Optimal. Leaf size=77 \[ \frac{a (2 A+3 B) \sin (c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+B)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.10831, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3996, 3787, 2635, 8, 2637} \[ \frac{a (2 A+3 B) \sin (c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+B)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3996
Rule 3787
Rule 2635
Rule 8
Rule 2637
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \int \cos ^2(c+d x) (-3 a (A+B)-a (2 A+3 B) \sec (c+d x)) \, dx\\ &=\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}+(a (A+B)) \int \cos ^2(c+d x) \, dx+\frac{1}{3} (a (2 A+3 B)) \int \cos (c+d x) \, dx\\ &=\frac{a (2 A+3 B) \sin (c+d x)}{3 d}+\frac{a (A+B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac{1}{2} (a (A+B)) \int 1 \, dx\\ &=\frac{1}{2} a (A+B) x+\frac{a (2 A+3 B) \sin (c+d x)}{3 d}+\frac{a (A+B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.169215, size = 65, normalized size = 0.84 \[ \frac{a (3 (3 A+4 B) \sin (c+d x)+3 (A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+6 A c+6 A d x+6 B c+6 B d x)}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 85, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{Aa \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Aa \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Ba \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Ba\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982049, size = 107, normalized size = 1.39 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 12 \, B a \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 = 0.464526, size = 146, normalized size = 1.9 \begin{align*} \frac{3 \,{\left (A + B\right )} a d x +{\left (2 \, A a \cos \left (d x + c\right )^{2} + 3 \,{\left (A + B\right )} a \cos \left (d x + c\right ) + 2 \,{\left (2 \, A + 3 \, B\right )} a\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 = 1.44524, size = 167, normalized size = 2.17 \begin{align*} \frac{3 \,{\left (A a + B a\right )}{\left (d x + c\right )} + \frac{2 \,{\left (3 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, B a \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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