Optimal. Leaf size=96 \[ \frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 A+C)+\frac{C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{3 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.298016, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3046, 2976, 2968, 3023, 2735, 3770} \[ \frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 A+C)+\frac{C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{3 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2976
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x))^2 (3 a A+2 a C \cos (c+d x)) \sec (c+d x) \, dx}{3 a}\\ &=\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x)) \left (6 a^2 A+6 a^2 (A+C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{\int \left (6 a^3 A+\left (6 a^3 A+6 a^3 (A+C)\right ) \cos (c+d x)+6 a^3 (A+C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{\int \left (6 a^3 A+6 a^3 (2 A+C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=a^2 (2 A+C) x+\frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\left (a^2 A\right ) \int \sec (c+d x) \, dx\\ &=a^2 (2 A+C) x+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.220093, size = 109, normalized size = 1.14 \[ \frac{a^2 \left (3 (4 A+7 C) \sin (c+d x)-12 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+24 A d x+6 C \sin (2 (c+d x))+C \sin (3 (c+d x))+12 C d x\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 128, normalized size = 1.3 \begin{align*}{\frac{A{a}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{3\,d}}+{\frac{5\,{a}^{2}C\sin \left ( dx+c \right ) }{3\,d}}+2\,A{a}^{2}x+2\,{\frac{A{a}^{2}c}{d}}+{\frac{{a}^{2}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+{a}^{2}Cx+{\frac{{a}^{2}Cc}{d}}+{\frac{A{a}^{2}\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 = 1.01176, size = 144, normalized size = 1.5 \begin{align*} \frac{12 \,{\left (d x + c\right )} A a^{2} - 2 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 6 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 6 \, A a^{2} \sin \left (d x + c\right ) + 6 \, C a^{2} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45344, size = 236, normalized size = 2.46 \begin{align*} \frac{6 \,{\left (2 \, A + C\right )} a^{2} d x + 3 \, A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (C a^{2} \cos \left (d x + c\right )^{2} + 3 \, C a^{2} \cos \left (d x + c\right ) +{\left (3 \, A + 5 \, 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 = 1.19238, size = 242, normalized size = 2.52 \begin{align*} \frac{3 \, A a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, A 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} + C a^{2}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (3 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, 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}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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