Optimal. Leaf size=39 \[ -\frac {2}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ -\frac {2}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{-a-x}+\frac {2 a}{(a+x)^2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {\log (1+\sin (c+d x))}{a^3 d}-\frac {2}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 58, normalized size = 1.49 \[ -\frac {\sin (c+d x) \log (\sin (c+d x)+1)+\log (\sin (c+d x)+1)+2}{a^3 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 41, normalized size = 1.05 \[ -\frac {{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2}{a^{3} d \sin \left (d x + c\right ) + a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 35, normalized size = 0.90 \[ -\frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} + \frac {2}{a^{3} {\left (\sin \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 37, normalized size = 0.95 \[ -\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a^{3} d}-\frac {2}{a^{3} d \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 37, normalized size = 0.95 \[ -\frac {\frac {2}{a^{3} \sin \left (d x + c\right ) + a^{3}} + \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 36, normalized size = 0.92 \[ -\frac {2}{a^3\,d\,\left (\sin \left (c+d\,x\right )+1\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.95, size = 299, normalized size = 7.67 \[ \begin {cases} - \frac {2 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {4 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {2 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {2 \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {\cos ^{2}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {2}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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