Optimal. Leaf size=102 \[ \frac{\tan ^5(c+d x) (6 a+5 b \sec (c+d x))}{30 d}-\frac{\tan ^3(c+d x) (8 a+5 b \sec (c+d x))}{24 d}+\frac{\tan (c+d x) (16 a+5 b \sec (c+d x))}{16 d}-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d} \]
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Rubi [A] time = 0.0953595, 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} \[ \frac{\tan ^5(c+d x) (6 a+5 b \sec (c+d x))}{30 d}-\frac{\tan ^3(c+d x) (8 a+5 b \sec (c+d x))}{24 d}+\frac{\tan (c+d x) (16 a+5 b \sec (c+d x))}{16 d}-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d} \]
Antiderivative was successfully verified.
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Rule 3881
Rule 3770
Rubi steps
\begin{align*} \int (a+b \sec (c+d x)) \tan ^6(c+d x) \, dx &=\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{6} \int (6 a+5 b \sec (c+d x)) \tan ^4(c+d x) \, dx\\ &=-\frac{(8 a+5 b \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}+\frac{1}{24} \int (24 a+15 b \sec (c+d x)) \tan ^2(c+d x) \, dx\\ &=\frac{(16 a+5 b \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 b \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{48} \int (48 a+15 b \sec (c+d x)) \, dx\\ &=-a x+\frac{(16 a+5 b \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 b \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{16} (5 b) \int \sec (c+d x) \, dx\\ &=-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{(16 a+5 b \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 b \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}\\ \end{align*}
Mathematica [A] time = 1.02845, size = 103, normalized size = 1.01 \[ \frac{\frac{1}{8} \tan (c+d x) \sec ^5(c+d x) (1168 a \cos (c+d x)+568 a \cos (3 (c+d x))+184 a \cos (5 (c+d x))+140 b \cos (2 (c+d x))+165 b \cos (4 (c+d x))+295 b)-240 a \tan ^{-1}(\tan (c+d x))-75 b \tanh ^{-1}(\sin (c+d x))}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 178, normalized size = 1.8 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}a}{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{b \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}-{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}b}{16\,d}}+{\frac{5\,b \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{5\,\sin \left ( dx+c \right ) b}{16\,d}}-{\frac{5\,b\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 = 1.46158, 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 \, b{\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 = 0.943063, size = 377, normalized size = 3.7 \begin{align*} -\frac{480 \, a d x \cos \left (d x + c\right )^{6} + 75 \, b \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 75 \, b \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 \, b \cos \left (d x + c\right )^{4} - 176 \, a \cos \left (d x + c\right )^{3} - 130 \, b \cos \left (d x + c\right )^{2} + 48 \, a \cos \left (d x + c\right ) + 40 \, b\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*} \int \left (a + b \sec{\left (c + d x \right )}\right ) \tan ^{6}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 5.0167, size = 308, normalized size = 3.02 \begin{align*} -\frac{240 \,{\left (d x + c\right )} a + 75 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 75 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (240 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 75 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1520 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 425 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 4128 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 990 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 4128 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 990 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1520 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 425 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 240 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 75 \, b \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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