Optimal. Leaf size=107 \[ \frac{\left (2 a^2 A+6 a b B+3 A b^2\right ) \sin (c+d x)}{3 d}+\frac{1}{2} x \left (a^2 B+2 a A b+2 b^2 B\right )+\frac{a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a (a B+2 A b) \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.215749, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {4024, 4047, 2637, 4045, 8} \[ \frac{\left (2 a^2 A+6 a b B+3 A b^2\right ) \sin (c+d x)}{3 d}+\frac{1}{2} x \left (a^2 B+2 a A b+2 b^2 B\right )+\frac{a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a (a B+2 A b) \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 4024
Rule 4047
Rule 2637
Rule 4045
Rule 8
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac{a^2 A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \int \cos ^2(c+d x) \left (-3 a (2 A b+a B)+\left (A \left (-2 a^2-3 b^2\right )-6 a b B\right ) \sec (c+d x)-3 b^2 B \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \int \cos ^2(c+d x) \left (-3 a (2 A b+a B)-3 b^2 B \sec ^2(c+d x)\right ) \, dx-\frac{1}{3} \left (-2 a^2 A-3 A b^2-6 a b B\right ) \int \cos (c+d x) \, dx\\ &=\frac{\left (2 a^2 A+3 A b^2+6 a b B\right ) \sin (c+d x)}{3 d}+\frac{a (2 A b+a B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a^2 A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{2} \left (-2 a A b-a^2 B-2 b^2 B\right ) \int 1 \, dx\\ &=\frac{1}{2} \left (2 a A b+a^2 B+2 b^2 B\right ) x+\frac{\left (2 a^2 A+3 A b^2+6 a b B\right ) \sin (c+d x)}{3 d}+\frac{a (2 A b+a B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a^2 A \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.223909, size = 90, normalized size = 0.84 \[ \frac{6 (c+d x) \left (a^2 B+2 a A b+2 b^2 B\right )+3 \left (3 a^2 A+8 a b B+4 A b^2\right ) \sin (c+d x)+a^2 A \sin (3 (c+d x))+3 a (a B+2 A b) \sin (2 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 114, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2}A \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,Aab \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +B{a}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +A{b}^{2}\sin \left ( dx+c \right ) +2\,Bab\sin \left ( dx+c \right ) +B{b}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968484, size = 146, normalized size = 1.36 \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 - 12 \,{\left (d x + c\right )} B b^{2} - 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 = 0.479716, size = 201, normalized size = 1.88 \begin{align*} \frac{3 \,{\left (B a^{2} + 2 \, A a b + 2 \, B b^{2}\right )} d x +{\left (2 \, A a^{2} \cos \left (d x + c\right )^{2} + 4 \, A 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 = 1.22963, size = 343, normalized size = 3.21 \begin{align*} \frac{3 \,{\left (B a^{2} + 2 \, A 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 \, 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} + 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 \, 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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