Optimal. Leaf size=161 \[ \frac{a^2 \tan ^7(c+d x)}{7 d}+\frac{a^2 \tan ^5(c+d x)}{5 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac{5 a^2 \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac{5 a^2 \tan (c+d x) \sec (c+d x)}{8 d}-a^2 x \]
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Rubi [A] time = 0.178566, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3886, 3473, 8, 2611, 3770, 2607, 30} \[ \frac{a^2 \tan ^7(c+d x)}{7 d}+\frac{a^2 \tan ^5(c+d x)}{5 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac{5 a^2 \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac{5 a^2 \tan (c+d x) \sec (c+d x)}{8 d}-a^2 x \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^2 \tan ^6(c+d x) \, dx &=\int \left (a^2 \tan ^6(c+d x)+2 a^2 \sec (c+d x) \tan ^6(c+d x)+a^2 \sec ^2(c+d x) \tan ^6(c+d x)\right ) \, dx\\ &=a^2 \int \tan ^6(c+d x) \, dx+a^2 \int \sec ^2(c+d x) \tan ^6(c+d x) \, dx+\left (2 a^2\right ) \int \sec (c+d x) \tan ^6(c+d x) \, dx\\ &=\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{a^2 \sec (c+d x) \tan ^5(c+d x)}{3 d}-a^2 \int \tan ^4(c+d x) \, dx-\frac{1}{3} \left (5 a^2\right ) \int \sec (c+d x) \tan ^4(c+d x) \, dx+\frac{a^2 \operatorname{Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{5 a^2 \sec (c+d x) \tan ^3(c+d x)}{12 d}+\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{a^2 \sec (c+d x) \tan ^5(c+d x)}{3 d}+\frac{a^2 \tan ^7(c+d x)}{7 d}+a^2 \int \tan ^2(c+d x) \, dx+\frac{1}{4} \left (5 a^2\right ) \int \sec (c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{a^2 \tan (c+d x)}{d}+\frac{5 a^2 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{5 a^2 \sec (c+d x) \tan ^3(c+d x)}{12 d}+\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{a^2 \sec (c+d x) \tan ^5(c+d x)}{3 d}+\frac{a^2 \tan ^7(c+d x)}{7 d}-\frac{1}{8} \left (5 a^2\right ) \int \sec (c+d x) \, dx-a^2 \int 1 \, dx\\ &=-a^2 x-\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{5 a^2 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{5 a^2 \sec (c+d x) \tan ^3(c+d x)}{12 d}+\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{a^2 \sec (c+d x) \tan ^5(c+d x)}{3 d}+\frac{a^2 \tan ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [B] time = 1.45914, size = 337, normalized size = 2.09 \[ \frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^7(c+d x) \left (33600 \cos ^7(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\sec (c) (-16240 \sin (2 c+d x)+2975 \sin (c+2 d x)+2975 \sin (3 c+2 d x)+14448 \sin (2 c+3 d x)-10080 \sin (4 c+3 d x)+980 \sin (3 c+4 d x)+980 \sin (5 c+4 d x)+6496 \sin (4 c+5 d x)-1680 \sin (6 c+5 d x)+1155 \sin (5 c+6 d x)+1155 \sin (7 c+6 d x)+1168 \sin (6 c+7 d x)-14700 d x \cos (2 c+d x)-8820 d x \cos (2 c+3 d x)-8820 d x \cos (4 c+3 d x)-2940 d x \cos (4 c+5 d x)-2940 d x \cos (6 c+5 d x)-420 d x \cos (6 c+7 d x)-420 d x \cos (8 c+7 d x)+24640 \sin (d x)-14700 d x \cos (d x))\right )}{215040 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 226, normalized size = 1.4 \begin{align*}{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{2}\tan \left ( dx+c \right ) }{d}}-{a}^{2}x-{\frac{{a}^{2}c}{d}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{12\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d}}+{\frac{5\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{5\,{a}^{2}\sin \left ( dx+c \right ) }{8\,d}}-{\frac{5\,{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77152, size = 204, normalized size = 1.27 \begin{align*} \frac{240 \, a^{2} \tan \left (d x + c\right )^{7} + 112 \,{\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{2} - 35 \, a^{2}{\left (\frac{2 \,{\left (33 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.913171, size = 446, normalized size = 2.77 \begin{align*} -\frac{1680 \, a^{2} d x \cos \left (d x + c\right )^{7} + 525 \, a^{2} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 525 \, a^{2} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (1168 \, a^{2} \cos \left (d x + c\right )^{6} + 1155 \, a^{2} \cos \left (d x + c\right )^{5} - 256 \, a^{2} \cos \left (d x + c\right )^{4} - 910 \, a^{2} \cos \left (d x + c\right )^{3} - 192 \, a^{2} \cos \left (d x + c\right )^{2} + 280 \, a^{2} \cos \left (d x + c\right ) + 120 \, a^{2}\right )} \sin \left (d x + c\right )}{1680 \, d \cos \left (d x + c\right )^{7}} \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*} a^{2} \left (\int 2 \tan ^{6}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.75821, size = 243, normalized size = 1.51 \begin{align*} -\frac{840 \,{\left (d x + c\right )} a^{2} + 525 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 525 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (315 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 2660 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 9863 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 21216 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 29673 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 9660 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1365 \, a^{2} \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 )}^{7}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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