Optimal. Leaf size=128 \[ \frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (3 A+4 B+2 C)-\frac{(A-2 C) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}+\frac{a^2 (B+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d} \]
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Rubi [A] time = 0.287309, antiderivative size = 128, 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, 4018, 3996, 3770} \[ \frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (3 A+4 B+2 C)-\frac{(A-2 C) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}+\frac{a^2 (B+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4018
Rule 3996
Rule 3770
Rubi steps
\begin{align*} \int \cos ^2(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 (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac{\int \cos (c+d x) (a+a \sec (c+d x))^2 (2 a (A+B)-a (A-2 C) \sec (c+d x)) \, dx}{2 a}\\ &=\frac{A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{(A-2 C) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{\int \cos (c+d x) (a+a \sec (c+d x)) \left (a^2 (3 A+2 B-2 C)+2 a^2 (B+2 C) \sec (c+d x)\right ) \, dx}{2 a}\\ &=\frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{(A-2 C) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}-\frac{\int \left (-a^3 (3 A+4 B+2 C)-2 a^3 (B+2 C) \sec (c+d x)\right ) \, dx}{2 a}\\ &=\frac{1}{2} a^2 (3 A+4 B+2 C) x+\frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{(A-2 C) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\left (a^2 (B+2 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^2 (3 A+4 B+2 C) x+\frac{a^2 (B+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{(A-2 C) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 3.65387, size = 329, normalized size = 2.57 \[ \frac{a^2 \cos ^2(c+d x) (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{4 (2 A+B) \sin (c) \cos (d x)}{d}+\frac{4 (2 A+B) \cos (c) \sin (d x)}{d}+2 x (3 A+4 B+2 C)+\frac{A \sin (2 c) \cos (2 d x)}{d}+\frac{A \cos (2 c) \sin (2 d x)}{d}-\frac{4 (B+2 C) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{4 (B+2 C) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{4 C \sin \left (\frac{d x}{2}\right )}{d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{4 C \sin \left (\frac{d x}{2}\right )}{d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}\right )}{8 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.091, size = 160, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}A\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{3\,{a}^{2}Ax}{2}}+{\frac{3\,{a}^{2}Ac}{2\,d}}+{\frac{B{a}^{2}\sin \left ( dx+c \right ) }{d}}+{a}^{2}Cx+{\frac{C{a}^{2}c}{d}}+2\,{\frac{{a}^{2}A\sin \left ( dx+c \right ) }{d}}+2\,{a}^{2}Bx+2\,{\frac{B{a}^{2}c}{d}}+2\,{\frac{{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{B{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}C\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 = 0.958634, size = 204, normalized size = 1.59 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 4 \,{\left (d x + c\right )} A a^{2} + 8 \,{\left (d x + c\right )} B a^{2} + 4 \,{\left (d x + c\right )} C a^{2} + 2 \, B a^{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 a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, A a^{2} \sin \left (d x + c\right ) + 4 \, B a^{2} \sin \left (d x + c\right ) + 4 \, C a^{2} \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 = 0.529299, size = 331, normalized size = 2.59 \begin{align*} \frac{{\left (3 \, A + 4 \, B + 2 \, C\right )} a^{2} d x \cos \left (d x + c\right ) +{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (A a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) + 2 \, C a^{2}\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 = 1.29523, size = 267, normalized size = 2.09 \begin{align*} -\frac{\frac{4 \, C a^{2} \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 (3 \, A a^{2} + 4 \, B a^{2} + 2 \, C a^{2}\right )}{\left (d x + c\right )} - 2 \,{\left (B a^{2} + 2 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 2 \,{\left (B a^{2} + 2 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, B 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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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