Optimal. Leaf size=124 \[ \frac{a \left (a^2 B+6 a b C+6 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a C+2 b B) \tan (c+d x)}{d}-\frac{b^2 (a B-2 b C) \sin (c+d x)}{2 d}+b^2 x (3 a C+b B)+\frac{a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.413213, antiderivative size = 124, 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, 3031, 3023, 2735, 3770} \[ \frac{a \left (a^2 B+6 a b C+6 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a C+2 b B) \tan (c+d x)}{d}-\frac{b^2 (a B-2 b C) \sin (c+d x)}{2 d}+b^2 x (3 a C+b B)+\frac{a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2989
Rule 3031
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 ^4(c+d x) \, dx &=\int (a+b \cos (c+d x))^3 (B+C \cos (c+d x)) \sec ^3(c+d x) \, dx\\ &=\frac{a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int (a+b \cos (c+d x)) \left (2 a (2 b B+a C)+\left (a^2 B+2 b^2 B+4 a b C\right ) \cos (c+d x)-b (a B-2 b C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{a^2 (2 b B+a C) \tan (c+d x)}{d}+\frac{a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac{1}{2} \int \left (-a \left (a^2 B+6 b^2 B+6 a b C\right )-2 b^2 (b B+3 a C) \cos (c+d x)+b^2 (a B-2 b C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b^2 (a B-2 b C) \sin (c+d x)}{2 d}+\frac{a^2 (2 b B+a C) \tan (c+d x)}{d}+\frac{a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac{1}{2} \int \left (-a \left (a^2 B+6 b^2 B+6 a b C\right )-2 b^2 (b B+3 a C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=b^2 (b B+3 a C) x-\frac{b^2 (a B-2 b C) \sin (c+d x)}{2 d}+\frac{a^2 (2 b B+a C) \tan (c+d x)}{d}+\frac{a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \left (a \left (a^2 B+6 b^2 B+6 a b C\right )\right ) \int \sec (c+d x) \, dx\\ &=b^2 (b B+3 a C) x+\frac{a \left (a^2 B+6 b^2 B+6 a b C\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 (a B-2 b C) \sin (c+d x)}{2 d}+\frac{a^2 (2 b B+a C) \tan (c+d x)}{d}+\frac{a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 2.16565, size = 277, normalized size = 2.23 \[ \frac{-2 a \left (a^2 B+6 a b C+6 b^2 B\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 a \left (a^2 B+6 a b C+6 b^2 B\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{4 a^2 (a C+3 b B) \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{4 a^2 (a C+3 b B) \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+\frac{a^3 B}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{a^3 B}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+4 b^2 (c+d x) (3 a C+b B)+4 b^3 C \sin (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 172, normalized size = 1.4 \begin{align*}{\frac{C{b}^{3}\sin \left ( dx+c \right ) }{d}}+{b}^{3}Bx+{\frac{B{b}^{3}c}{d}}+3\,a{b}^{2}Cx+3\,{\frac{Ca{b}^{2}c}{d}}+3\,{\frac{a{b}^{2}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{2}bC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{2}bB\tan \left ( dx+c \right ) }{d}}+{\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{{a}^{3}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01802, size = 228, normalized size = 1.84 \begin{align*} \frac{12 \,{\left (d x + c\right )} C a b^{2} + 4 \,{\left (d x + c\right )} B b^{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 \, C 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 )} + 6 \, B a b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C b^{3} \sin \left (d x + c\right ) + 4 \, C a^{3} \tan \left (d x + c\right ) + 12 \, B a^{2} b \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 = 1.5192, size = 401, normalized size = 3.23 \begin{align*} \frac{4 \,{\left (3 \, C a b^{2} + B b^{3}\right )} d x \cos \left (d x + c\right )^{2} +{\left (B a^{3} + 6 \, C a^{2} b + 6 \, B a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (B a^{3} + 6 \, C a^{2} b + 6 \, B a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \, C b^{3} \cos \left (d x + c\right )^{2} + B a^{3} + 2 \,{\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )\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 [B] time = 1.33928, size = 323, normalized size = 2.6 \begin{align*} \frac{\frac{4 \, C b^{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 (3 \, C a b^{2} + B b^{3}\right )}{\left (d x + c\right )} +{\left (B a^{3} + 6 \, C a^{2} b + 6 \, B a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (B a^{3} + 6 \, C a^{2} b + 6 \, B a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (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} - 6 \, B a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 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 ) + 6 \, B a^{2} 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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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