Optimal. Leaf size=116 \[ \frac {\sin ^2(c+d x)}{2 a^4 d}-\frac {4 \sin (c+d x)}{a^4 d}+\frac {10}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {10 \log (\sin (c+d x)+1)}{a^4 d}-\frac {5}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.10, antiderivative size = 116, 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} \[ \frac {\sin ^2(c+d x)}{2 a^4 d}-\frac {4 \sin (c+d x)}{a^4 d}+\frac {10}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {5}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {10 \log (\sin (c+d x)+1)}{a^4 d}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
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*}
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Mathematica [A] time = 0.88, 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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fricas [A] time = 0.52, size = 144, normalized size = 1.24 \[ \frac {30 \, \cos \left (d x + c\right )^{4} - 87 \, \cos \left (d x + c\right )^{2} + 120 \, {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, \cos \left (d x + c\right )^{4} + 39 \, \cos \left (d x + c\right )^{2} + 10\right )} \sin \left (d x + c\right ) - 34}{12 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 84, normalized size = 0.72 \[ \frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} + \frac {60 \, \sin \left (d x + c\right )^{2} + 105 \, \sin \left (d x + c\right ) + 47}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, {\left (a^{4} \sin \left (d x + c\right )^{2} - 8 \, a^{4} \sin \left (d x + c\right )\right )}}{a^{8}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 103, normalized size = 0.89 \[ \frac {\sin ^{2}\left (d x +c \right )}{2 a^{4} d}-\frac {4 \sin \left (d x +c \right )}{a^{4} d}+\frac {1}{3 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{2 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {10 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{4} d}+\frac {10}{d \,a^{4} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 105, normalized size = 0.91 \[ \frac {\frac {60 \, \sin \left (d x + c\right )^{2} + 105 \, \sin \left (d x + c\right ) + 47}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} + \frac {3 \, {\left (\sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right )\right )}}{a^{4}} + \frac {60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.53, size = 114, normalized size = 0.98 \[ \frac {10\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^4\,d}-\frac {4\,\sin \left (c+d\,x\right )}{a^4\,d}+\frac {10\,{\sin \left (c+d\,x\right )}^2+\frac {35\,\sin \left (c+d\,x\right )}{2}+\frac {47}{6}}{d\,\left (a^4\,{\sin \left (c+d\,x\right )}^3+3\,a^4\,{\sin \left (c+d\,x\right )}^2+3\,a^4\,\sin \left (c+d\,x\right )+a^4\right )}+\frac {{\sin \left (c+d\,x\right )}^2}{2\,a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.05, size = 588, normalized size = 5.07 \[ \begin {cases} \frac {60 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{3}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {180 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {180 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {60 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {3 \sin ^{5}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} - \frac {15 \sin ^{4}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {180 \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {270 \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {110}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{5}{\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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