Optimal. Leaf size=131 \[ -\frac{b \left (2 a^2 B-3 a b C-b^2 B\right ) \sin (c+d x)}{d}+\frac{1}{2} b x \left (6 a^2 C+6 a b B+b^2 C\right )+\frac{a^2 (a C+3 b B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a B-b C) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a B \tan (c+d x) (a+b \cos (c+d x))^2}{d} \]
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Rubi [A] time = 0.464639, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3029, 2989, 3033, 3023, 2735, 3770} \[ -\frac{b \left (2 a^2 B-3 a b C-b^2 B\right ) \sin (c+d x)}{d}+\frac{1}{2} b x \left (6 a^2 C+6 a b B+b^2 C\right )+\frac{a^2 (a C+3 b B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a B-b C) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a B \tan (c+d x) (a+b \cos (c+d x))^2}{d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2989
Rule 3033
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\int (a+b \cos (c+d x))^3 (B+C \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac{a B (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\int (a+b \cos (c+d x)) \left (a (3 b B+a C)+b (b B+2 a C) \cos (c+d x)-b (2 a B-b C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b^2 (2 a B-b C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a B (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{1}{2} \int \left (2 a^2 (3 b B+a C)+b \left (6 a b B+6 a^2 C+b^2 C\right ) \cos (c+d x)-2 b \left (2 a^2 B-b^2 B-3 a b C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b \left (2 a^2 B-b^2 B-3 a b C\right ) \sin (c+d x)}{d}-\frac{b^2 (2 a B-b C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a B (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{1}{2} \int \left (2 a^2 (3 b B+a C)+b \left (6 a b B+6 a^2 C+b^2 C\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} b \left (6 a b B+6 a^2 C+b^2 C\right ) x-\frac{b \left (2 a^2 B-b^2 B-3 a b C\right ) \sin (c+d x)}{d}-\frac{b^2 (2 a B-b C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a B (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\left (a^2 (3 b B+a C)\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} b \left (6 a b B+6 a^2 C+b^2 C\right ) x+\frac{a^2 (3 b B+a C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \left (2 a^2 B-b^2 B-3 a b C\right ) \sin (c+d x)}{d}-\frac{b^2 (2 a B-b C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a B (a+b \cos (c+d x))^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.632534, size = 217, normalized size = 1.66 \[ \frac{2 b (c+d x) \left (6 a^2 C+6 a b B+b^2 C\right )-4 a^2 (a C+3 b B) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 a^2 (a C+3 b B) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{4 a^3 B \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{4 a^3 B \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+4 b^2 (3 a C+b B) \sin (c+d x)+b^3 C \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 168, normalized size = 1.3 \begin{align*}{\frac{C{b}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{3}Cx}{2}}+{\frac{C{b}^{3}c}{2\,d}}+{\frac{{b}^{3}B\sin \left ( dx+c \right ) }{d}}+3\,{\frac{Ca{b}^{2}\sin \left ( dx+c \right ) }{d}}+3\,a{b}^{2}Bx+3\,{\frac{Ba{b}^{2}c}{d}}+3\,{a}^{2}bCx+3\,{\frac{C{a}^{2}bc}{d}}+3\,{\frac{{a}^{2}bB\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{3}B\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01296, size = 194, normalized size = 1.48 \begin{align*} \frac{12 \,{\left (d x + c\right )} C a^{2} b + 12 \,{\left (d x + c\right )} B a b^{2} +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} + 2 \, 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 )} + 6 \, B a^{2} b{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, C a b^{2} \sin \left (d x + c\right ) + 4 \, B b^{3} \sin \left (d x + c\right ) + 4 \, B 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 = 1.45581, size = 369, normalized size = 2.82 \begin{align*} \frac{{\left (6 \, C a^{2} b + 6 \, B a b^{2} + C b^{3}\right )} d x \cos \left (d x + c\right ) +{\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (C b^{3} \cos \left (d x + c\right )^{2} + 2 \, B a^{3} + 2 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \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 = 1.43093, size = 316, normalized size = 2.41 \begin{align*} -\frac{\frac{4 \, B 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} -{\left (6 \, C a^{2} b + 6 \, B a b^{2} + C b^{3}\right )}{\left (d x + c\right )} - 2 \,{\left (C a^{3} + 3 \, B a^{2} b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 2 \,{\left (C a^{3} + 3 \, B a^{2} b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (6 \, C a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, C a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, B b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C b^{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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