Optimal. Leaf size=114 \[ \frac{a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 B+C) \tan (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}-\frac{5 a^3 B \sin (c+d x)}{2 d}+a^3 x (B+3 C)+\frac{a B \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d} \]
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Rubi [A] time = 0.425466, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3029, 2975, 2968, 3023, 2735, 3770} \[ \frac{a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 B+C) \tan (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}-\frac{5 a^3 B \sin (c+d x)}{2 d}+a^3 x (B+3 C)+\frac{a B \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2975
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\int (a+a \cos (c+d x))^3 (B+C \cos (c+d x)) \sec ^3(c+d x) \, dx\\ &=\frac{a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int (a+a \cos (c+d x))^2 (2 a (2 B+C)-a (B-2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac{(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int (a+a \cos (c+d x)) \left (a^2 (7 B+6 C)-5 a^2 B \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int \left (a^3 (7 B+6 C)+\left (-5 a^3 B+a^3 (7 B+6 C)\right ) \cos (c+d x)-5 a^3 B \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{5 a^3 B \sin (c+d x)}{2 d}+\frac{(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int \left (a^3 (7 B+6 C)+2 a^3 (B+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=a^3 (B+3 C) x-\frac{5 a^3 B \sin (c+d x)}{2 d}+\frac{(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \left (a^3 (7 B+6 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 (B+3 C) x+\frac{a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{5 a^3 B \sin (c+d x)}{2 d}+\frac{(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.83738, size = 208, normalized size = 1.82 \[ \frac{a^3 \left (4 (3 B+C) \tan (c+d x)+\frac{B}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{B}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-14 B \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+14 B \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+4 B c+4 B d x+4 C \sin (c+d x)-12 C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+12 c C+12 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 144, normalized size = 1.3 \begin{align*}{\frac{{a}^{3}C\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{7\,{a}^{3}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{3}B\tan \left ( dx+c \right ) }{d}}+3\,{a}^{3}Cx+3\,{\frac{C{a}^{3}c}{d}}+{\frac{{a}^{3}C\sin \left ( dx+c \right ) }{d}}+{a}^{3}Bx+{\frac{B{a}^{3}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07551, size = 223, normalized size = 1.96 \begin{align*} \frac{4 \,{\left (d x + c\right )} B a^{3} + 12 \,{\left (d x + c\right )} C a^{3} - B 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 )} + 6 \, B a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C a^{3} \sin \left (d x + c\right ) + 12 \, B a^{3} \tan \left (d x + c\right ) + 4 \, C a^{3} \tan \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 = 1.99004, size = 342, normalized size = 3. \begin{align*} \frac{4 \,{\left (B + 3 \, C\right )} a^{3} d x \cos \left (d x + c\right )^{2} +{\left (7 \, B + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (7 \, B + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \, C a^{3} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right ) + B a^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \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 = 1.36995, size = 259, normalized size = 2.27 \begin{align*} \frac{\frac{4 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 2 \,{\left (B a^{3} + 3 \, C a^{3}\right )}{\left (d x + c\right )} +{\left (7 \, B a^{3} + 6 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (7 \, B a^{3} + 6 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (5 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 7 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, C 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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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