Optimal. Leaf size=83 \[ \frac {3}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {\log (\sin (c+d x)+1)}{a^4 d}-\frac {3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.09, antiderivative size = 83, 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} \[ \frac {3}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {\log (\sin (c+d x)+1)}{a^4 d}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 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))^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*}
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Mathematica [A] time = 0.37, 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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fricas [A] time = 0.51, size = 112, normalized size = 1.35 \[ \frac {18 \, \cos \left (d x + c\right )^{2} + 6 \, {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 27 \, \sin \left (d x + c\right ) - 29}{6 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 55, normalized size = 0.66 \[ \frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} + \frac {18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 72, normalized size = 0.87 \[ \frac {1}{3 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {3}{2 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a^{4} d}+\frac {3}{d \,a^{4} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 83, normalized size = 1.00 \[ \frac {\frac {18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} + \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 54, normalized size = 0.65 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^4\,d}+\frac {3\,{\sin \left (c+d\,x\right )}^2+\frac {9\,\sin \left (c+d\,x\right )}{2}+\frac {11}{6}}{a^4\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.48, size = 466, normalized size = 5.61 \[ \begin {cases} \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{3}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {27 \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {11}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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