Optimal. Leaf size=141 \[ \frac{\sin (c+d x) \left (a^2 (2 A+3 C)+6 a b B+2 A b^2\right )}{3 d}+\frac{1}{2} x \left (a^2 B+2 a b (A+2 C)+2 b^2 B\right )+\frac{a (3 a B+2 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{b^2 C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.365023, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {4094, 4074, 4047, 8, 4045, 3770} \[ \frac{\sin (c+d x) \left (a^2 (2 A+3 C)+6 a b B+2 A b^2\right )}{3 d}+\frac{1}{2} x \left (a^2 B+2 a b (A+2 C)+2 b^2 B\right )+\frac{a (3 a B+2 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{b^2 C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4074
Rule 4047
Rule 8
Rule 4045
Rule 3770
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \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+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac{1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (2 A b+3 a B+(2 a A+3 b B+3 a C) \sec (c+d x)+3 b C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a (2 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac{A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}-\frac{1}{6} \int \cos (c+d x) \left (-2 \left (2 A b^2+6 a b B+a^2 (2 A+3 C)\right )-3 \left (a^2 B+2 b^2 B+2 a b (A+2 C)\right ) \sec (c+d x)-6 b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a (2 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac{A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}-\frac{1}{6} \int \cos (c+d x) \left (-2 \left (2 A b^2+6 a b B+a^2 (2 A+3 C)\right )-6 b^2 C \sec ^2(c+d x)\right ) \, dx-\frac{1}{2} \left (-a^2 B-2 b^2 B-2 a b (A+2 C)\right ) \int 1 \, dx\\ &=\frac{1}{2} \left (a^2 B+2 b^2 B+2 a b (A+2 C)\right ) x+\frac{\left (2 A b^2+6 a b B+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}+\frac{a (2 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac{A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\left (b^2 C\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} \left (a^2 B+2 b^2 B+2 a b (A+2 C)\right ) x+\frac{b^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\left (2 A b^2+6 a b B+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}+\frac{a (2 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac{A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.521084, size = 157, normalized size = 1.11 \[ \frac{6 (c+d x) \left (a^2 B+2 a b (A+2 C)+2 b^2 B\right )+3 \sin (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right )+a^2 A \sin (3 (c+d x))+3 a (a B+2 A b) \sin (2 (c+d x))-12 b^2 C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 b^2 C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 204, normalized size = 1.5 \begin{align*}{\frac{A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{3\,d}}+{\frac{2\,{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{{a}^{2}Bx}{2}}+{\frac{B{a}^{2}c}{2\,d}}+{\frac{{a}^{2}C\sin \left ( dx+c \right ) }{d}}+{\frac{Aab\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+aAbx+{\frac{Aabc}{d}}+2\,{\frac{Bab\sin \left ( dx+c \right ) }{d}}+2\,abCx+2\,{\frac{Cabc}{d}}+{\frac{A{b}^{2}\sin \left ( dx+c \right ) }{d}}+B{b}^{2}x+{\frac{B{b}^{2}c}{d}}+{\frac{{b}^{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 = 1.02046, size = 212, normalized size = 1.5 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + 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} - 6 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b - 24 \,{\left (d x + c\right )} C a b - 12 \,{\left (d x + c\right )} B b^{2} - 6 \, C b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{2} \sin \left (d x + c\right ) - 24 \, B a b \sin \left (d x + c\right ) - 12 \, A b^{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 = 0.548668, size = 313, normalized size = 2.22 \begin{align*} \frac{3 \, C b^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, C b^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (B a^{2} + 2 \,{\left (A + 2 \, C\right )} a b + 2 \, B b^{2}\right )} d x +{\left (2 \, A a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (2 \, A + 3 \, C\right )} a^{2} + 12 \, B a b + 6 \, A b^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\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 [B] time = 1.20832, size = 467, normalized size = 3.31 \begin{align*} \frac{6 \, C b^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, C b^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 3 \,{\left (B a^{2} + 2 \, A a b + 4 \, C a b + 2 \, B b^{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} - 3 \, 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} - 6 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, A 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} + 24 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, 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 ) + 6 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, A b^{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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