Optimal. Leaf size=51 \[ \frac{B (a \sin (c+d x)+a)^5}{5 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{4 a d} \]
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Rubi [A] time = 0.062236, 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} \[ \frac{B (a \sin (c+d x)+a)^5}{5 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{4 a d} \]
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))^3 (A+B \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^3 \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)^3+\frac{B (a+x)^4}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{(A-B) (a+a \sin (c+d x))^4}{4 a d}+\frac{B (a+a \sin (c+d x))^5}{5 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0882942, size = 36, normalized size = 0.71 \[ \frac{a^3 (\sin (c+d x)+1)^4 (5 A+4 B \sin (c+d x)-B)}{20 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 98, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({\frac{B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ({a}^{3}A+3\,B{a}^{3} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{ \left ( 3\,{a}^{3}A+3\,B{a}^{3} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( 3\,{a}^{3}A+B{a}^{3} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2}}+{a}^{3}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 = 1.05471, size = 113, normalized size = 2.22 \begin{align*} \frac{4 \, B a^{3} \sin \left (d x + c\right )^{5} + 5 \,{\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{4} + 20 \,{\left (A + B\right )} a^{3} \sin \left (d x + c\right )^{3} + 10 \,{\left (3 \, A + B\right )} a^{3} \sin \left (d x + c\right )^{2} + 20 \, A a^{3} \sin \left (d x + c\right )}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68586, size = 224, normalized size = 4.39 \begin{align*} \frac{5 \,{\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 40 \,{\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 4 \,{\left (B a^{3} \cos \left (d x + c\right )^{4} -{\left (5 \, A + 7 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 2 \,{\left (5 \, A + 3 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.70239, size = 204, normalized size = 4. \begin{align*} \begin{cases} \frac{A a^{3} \sin ^{3}{\left (c + d x \right )}}{d} - \frac{A a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{A a^{3} \sin{\left (c + d x \right )}}{d} - \frac{A a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac{3 A a^{3} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{B a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac{B a^{3} \sin ^{3}{\left (c + d x \right )}}{d} - \frac{3 B a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{3 B a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac{B a^{3} \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 )^{3} \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 [B] time = 1.32333, size = 157, normalized size = 3.08 \begin{align*} \frac{4 \, B a^{3} \sin \left (d x + c\right )^{5} + 5 \, A a^{3} \sin \left (d x + c\right )^{4} + 15 \, B a^{3} \sin \left (d x + c\right )^{4} + 20 \, A a^{3} \sin \left (d x + c\right )^{3} + 20 \, B a^{3} \sin \left (d x + c\right )^{3} + 30 \, A a^{3} \sin \left (d x + c\right )^{2} + 10 \, B a^{3} \sin \left (d x + c\right )^{2} + 20 \, A a^{3} \sin \left (d x + c\right )}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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