Optimal. Leaf size=54 \[ \frac{(A b-a B) (a+b \sin (c+d x))^3}{3 b^2 d}+\frac{B (a+b \sin (c+d x))^4}{4 b^2 d} \]
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Rubi [A] time = 0.07638, antiderivative size = 54, 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} \[ \frac{(A b-a B) (a+b \sin (c+d x))^3}{3 b^2 d}+\frac{B (a+b \sin (c+d x))^4}{4 b^2 d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \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}{b}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{(A b-a B) (a+x)^2}{b}+\frac{B (a+x)^3}{b}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{(A b-a B) (a+b \sin (c+d x))^3}{3 b^2 d}+\frac{B (a+b \sin (c+d x))^4}{4 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.0670486, size = 41, normalized size = 0.76 \[ \frac{(a+b \sin (c+d x))^3 (-a B+4 A b+3 b B \sin (c+d x))}{12 b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 73, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({\frac{B{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{ \left ( A{b}^{2}+2\,Bab \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( 2\,Aab+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 = 0.976322, size = 100, normalized size = 1.85 \begin{align*} \frac{3 \, B b^{2} \sin \left (d x + c\right )^{4} + 12 \, A a^{2} \sin \left (d x + c\right ) + 4 \,{\left (2 \, B a b + A b^{2}\right )} \sin \left (d x + c\right )^{3} + 6 \,{\left (B a^{2} + 2 \, A a b\right )} \sin \left (d x + c\right )^{2}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38124, size = 213, normalized size = 3.94 \begin{align*} \frac{3 \, B b^{2} \cos \left (d x + c\right )^{4} - 6 \,{\left (B a^{2} + 2 \, A a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (3 \, A a^{2} + 2 \, B a b + A b^{2} -{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{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 = 1.30888, size = 143, normalized size = 2.65 \begin{align*} \begin{cases} \frac{A a^{2} \sin{\left (c + d x \right )}}{d} - \frac{A a b \cos ^{2}{\left (c + d x \right )}}{d} + \frac{A b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{B a^{2} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{2 B a b \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{B b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{B b^{2} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\x \left (A + B \sin{\left (c \right )}\right ) \left (a + b \sin{\left (c \right )}\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 = 1.23297, size = 116, normalized size = 2.15 \begin{align*} \frac{3 \, B b^{2} \sin \left (d x + c\right )^{4} + 8 \, B a b \sin \left (d x + c\right )^{3} + 4 \, A b^{2} \sin \left (d x + c\right )^{3} + 6 \, B a^{2} \sin \left (d x + c\right )^{2} + 12 \, A a b \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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