Optimal. Leaf size=51 \[ \frac{B (a \sin (c+d x)+a)^4}{4 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^3}{3 a d} \]
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Rubi [A] time = 0.069862, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2833, 43} \[ \frac{B (a \sin (c+d x)+a)^4}{4 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^3}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^2 \left (A+\frac{B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left ((A-B) (a+x)^2+\frac{B (a+x)^3}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{(A-B) (a+a \sin (c+d x))^3}{3 a d}+\frac{B (a+a \sin (c+d x))^4}{4 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0932027, size = 49, normalized size = 0.96 \[ \frac{\frac{1}{3} (A-B) (a \sin (c+d x)+a)^3+\frac{B (a \sin (c+d x)+a)^4}{4 a}}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 75, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{ \left ({a}^{2}A+2\,B{a}^{2} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}A+B{a}^{2} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2}}+{a}^{2}A\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 = 1.06434, size = 92, normalized size = 1.8 \begin{align*} \frac{3 \, B a^{2} \sin \left (d x + c\right )^{4} + 4 \,{\left (A + 2 \, B\right )} a^{2} \sin \left (d x + c\right )^{3} + 6 \,{\left (2 \, A + B\right )} a^{2} \sin \left (d x + c\right )^{2} + 12 \, A a^{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 = 1.83226, size = 177, normalized size = 3.47 \begin{align*} \frac{3 \, B a^{2} \cos \left (d x + c\right )^{4} - 12 \,{\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{2} - 4 \,{\left ({\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 2 \,{\left (2 \, A + B\right )} a^{2}\right )} \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.36024, size = 143, normalized size = 2.8 \begin{align*} \begin{cases} \frac{A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{A a^{2} \sin{\left (c + d x \right )}}{d} - \frac{A a^{2} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{2 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{B a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{B a^{2} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac{B a^{2} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (A + B \sin{\left (c \right )}\right ) \left (a \sin{\left (c \right )} + a\right )^{2} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36954, size = 119, normalized size = 2.33 \begin{align*} \frac{3 \, B a^{2} \sin \left (d x + c\right )^{4} + 4 \, A a^{2} \sin \left (d x + c\right )^{3} + 8 \, B a^{2} \sin \left (d x + c\right )^{3} + 12 \, A a^{2} \sin \left (d x + c\right )^{2} + 6 \, B a^{2} \sin \left (d x + c\right )^{2} + 12 \, A a^{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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