Optimal. Leaf size=74 \[ \frac {\sin (c+d x)}{a^3 d}-\frac {3}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {3 \log (\sin (c+d x)+1)}{a^3 d}+\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, 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} \[ \frac {\sin (c+d x)}{a^3 d}-\frac {3}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {3 \log (\sin (c+d x)+1)}{a^3 d}+\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{a^3 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {a^3}{(a+x)^3}+\frac {3 a^2}{(a+x)^2}-\frac {3 a}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=-\frac {3 \log (1+\sin (c+d x))}{a^3 d}+\frac {\sin (c+d x)}{a^3 d}+\frac {1}{2 a d (a+a \sin (c+d x))^2}-\frac {3}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 70, normalized size = 0.95 \[ \frac {\frac {\sin ^2(c+d x)}{(\sin (c+d x)+1)^2}+4 \sin (c+d x)+\frac {-10 \sin (c+d x)-9}{(\sin (c+d x)+1)^2}-12 \log (\sin (c+d x)+1)}{4 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 95, normalized size = 1.28 \[ \frac {4 \, \cos \left (d x + c\right )^{2} - 6 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (\cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 1}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 56, normalized size = 0.76 \[ -\frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac {2 \, \sin \left (d x + c\right )}{a^{3}} + \frac {6 \, \sin \left (d x + c\right ) + 5}{a^{3} {\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 68, normalized size = 0.92 \[ \frac {\sin \left (d x +c \right )}{a^{3} d}+\frac {1}{2 d \,a^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{3} d}-\frac {3}{d \,a^{3} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 71, normalized size = 0.96 \[ -\frac {\frac {6 \, \sin \left (d x + c\right ) + 5}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} + \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {2 \, \sin \left (d x + c\right )}{a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 59, normalized size = 0.80 \[ \frac {\sin \left (c+d\,x\right )}{a^3\,d}-\frac {3\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^3\,d}-\frac {3\,\sin \left (c+d\,x\right )+\frac {5}{2}}{a^3\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.06, size = 303, normalized size = 4.09 \[ \begin {cases} - \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {2 \sin ^{3}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {12 \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {9}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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