Optimal. Leaf size=116 \[ \frac{\sin ^2(c+d x)}{2 a^4 d}-\frac{4 \sin (c+d x)}{a^4 d}+\frac{10}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{5}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac{10 \log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.104187, antiderivative size = 116, normalized size of antiderivative = 1., 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} \[ \frac{\sin ^2(c+d x)}{2 a^4 d}-\frac{4 \sin (c+d x)}{a^4 d}+\frac{10}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{5}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac{10 \log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{a^5 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-4 a+x-\frac{a^5}{(a+x)^4}+\frac{5 a^4}{(a+x)^3}-\frac{10 a^3}{(a+x)^2}+\frac{10 a^2}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac{10 \log (1+\sin (c+d x))}{a^4 d}-\frac{4 \sin (c+d x)}{a^4 d}+\frac{\sin ^2(c+d x)}{2 a^4 d}+\frac{1}{3 a d (a+a \sin (c+d x))^3}-\frac{5}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac{10}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.842507, size = 119, normalized size = 1.03 \[ \frac{3 \sin ^5(c+d x)-15 \sin ^4(c+d x)+\sin ^3(c+d x) (60 \log (\sin (c+d x)+1)-63)+9 \sin ^2(c+d x) (20 \log (\sin (c+d x)+1)-1)+9 \sin (c+d x) (20 \log (\sin (c+d x)+1)+9)+60 \log (\sin (c+d x)+1)+47}{6 a^4 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 103, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{a}^{4}d}}-4\,{\frac{\sin \left ( dx+c \right ) }{{a}^{4}d}}-{\frac{5}{2\,{a}^{4}d \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{3\,{a}^{4}d \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+10\,{\frac{1}{{a}^{4}d \left ( 1+\sin \left ( dx+c \right ) \right ) }}+10\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{4}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00361, size = 142, normalized size = 1.22 \begin{align*} \frac{\frac{60 \, \sin \left (d x + c\right )^{2} + 105 \, \sin \left (d x + c\right ) + 47}{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}} + \frac{3 \,{\left (\sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right )\right )}}{a^{4}} + \frac{60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60126, size = 381, normalized size = 3.28 \begin{align*} \frac{30 \, \cos \left (d x + c\right )^{4} - 87 \, \cos \left (d x + c\right )^{2} + 120 \,{\left (3 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (2 \, \cos \left (d x + c\right )^{4} + 39 \, \cos \left (d x + c\right )^{2} + 10\right )} \sin \left (d x + c\right ) - 34}{12 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.5586, size = 796, normalized size = 6.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19353, size = 113, normalized size = 0.97 \begin{align*} \frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} + \frac{60 \, \sin \left (d x + c\right )^{2} + 105 \, \sin \left (d x + c\right ) + 47}{a^{4}{\left (\sin \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \,{\left (a^{4} \sin \left (d x + c\right )^{2} - 8 \, a^{4} \sin \left (d x + c\right )\right )}}{a^{8}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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