Optimal. Leaf size=70 \[ \frac {\sin ^2(c+d x)}{2 a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {3 \log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 70, 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} \[ \frac {\sin ^2(c+d x)}{2 a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {3 \log (\sin (c+d x)+1)}{a^2 d} \]
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))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{a^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2 a+x-\frac {a^3}{(a+x)^2}+\frac {3 a^2}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {3 \log (1+\sin (c+d x))}{a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {\sin ^2(c+d x)}{2 a^2 d}+\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 71, normalized size = 1.01 \[ \frac {\sin ^3(c+d x)-3 \sin ^2(c+d x)+\sin (c+d x) (6 \log (\sin (c+d x)+1)-4)+6 \log (\sin (c+d x)+1)+2}{2 a^2 d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 72, normalized size = 1.03 \[ \frac {6 \, \cos \left (d x + c\right )^{2} + 12 \, {\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, \cos \left (d x + c\right )^{2} + 7\right )} \sin \left (d x + c\right ) - 3}{4 \, {\left (a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 90, normalized size = 1.29 \[ -\frac {\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left (\frac {6 \, a}{a \sin \left (d x + c\right ) + a} - 1\right )}}{a^{4}} + \frac {6 \, \log \left (\frac {{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left | a \right |}}\right )}{a^{2}} - \frac {2}{{\left (a \sin \left (d x + c\right ) + a\right )} a}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 66, normalized size = 0.94 \[ \frac {\sin ^{2}\left (d x +c \right )}{2 a^{2} d}-\frac {2 \sin \left (d x +c \right )}{a^{2} d}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{2} d}+\frac {1}{d \,a^{2} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 59, normalized size = 0.84 \[ \frac {\frac {2}{a^{2} \sin \left (d x + c\right ) + a^{2}} + \frac {\sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{a^{2}} + \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.47, size = 59, normalized size = 0.84 \[ \frac {\frac {1}{a^2\,\sin \left (c+d\,x\right )+a^2}+\frac {3\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^2}-\frac {2\,\sin \left (c+d\,x\right )}{a^2}+\frac {{\sin \left (c+d\,x\right )}^2}{2\,a^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.81, size = 170, normalized size = 2.43 \[ \begin {cases} \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{2 a^{2} d \sin {\left (c + d x \right )} + 2 a^{2} d} + \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{2 a^{2} d \sin {\left (c + d x \right )} + 2 a^{2} d} + \frac {\sin ^{3}{\left (c + d x \right )}}{2 a^{2} d \sin {\left (c + d x \right )} + 2 a^{2} d} - \frac {3 \sin ^{2}{\left (c + d x \right )}}{2 a^{2} d \sin {\left (c + d x \right )} + 2 a^{2} d} + \frac {6}{2 a^{2} d \sin {\left (c + d x \right )} + 2 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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