Optimal. Leaf size=46 \[ \frac {1}{3 a d (a \sin (c+d x)+a)^3}-\frac {1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ \frac {1}{3 a d (a \sin (c+d x)+a)^3}-\frac {1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2} \]
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 (c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{a (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{(a+x)^4}+\frac {1}{(a+x)^3}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {1}{3 a d (a+a \sin (c+d x))^3}-\frac {1}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 30, normalized size = 0.65 \[ -\frac {3 \sin (c+d x)+1}{6 a^4 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 62, normalized size = 1.35 \[ \frac {3 \, \sin \left (d x + c\right ) + 1}{6 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 28, normalized size = 0.61 \[ -\frac {3 \, \sin \left (d x + c\right ) + 1}{6 \, a^{4} d {\left (\sin \left (d x + c\right ) + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 33, normalized size = 0.72 \[ \frac {-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 57, normalized size = 1.24 \[ -\frac {3 \, \sin \left (d x + c\right ) + 1}{6 \, {\left (a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.46, size = 37, normalized size = 0.80 \[ \frac {1}{3\,a^4\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^3}-\frac {1}{2\,a^4\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.49, size = 129, normalized size = 2.80 \[ \begin {cases} - \frac {3 \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} - \frac {1}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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