Optimal. Leaf size=125 \[ \frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{(5 A+3 B) \sin (c+d x) \cos (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+\frac{1}{2} a^3 x (5 A+7 B)+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.270957, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {4017, 3996, 3770} \[ \frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{(5 A+3 B) \sin (c+d x) \cos (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+\frac{1}{2} a^3 x (5 A+7 B)+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 4017
Rule 3996
Rule 3770
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac{a A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{1}{3} \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 (a (5 A+3 B)+3 a B \sec (c+d x)) \, dx\\ &=\frac{a A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(5 A+3 B) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{1}{6} \int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^2 (A+B)+6 a^2 B \sec (c+d x)\right ) \, dx\\ &=\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(5 A+3 B) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}-\frac{1}{6} \int \left (-3 a^3 (5 A+7 B)-6 a^3 B \sec (c+d x)\right ) \, dx\\ &=\frac{1}{2} a^3 (5 A+7 B) x+\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(5 A+3 B) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^3 B\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^3 (5 A+7 B) x+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(5 A+3 B) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.238765, size = 113, normalized size = 0.9 \[ \frac{a^3 \left (9 (5 A+4 B) \sin (c+d x)+3 (3 A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+30 A d x-12 B \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 B \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+42 B d x\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 153, normalized size = 1.2 \begin{align*}{\frac{A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{3}}{3\,d}}+{\frac{11\,A{a}^{3}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{B{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{7\,B{a}^{3}x}{2}}+{\frac{7\,B{a}^{3}c}{2\,d}}+{\frac{3\,A{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{5\,{a}^{3}Ax}{2}}+{\frac{5\,A{a}^{3}c}{2\,d}}+3\,{\frac{B{a}^{3}\sin \left ( dx+c \right ) }{d}}+{\frac{B{a}^{3}\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 = 1.02456, size = 200, normalized size = 1.6 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 9 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 12 \,{\left (d x + c\right )} A a^{3} - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 36 \,{\left (d x + c\right )} B a^{3} - 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 )} - 36 \, A a^{3} \sin \left (d x + c\right ) - 36 \, B a^{3} \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 = 0.501243, size = 254, normalized size = 2.03 \begin{align*} \frac{3 \,{\left (5 \, A + 7 \, B\right )} a^{3} d x + 3 \, B a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, B a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, A a^{3} \cos \left (d x + c\right )^{2} + 3 \,{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 2 \,{\left (11 \, A + 9 \, B\right )} a^{3}\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 = 1.37528, size = 243, normalized size = 1.94 \begin{align*} \frac{6 \, B a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, B a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 3 \,{\left (5 \, A a^{3} + 7 \, B a^{3}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (15 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 36 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 33 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21 \, B 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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