Optimal. Leaf size=98 \[ \frac{a^3 \tan ^3(c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}-\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{11 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-a^3 x \]
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Rubi [A] time = 0.158117, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3886, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ \frac{a^3 \tan ^3(c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}-\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{11 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-a^3 x \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rule 3768
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^3 \tan ^2(c+d x) \, dx &=\int \left (a^3 \tan ^2(c+d x)+3 a^3 \sec (c+d x) \tan ^2(c+d x)+3 a^3 \sec ^2(c+d x) \tan ^2(c+d x)+a^3 \sec ^3(c+d x) \tan ^2(c+d x)\right ) \, dx\\ &=a^3 \int \tan ^2(c+d x) \, dx+a^3 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx+\left (3 a^3\right ) \int \sec (c+d x) \tan ^2(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{a^3 \tan (c+d x)}{d}+\frac{3 a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{4} a^3 \int \sec ^3(c+d x) \, dx-a^3 \int 1 \, dx-\frac{1}{2} \left (3 a^3\right ) \int \sec (c+d x) \, dx+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{d}\\ &=-a^3 x-\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{11 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a^3 \tan ^3(c+d x)}{d}-\frac{1}{8} a^3 \int \sec (c+d x) \, dx\\ &=-a^3 x-\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{11 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a^3 \tan ^3(c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 0.834652, size = 230, normalized size = 2.35 \[ -\frac{a^3 \sec ^4(c+d x) \left (-38 \sin (c+d x)-32 \sin (2 (c+d x))-22 \sin (3 (c+d x))-39 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+39 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+4 \cos (2 (c+d x)) \left (-13 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+13 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 d x\right )+\cos (4 (c+d x)) \left (-13 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+13 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 d x\right )+24 d x\right )}{64 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 137, normalized size = 1.4 \begin{align*} -{a}^{3}x+{\frac{{a}^{3}\tan \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}c}{d}}+{\frac{13\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{13\,{a}^{3}\sin \left ( dx+c \right ) }{8\,d}}-{\frac{13\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.89841, size = 198, normalized size = 2.02 \begin{align*} \frac{16 \, a^{3} \tan \left (d x + c\right )^{3} - 16 \,{\left (d x + c - \tan \left (d x + c\right )\right )} a^{3} + a^{3}{\left (\frac{2 \,{\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18766, size = 293, normalized size = 2.99 \begin{align*} -\frac{16 \, a^{3} d x \cos \left (d x + c\right )^{4} + 13 \, a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 13 \, a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (11 \, a^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{16 \, 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*} a^{3} \left (\int 3 \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int 3 \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \tan ^{2}{\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 = 1.66335, size = 178, normalized size = 1.82 \begin{align*} -\frac{8 \,{\left (d x + c\right )} a^{3} + 13 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 13 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 13 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 21 \, a^{3} \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}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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