Optimal. Leaf size=50 \[ \frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.0499715, antiderivative size = 50, normalized size of antiderivative = 1., 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} \[ \frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-3 a+x+\frac{4 a^2}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{4 \log (1+\sin (c+d x))}{a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{\sin ^2(c+d x)}{2 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.0500167, size = 38, normalized size = 0.76 \[ \frac{\sin ^2(c+d x)-6 \sin (c+d x)+8 \log (\sin (c+d x)+1)}{2 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 49, normalized size = 1. \begin{align*} 4\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-3\,{\frac{\sin \left ( dx+c \right ) }{{a}^{3}d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{a}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942104, size = 55, normalized size = 1.1 \begin{align*} \frac{\frac{\sin \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right )}{a^{3}} + \frac{8 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01086, size = 100, normalized size = 2. \begin{align*} -\frac{\cos \left (d x + c\right )^{2} - 8 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, \sin \left (d x + c\right )}{2 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16831, size = 155, normalized size = 3.1 \begin{align*} -\frac{2 \,{\left (\frac{2 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac{4 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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