Optimal. Leaf size=88 \[ \frac{a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac{B \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}+\frac{1}{2} a^2 x (3 B+4 C)+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.219644, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4072, 4017, 3996, 3770} \[ \frac{a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac{B \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}+\frac{1}{2} a^2 x (3 B+4 C)+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 4017
Rule 3996
Rule 3770
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 (B+C \sec (c+d x)) \, dx\\ &=\frac{B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{1}{2} \int \cos (c+d x) (a+a \sec (c+d x)) (a (3 B+2 C)+2 a C \sec (c+d x)) \, dx\\ &=\frac{a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac{B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}-\frac{1}{2} \int \left (-a^2 (3 B+4 C)-2 a^2 C \sec (c+d x)\right ) \, dx\\ &=\frac{1}{2} a^2 (3 B+4 C) x+\frac{a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac{B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\left (a^2 C\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^2 (3 B+4 C) x+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac{B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.151612, size = 96, normalized size = 1.09 \[ \frac{a^2 \left (4 (2 B+C) \sin (c+d x)+B \sin (2 (c+d x))+6 B d x-4 C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 108, normalized size = 1.2 \begin{align*}{\frac{B{a}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{3\,{a}^{2}Bx}{2}}+{\frac{3\,B{a}^{2}c}{2\,d}}+{\frac{{a}^{2}C\sin \left ( dx+c \right ) }{d}}+2\,{\frac{B{a}^{2}\sin \left ( dx+c \right ) }{d}}+2\,{a}^{2}Cx+2\,{\frac{C{a}^{2}c}{d}}+{\frac{{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941587, size = 136, normalized size = 1.55 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 4 \,{\left (d x + c\right )} B a^{2} + 8 \,{\left (d x + c\right )} C a^{2} + 2 \, C a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, B a^{2} \sin \left (d x + c\right ) + 4 \, C a^{2} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.516076, size = 194, normalized size = 2.2 \begin{align*} \frac{{\left (3 \, B + 4 \, C\right )} a^{2} d x + C a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - C a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (B a^{2} \cos \left (d x + c\right ) + 2 \,{\left (2 \, B + C\right )} a^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17739, size = 196, normalized size = 2.23 \begin{align*} \frac{2 \, C a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, C a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) +{\left (3 \, B a^{2} + 4 \, C a^{2}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (3 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, C 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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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