Optimal. Leaf size=78 \[ \frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{\tan ^3(c+d x) (4-3 \sec (c+d x))}{12 a d}+\frac{\tan (c+d x) (8-3 \sec (c+d x))}{8 a d}-\frac{x}{a} \]
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Rubi [A] time = 0.108635, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3888, 3881, 3770} \[ \frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{\tan ^3(c+d x) (4-3 \sec (c+d x))}{12 a d}+\frac{\tan (c+d x) (8-3 \sec (c+d x))}{8 a d}-\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3881
Rule 3770
Rubi steps
\begin{align*} \int \frac{\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx &=\frac{\int (-a+a \sec (c+d x)) \tan ^4(c+d x) \, dx}{a^2}\\ &=-\frac{(4-3 \sec (c+d x)) \tan ^3(c+d x)}{12 a d}-\frac{\int (-4 a+3 a \sec (c+d x)) \tan ^2(c+d x) \, dx}{4 a^2}\\ &=\frac{(8-3 \sec (c+d x)) \tan (c+d x)}{8 a d}-\frac{(4-3 \sec (c+d x)) \tan ^3(c+d x)}{12 a d}+\frac{\int (-8 a+3 a \sec (c+d x)) \, dx}{8 a^2}\\ &=-\frac{x}{a}+\frac{(8-3 \sec (c+d x)) \tan (c+d x)}{8 a d}-\frac{(4-3 \sec (c+d x)) \tan ^3(c+d x)}{12 a d}+\frac{3 \int \sec (c+d x) \, dx}{8 a}\\ &=-\frac{x}{a}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}+\frac{(8-3 \sec (c+d x)) \tan (c+d x)}{8 a d}-\frac{(4-3 \sec (c+d x)) \tan ^3(c+d x)}{12 a d}\\ \end{align*}
Mathematica [B] time = 6.42618, size = 893, normalized size = 11.45 \[ -\frac{2 x \sec (c+d x) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{\sec (c+d x) a+a}-\frac{3 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec (c+d x) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d (\sec (c+d x) a+a)}+\frac{3 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec (c+d x) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d (\sec (c+d x) a+a)}+\frac{8 \sec (c+d x) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\sec (c+d x) a+a) \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{8 \sec (c+d x) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\sec (c+d x) a+a) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{\sec (c+d x) \left (11 \sin \left (\frac{c}{2}\right )-19 \cos \left (\frac{c}{2}\right )\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 d (\sec (c+d x) a+a) \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}+\frac{\sec (c+d x) \left (19 \cos \left (\frac{c}{2}\right )+11 \sin \left (\frac{c}{2}\right )\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 d (\sec (c+d x) a+a) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}-\frac{\sec (c+d x) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\sec (c+d x) a+a) \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3}-\frac{\sec (c+d x) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\sec (c+d x) a+a) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3}+\frac{\sec (c+d x) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{8 d (\sec (c+d x) a+a) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^4}-\frac{\sec (c+d x) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{8 d (\sec (c+d x) a+a) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.076, size = 228, normalized size = 2.9 \begin{align*} -2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}-{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-4}}+{\frac{5}{6\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{3}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{11}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3}{8\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-4}}+{\frac{5}{6\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}+{\frac{3}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{11}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{3}{8\,da}\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 = 1.7429, size = 333, normalized size = 4.27 \begin{align*} \frac{\frac{2 \,{\left (\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{71 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{137 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{33 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a - \frac{4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{48 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{9 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac{9 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20081, size = 288, normalized size = 3.69 \begin{align*} -\frac{48 \, d x \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) + 9 \, \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (32 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )^{2} - 8 \, \cos \left (d x + c\right ) + 6\right )} \sin \left (d x + c\right )}{48 \, a 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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan ^{6}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 5.1992, size = 166, normalized size = 2.13 \begin{align*} -\frac{\frac{24 \,{\left (d x + c\right )}}{a} - \frac{9 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} + \frac{9 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a} + \frac{2 \,{\left (33 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 137 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 71 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, \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}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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