Optimal. Leaf size=78 \[ -\frac{(A-3 B) (a \sin (c+d x)+a)^5}{5 a^3 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{2 a^2 d}-\frac{B (a \sin (c+d x)+a)^6}{6 a^4 d} \]
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Rubi [A] time = 0.109171, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2836, 77} \[ -\frac{(A-3 B) (a \sin (c+d x)+a)^5}{5 a^3 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{2 a^2 d}-\frac{B (a \sin (c+d x)+a)^6}{6 a^4 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 77
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a-x) (a+x)^3 \left (A+\frac{B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (A-B) (a+x)^3+(-A+3 B) (a+x)^4-\frac{B (a+x)^5}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{(A-B) (a+a \sin (c+d x))^4}{2 a^2 d}-\frac{(A-3 B) (a+a \sin (c+d x))^5}{5 a^3 d}-\frac{B (a+a \sin (c+d x))^6}{6 a^4 d}\\ \end{align*}
Mathematica [A] time = 0.405951, size = 66, normalized size = 0.85 \[ \frac{a^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8 (-4 (3 A-4 B) \sin (c+d x)+18 A+5 B \cos (2 (c+d x))-9 B)}{60 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.068, size = 171, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ({a}^{2}A \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{ \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{15}} \right ) +B{a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{6}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{12}} \right ) -{\frac{{a}^{2}A \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{2}}+2\,B{a}^{2} \left ( -1/5\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1/15\, \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) +{\frac{{a}^{2}A \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}-{\frac{B{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978895, size = 130, normalized size = 1.67 \begin{align*} -\frac{5 \, B a^{2} \sin \left (d x + c\right )^{6} + 6 \,{\left (A + 2 \, B\right )} a^{2} \sin \left (d x + c\right )^{5} + 15 \, A a^{2} \sin \left (d x + c\right )^{4} - 20 \, B a^{2} \sin \left (d x + c\right )^{3} - 15 \,{\left (2 \, A + B\right )} a^{2} \sin \left (d x + c\right )^{2} - 30 \, A a^{2} \sin \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79339, size = 224, normalized size = 2.87 \begin{align*} \frac{5 \, B a^{2} \cos \left (d x + c\right )^{6} - 15 \,{\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{4} - 2 \,{\left (3 \,{\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 2 \,{\left (3 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{2} - 4 \,{\left (3 \, A + B\right )} a^{2}\right )} \sin \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.96456, size = 228, normalized size = 2.92 \begin{align*} \begin{cases} \frac{2 A a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{2 A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{A a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{A a^{2} \cos ^{4}{\left (c + d x \right )}}{2 d} + \frac{B a^{2} \sin ^{6}{\left (c + d x \right )}}{12 d} + \frac{4 B a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{B a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4 d} + \frac{2 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac{B a^{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 \sin{\left (c \right )} + a\right )^{2} \cos ^{3}{\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 = 1.28829, size = 157, normalized size = 2.01 \begin{align*} -\frac{5 \, B a^{2} \sin \left (d x + c\right )^{6} + 6 \, A a^{2} \sin \left (d x + c\right )^{5} + 12 \, B a^{2} \sin \left (d x + c\right )^{5} + 15 \, A a^{2} \sin \left (d x + c\right )^{4} - 20 \, B a^{2} \sin \left (d x + c\right )^{3} - 30 \, A a^{2} \sin \left (d x + c\right )^{2} - 15 \, B a^{2} \sin \left (d x + c\right )^{2} - 30 \, A a^{2} \sin \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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