Optimal. Leaf size=134 \[ \frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{(2 A+3 B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{6 d}+\frac{1}{2} a^2 x (2 A+3 B+4 C)+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.291759, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {4086, 4017, 3996, 3770} \[ \frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{(2 A+3 B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{6 d}+\frac{1}{2} a^2 x (2 A+3 B+4 C)+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4017
Rule 3996
Rule 3770
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 (a (2 A+3 B)+3 a C \sec (c+d x)) \, dx}{3 a}\\ &=\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(2 A+3 B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{\int \cos (c+d x) (a+a \sec (c+d x)) \left (3 a^2 (2 A+3 B+2 C)+6 a^2 C \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(2 A+3 B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{6 d}-\frac{\int \left (-3 a^3 (2 A+3 B+4 C)-6 a^3 C \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac{1}{2} a^2 (2 A+3 B+4 C) x+\frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(2 A+3 B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^2 C\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^2 (2 A+3 B+4 C) x+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(2 A+3 B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.290327, size = 121, normalized size = 0.9 \[ \frac{a^2 \left (3 (7 A+8 B+4 C) \sin (c+d x)+3 (2 A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+12 A d x+18 B d x-12 C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+24 C d x\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 181, normalized size = 1.4 \begin{align*}{\frac{A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ){a}^{2}}{3\,d}}+{\frac{5\,{a}^{2}A\sin \left ( dx+c \right ) }{3\,d}}+{\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}}+{\frac{{a}^{2}A\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+{a}^{2}Ax+{\frac{{a}^{2}Ac}{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 = 0.946821, size = 216, normalized size = 1.61 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 6 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 12 \,{\left (d x + c\right )} B a^{2} - 24 \,{\left (d x + c\right )} C a^{2} - 6 \, 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 )} - 12 \, A a^{2} \sin \left (d x + c\right ) - 24 \, B a^{2} \sin \left (d x + c\right ) - 12 \, C a^{2} \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.528278, size = 269, normalized size = 2.01 \begin{align*} \frac{3 \,{\left (2 \, A + 3 \, B + 4 \, C\right )} a^{2} d x + 3 \, C a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, C a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, A a^{2} \cos \left (d x + c\right )^{2} + 3 \,{\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (5 \, A + 6 \, B + 3 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{6 \, 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 = 1.27996, size = 317, normalized size = 2.37 \begin{align*} \frac{6 \, C a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, C a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 3 \,{\left (2 \, A a^{2} + 3 \, B a^{2} + 4 \, C a^{2}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (6 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 16 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 24 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 18 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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