec (c+d x) \tan ^5(c+d x)}{2 a^3 d}+\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a^3 d}-\frac{3 \tan ^7(c+d x)}{7 a^3 d}-\frac{5 \int \sec (c+d x) \, dx}{128 a^3}\\ &=\frac{x}{a^3}-\frac{125 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{115 \sec (c+d x) \tan (c+d x)}{128 a^3 d}+\frac{5 \sec ^3(c+d x) \tan (c+d x)}{64 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{5 \sec (c+d x) \tan ^3(c+d x)}{8 a^3 d}-\frac{5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\sec (c+d x) \tan ^5(c+d x)}{2 a^3 d}+\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a^3 d}-\frac{3 \tan ^7(c+d x)}{7 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.3231, size = 362, normalized size = 1.53 \[ \frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (1680000 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\sec (c) \sec ^8(c+d x) (133175 \sin (2 c+d x)-544768 \sin (c+2 d x)+286720 \sin (3 c+2 d x)+63595 \sin (2 c+3 d x)+63595 \sin (4 c+3 d x)-254464 \sin (3 c+4 d x)+161280 \sin (5 c+4 d x)+65135 \sin (4 c+5 d x)+65135 \sin (6 c+5 d x)-118784 \sin (5 c+6 d x)+27195 \sin (6 c+7 d x)+27195 \sin (8 c+7 d x)-14848 \sin (7 c+8 d x)+470400 d x \cos (c)+376320 d x \cos (c+2 d x)+376320 d x \cos (3 c+2 d x)+188160 d x \cos (3 c+4 d x)+188160 d x \cos (5 c+4 d x)+53760 d x \cos (5 c+6 d x)+53760 d x \cos (7 c+6 d x)+6720 d x \cos (7 c+8 d x)+6720 d x \cos (9 c+8 d x)+519680 \sin (c)+133175 \sin (d x))\right )}{215040 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.115, size = 396, normalized size = 1.7 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-{\frac{1}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-8}}+{\frac{13}{14\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-7}}-{\frac{65}{24\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-6}}+{\frac{143}{40\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}-{\frac{79}{64\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-4}}-{\frac{49}{32\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{29}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{253}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{125}{128\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-8}}+{\frac{13}{14\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-7}}+{\frac{65}{24\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-6}}+{\frac{143}{40\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-5}}+{\frac{79}{64\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-4}}-{\frac{49}{32\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{29}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{253}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{125}{128\,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 [A] time = 1.59826, size = 579, normalized size = 2.44 \begin{align*} -\frac{\frac{2 \,{\left (\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{11375 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{79723 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{269879 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{550089 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{749973 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac{212625 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac{26565 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}\right )}}{a^{3} - \frac{8 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{28 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{56 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{70 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{56 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{28 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac{8 \, a^{3} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac{a^{3} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac{26880 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac{13125 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} - \frac{13125 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{13440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29227, size = 446, normalized size = 1.88 \begin{align*} \frac{26880 \, d x \cos \left (d x + c\right )^{8} - 13125 \, \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) + 13125 \, \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (14848 \, \cos \left (d x + c\right )^{7} - 27195 \, \cos \left (d x + c\right )^{6} + 7424 \, \cos \left (d x + c\right )^{5} + 17710 \, \cos \left (d x + c\right )^{4} - 14592 \, \cos \left (d x + c\right )^{3} - 1960 \, \cos \left (d x + c\right )^{2} + 5760 \, \cos \left (d x + c\right ) - 1680\right )} \sin \left (d x + c\right )}{26880 \, a^{3} d \cos \left (d x + c\right )^{8}} \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 = 69.7811, size = 236, normalized size = 1. \begin{align*} \frac{\frac{13440 \,{\left (d x + c\right )}}{a^{3}} - \frac{13125 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac{13125 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{2 \,{\left (26565 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} - 212625 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 749973 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 550089 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 269879 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 79723 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 11375 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 315 \, \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 )}^{8} a^{3}}}{13440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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