Integrand size = 27, antiderivative size = 116 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {10 \log (1+\sin (c+d x))}{a^4 d}-\frac {4 \sin (c+d x)}{a^4 d}+\frac {\sin ^2(c+d x)}{2 a^4 d}+\frac {1}{3 a d (a+a \sin (c+d x))^3}-\frac {5}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {10}{d \left (a^4+a^4 \sin (c+d x)\right )} \]
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Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 45} \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\sin ^2(c+d x)}{2 a^4 d}-\frac {4 \sin (c+d x)}{a^4 d}+\frac {10}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {10 \log (\sin (c+d x)+1)}{a^4 d}-\frac {5}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^5}{a^5 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \frac {x^5}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^6 d} \\ & = \frac {\text {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*}
Time = 0.62 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {47+60 \log (1+\sin (c+d x))+9 (9+20 \log (1+\sin (c+d x))) \sin (c+d x)+9 (-1+20 \log (1+\sin (c+d x))) \sin ^2(c+d x)+(-63+60 \log (1+\sin (c+d x))) \sin ^3(c+d x)-15 \sin ^4(c+d x)+3 \sin ^5(c+d x)}{6 a^4 d (1+\sin (c+d x))^3} \]
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Time = 0.54 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-4 \sin \left (d x +c \right )+\frac {10}{1+\sin \left (d x +c \right )}+10 \ln \left (1+\sin \left (d x +c \right )\right )+\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{4}}\) | \(74\) |
default | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-4 \sin \left (d x +c \right )+\frac {10}{1+\sin \left (d x +c \right )}+10 \ln \left (1+\sin \left (d x +c \right )\right )+\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{4}}\) | \(74\) |
parallelrisch | \(\frac {\left (-1440 \cos \left (2 d x +2 c \right )+3600 \sin \left (d x +c \right )-240 \sin \left (3 d x +3 c \right )+2400\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2880 \cos \left (2 d x +2 c \right )-7200 \sin \left (d x +c \right )+480 \sin \left (3 d x +3 c \right )-4800\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-1320 \cos \left (2 d x +2 c \right )+30 \cos \left (4 d x +4 c \right )+2250 \sin \left (d x +c \right )-425 \sin \left (3 d x +3 c \right )-3 \sin \left (5 d x +5 c \right )+1290}{24 d \,a^{4} \left (-10+6 \cos \left (2 d x +2 c \right )+\sin \left (3 d x +3 c \right )-15 \sin \left (d x +c \right )\right )}\) | \(185\) |
risch | \(-\frac {10 i x}{a^{4}}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{4}}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )}}{d \,a^{4}}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )}}{d \,a^{4}}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{4}}-\frac {20 i c}{d \,a^{4}}+\frac {2 i \left (-154 \,{\mathrm e}^{3 i \left (d x +c \right )}-105 i {\mathrm e}^{2 i \left (d x +c \right )}+105 i {\mathrm e}^{4 i \left (d x +c \right )}+30 \,{\mathrm e}^{5 i \left (d x +c \right )}+30 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{6}}+\frac {20 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{4}}\) | \(191\) |
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Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.24 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {30 \, \cos \left (d x + c\right )^{4} - 87 \, \cos \left (d x + c\right )^{2} + 120 \, {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, \cos \left (d x + c\right )^{4} + 39 \, \cos \left (d x + c\right )^{2} + 10\right )} \sin \left (d x + c\right ) - 34}{12 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (100) = 200\).
Time = 1.85 (sec) , antiderivative size = 588, normalized size of antiderivative = 5.07 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\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}{\left (c \right )} \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {60 \, \sin \left (d x + c\right )^{2} + 105 \, \sin \left (d x + c\right ) + 47}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} + \frac {3 \, {\left (\sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right )\right )}}{a^{4}} + \frac {60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}}}{6 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} + \frac {60 \, \sin \left (d x + c\right )^{2} + 105 \, \sin \left (d x + c\right ) + 47}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, {\left (a^{4} \sin \left (d x + c\right )^{2} - 8 \, a^{4} \sin \left (d x + c\right )\right )}}{a^{8}}}{6 \, d} \]
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Time = 10.37 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.98 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {10\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^4\,d}-\frac {4\,\sin \left (c+d\,x\right )}{a^4\,d}+\frac {10\,{\sin \left (c+d\,x\right )}^2+\frac {35\,\sin \left (c+d\,x\right )}{2}+\frac {47}{6}}{d\,\left (a^4\,{\sin \left (c+d\,x\right )}^3+3\,a^4\,{\sin \left (c+d\,x\right )}^2+3\,a^4\,\sin \left (c+d\,x\right )+a^4\right )}+\frac {{\sin \left (c+d\,x\right )}^2}{2\,a^4\,d} \]
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