Optimal. Leaf size=82 \[ \frac{\sin ^4(c+d x)}{4 a^3 d}-\frac{\sin ^3(c+d x)}{a^3 d}+\frac{2 \sin ^2(c+d x)}{a^3 d}-\frac{4 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.118477, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ \frac{\sin ^4(c+d x)}{4 a^3 d}-\frac{\sin ^3(c+d x)}{a^3 d}+\frac{2 \sin ^2(c+d x)}{a^3 d}-\frac{4 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 x^2}{a^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 x^2}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-4 a^3+4 a^2 x-3 a x^2+x^3+\frac{4 a^4}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{4 \log (1+\sin (c+d x))}{a^3 d}-\frac{4 \sin (c+d x)}{a^3 d}+\frac{2 \sin ^2(c+d x)}{a^3 d}-\frac{\sin ^3(c+d x)}{a^3 d}+\frac{\sin ^4(c+d x)}{4 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.932708, size = 59, normalized size = 0.72 \[ \frac{-152 \sin (c+d x)+8 \sin (3 (c+d x))-36 \cos (2 (c+d x))+\cos (4 (c+d x))+128 \log (\sin (c+d x)+1)+35}{32 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 81, normalized size = 1. \begin{align*} 4\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-4\,{\frac{\sin \left ( dx+c \right ) }{{a}^{3}d}}+2\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{{a}^{3}d}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{{a}^{3}d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,{a}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07935, size = 82, normalized size = 1. \begin{align*} \frac{\frac{\sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 8 \, \sin \left (d x + c\right )^{2} - 16 \, \sin \left (d x + c\right )}{a^{3}} + \frac{16 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1258, size = 155, normalized size = 1.89 \begin{align*} \frac{\cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} + 4 \,{\left (\cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) + 16 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{4 \, 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 = 1.32593, size = 225, normalized size = 2.74 \begin{align*} -\frac{\frac{12 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac{24 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{25 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 124 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 96 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 96 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 124 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 25}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} a^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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