Optimal. Leaf size=124 \[ \frac{1}{2} a x \left (a^2 A+6 a b B+6 A b^2\right )+\frac{a^2 (a B+2 A b) \sin (c+d x)}{d}-\frac{b^2 (a A-2 b B) \tan (c+d x)}{2 d}+\frac{b^2 (3 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.333451, antiderivative size = 124, 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 = {4025, 4076, 4047, 8, 4045, 3770} \[ \frac{1}{2} a x \left (a^2 A+6 a b B+6 A b^2\right )+\frac{a^2 (a B+2 A b) \sin (c+d x)}{d}-\frac{b^2 (a A-2 b B) \tan (c+d x)}{2 d}+\frac{b^2 (3 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 4025
Rule 4076
Rule 4047
Rule 8
Rule 4045
Rule 3770
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac{a A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{1}{2} \int \cos (c+d x) (a+b \sec (c+d x)) \left (-2 a (2 A b+a B)-\left (a^2 A+2 A b^2+4 a b B\right ) \sec (c+d x)+b (a A-2 b B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{b^2 (a A-2 b B) \tan (c+d x)}{2 d}-\frac{1}{2} \int \cos (c+d x) \left (-2 a^2 (2 A b+a B)-a \left (a^2 A+6 A b^2+6 a b B\right ) \sec (c+d x)-2 b^2 (A b+3 a B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{b^2 (a A-2 b B) \tan (c+d x)}{2 d}-\frac{1}{2} \int \cos (c+d x) \left (-2 a^2 (2 A b+a B)-2 b^2 (A b+3 a B) \sec ^2(c+d x)\right ) \, dx+\frac{1}{2} \left (a \left (a^2 A+6 A b^2+6 a b B\right )\right ) \int 1 \, dx\\ &=\frac{1}{2} a \left (a^2 A+6 A b^2+6 a b B\right ) x+\frac{a^2 (2 A b+a B) \sin (c+d x)}{d}+\frac{a A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{b^2 (a A-2 b B) \tan (c+d x)}{2 d}+\left (b^2 (A b+3 a B)\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a \left (a^2 A+6 A b^2+6 a b B\right ) x+\frac{b^2 (A b+3 a B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 (2 A b+a B) \sin (c+d x)}{d}+\frac{a A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{b^2 (a A-2 b B) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.681226, size = 217, normalized size = 1.75 \[ \frac{2 a (c+d x) \left (a^2 A+6 a b B+6 A b^2\right )+4 a^2 (a B+3 A b) \sin (c+d x)+a^3 A \sin (2 (c+d x))-4 b^2 (3 a B+A b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 b^2 (3 a B+A b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{4 b^3 B \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{4 b^3 B \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 168, normalized size = 1.4 \begin{align*}{\frac{A{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}Ax}{2}}+{\frac{A{a}^{3}c}{2\,d}}+{\frac{B{a}^{3}\sin \left ( dx+c \right ) }{d}}+3\,{\frac{A{a}^{2}b\sin \left ( dx+c \right ) }{d}}+3\,B{a}^{2}bx+3\,{\frac{B{a}^{2}bc}{d}}+3\,Aa{b}^{2}x+3\,{\frac{Aa{b}^{2}c}{d}}+3\,{\frac{Ba{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{A{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{B{b}^{3}\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 = 0.980297, size = 194, normalized size = 1.56 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 12 \,{\left (d x + c\right )} B a^{2} b + 12 \,{\left (d x + c\right )} A a b^{2} + 6 \, B a b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A b^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, B a^{3} \sin \left (d x + c\right ) + 12 \, A a^{2} b \sin \left (d x + c\right ) + 4 \, B b^{3} \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 = 0.529613, size = 369, normalized size = 2.98 \begin{align*} \frac{{\left (A a^{3} + 6 \, B a^{2} b + 6 \, A a b^{2}\right )} d x \cos \left (d x + c\right ) +{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (A a^{3} \cos \left (d x + c\right )^{2} + 2 \, B b^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\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 = 1.29103, size = 316, normalized size = 2.55 \begin{align*} -\frac{\frac{4 \, B b^{3} \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 (A a^{3} + 6 \, B a^{2} b + 6 \, A a b^{2}\right )}{\left (d x + c\right )} - 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, A a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, A a^{2} b \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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