Optimal. Leaf size=102 \[ -\frac{5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac{\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{24 d}+\frac{\tan (c+d x) (5 a \sec (c+d x)+16 a)}{16 d}-a x \]
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Rubi [A] time = 0.0944652, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3881, 3770} \[ -\frac{5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac{\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{24 d}+\frac{\tan (c+d x) (5 a \sec (c+d x)+16 a)}{16 d}-a x \]
Antiderivative was successfully verified.
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Rule 3881
Rule 3770
Rubi steps
\begin{align*} \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx &=\frac{(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{6} \int (6 a+5 a \sec (c+d x)) \tan ^4(c+d x) \, dx\\ &=-\frac{(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}+\frac{1}{24} \int (24 a+15 a \sec (c+d x)) \tan ^2(c+d x) \, dx\\ &=\frac{(16 a+5 a \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{48} \int (48 a+15 a \sec (c+d x)) \, dx\\ &=-a x+\frac{(16 a+5 a \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{16} (5 a) \int \sec (c+d x) \, dx\\ &=-a x-\frac{5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{(16 a+5 a \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}\\ \end{align*}
Mathematica [A] time = 1.23455, size = 95, normalized size = 0.93 \[ -\frac{a \left (240 \tan ^{-1}(\tan (c+d x))+75 \tanh ^{-1}(\sin (c+d x))-\frac{1}{8} (1168 \cos (c+d x)+140 \cos (2 (c+d x))+568 \cos (3 (c+d x))+165 \cos (4 (c+d x))+184 \cos (5 (c+d x))+295) \tan (c+d x) \sec ^5(c+d x)\right )}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 178, normalized size = 1.8 \begin{align*}{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{a\tan \left ( dx+c \right ) }{d}}-ax-{\frac{ac}{d}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{16\,d}}+{\frac{5\,a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{5\,a\sin \left ( dx+c \right ) }{16\,d}}-{\frac{5\,a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70835, size = 181, normalized size = 1.77 \begin{align*} \frac{32 \,{\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 - 5 \, a{\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 )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05807, size = 377, normalized size = 3.7 \begin{align*} -\frac{480 \, a d x \cos \left (d x + c\right )^{6} + 75 \, a \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 75 \, a \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (368 \, a \cos \left (d x + c\right )^{5} + 165 \, a \cos \left (d x + c\right )^{4} - 176 \, a \cos \left (d x + c\right )^{3} - 130 \, a \cos \left (d x + c\right )^{2} + 48 \, a \cos \left (d x + c\right ) + 40 \, a\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \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 \left (\int \tan ^{6}{\left (c + d x \right )} \sec{\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 = 4.6186, size = 197, normalized size = 1.93 \begin{align*} -\frac{240 \,{\left (d x + c\right )} a + 75 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 75 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (165 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1095 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 3138 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 5118 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1945 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 315 \, a \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 )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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