Optimal. Leaf size=83 \[ \frac{3}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.0928894, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac{3}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{a^3 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^3}{(a+x)^4}+\frac{3 a^2}{(a+x)^3}-\frac{3 a}{(a+x)^2}+\frac{1}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{\log (1+\sin (c+d x))}{a^4 d}+\frac{1}{3 a d (a+a \sin (c+d x))^3}-\frac{3}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac{3}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.357063, size = 61, normalized size = 0.73 \[ \frac{18 \sin ^2(c+d x)+27 \sin (c+d x)+6 (\sin (c+d x)+1)^3 \log (\sin (c+d x)+1)+11}{6 a^4 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 72, normalized size = 0.9 \begin{align*} -{\frac{3}{2\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{3\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+3\,{\frac{1}{d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11051, size = 112, normalized size = 1.35 \begin{align*} \frac{\frac{18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} + \frac{6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53932, size = 292, normalized size = 3.52 \begin{align*} \frac{18 \, \cos \left (d x + c\right )^{2} + 6 \,{\left (3 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 27 \, \sin \left (d x + c\right ) - 29}{6 \,{\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d +{\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.08603, size = 466, normalized size = 5.61 \begin{align*} \begin{cases} \frac{6 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin ^{3}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{18 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{18 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{6 \log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} - \frac{6 \sin ^{3}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{9 \sin{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{5}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \sin ^{3}{\left (c \right )} \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23416, size = 74, normalized size = 0.89 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} + \frac{18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4}{\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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