Optimal. Leaf size=30 \[ \frac {\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 37} \[ \frac {\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{a^2 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\sin ^3(c+d x)}{3 a d (a+a \sin (c+d x))^3}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 53, normalized size = 1.77 \[ \frac {-6 \sin (c+d x)+3 \cos (2 (c+d x))-5}{6 a^4 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 72, normalized size = 2.40 \[ -\frac {3 \, \cos \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) - 4}{3 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 38, normalized size = 1.27 \[ -\frac {3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{4} d {\left (\sin \left (d x + c\right ) + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 43, normalized size = 1.43 \[ \frac {-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {1}{\left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{1+\sin \left (d x +c \right )}}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 67, normalized size = 2.23 \[ -\frac {3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) + 1}{3 \, {\left (a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.48, size = 54, normalized size = 1.80 \[ \frac {1}{a^4\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^2}-\frac {1}{a^4\,d\,\left (\sin \left (c+d\,x\right )+1\right )}-\frac {1}{3\,a^4\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.26, size = 192, normalized size = 6.40 \[ \begin {cases} - \frac {3 \sin ^{2}{\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} - \frac {3 \sin {\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} - \frac {1}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{2}{\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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