Optimal. Leaf size=30 \[ \frac {\sin ^2(c+d x)}{2 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 37} \[ \frac {\sin ^2(c+d x)}{2 a d (a \sin (c+d x)+a)^2} \]
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 (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{a (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\sin ^2(c+d x)}{2 a d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 30, normalized size = 1.00 \[ \frac {\sin ^2(c+d x)}{2 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 46, normalized size = 1.53 \[ \frac {2 \, \sin \left (d x + c\right ) + 1}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 28, normalized size = 0.93 \[ -\frac {2 \, \sin \left (d x + c\right ) + 1}{2 \, a^{3} d {\left (\sin \left (d x + c\right ) + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 33, normalized size = 1.10 \[ \frac {\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{1+\sin \left (d x +c \right )}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 44, normalized size = 1.47 \[ -\frac {2 \, \sin \left (d x + c\right ) + 1}{2 \, {\left (a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 37, normalized size = 1.23 \[ \frac {1}{2\,a^3\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^2}-\frac {1}{a^3\,d\,\left (\sin \left (c+d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.70, size = 99, normalized size = 3.30 \[ \begin {cases} - \frac {2 \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {1}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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