Integrand size = 27, antiderivative size = 93 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6 \log (1+\sin (c+d x))}{a^3 d}-\frac {3 \sin (c+d x)}{a^3 d}+\frac {\sin ^2(c+d x)}{2 a^3 d}-\frac {1}{2 a d (a+a \sin (c+d x))^2}+\frac {4}{d \left (a^3+a^3 \sin (c+d x)\right )} \]
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Time = 0.07 (sec) , antiderivative size = 93, 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} \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sin ^2(c+d x)}{2 a^3 d}-\frac {3 \sin (c+d x)}{a^3 d}+\frac {4}{d \left (a^3 \sin (c+d x)+a^3\right )}+\frac {6 \log (\sin (c+d x)+1)}{a^3 d}-\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{a^4 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \frac {x^4}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \left (-3 a+x+\frac {a^4}{(a+x)^3}-\frac {4 a^3}{(a+x)^2}+\frac {6 a^2}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {6 \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}-\frac {1}{2 a d (a+a \sin (c+d x))^2}+\frac {4}{d \left (a^3+a^3 \sin (c+d x)\right )} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {7+12 \log (1+\sin (c+d x))+(2+24 \log (1+\sin (c+d x))) \sin (c+d x)+(-11+12 \log (1+\sin (c+d x))) \sin ^2(c+d x)-4 \sin ^3(c+d x)+\sin ^4(c+d x)}{2 a^3 d (1+\sin (c+d x))^2} \]
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Time = 0.44 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \sin \left (d x +c \right )+6 \ln \left (1+\sin \left (d x +c \right )\right )+\frac {4}{1+\sin \left (d x +c \right )}-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{3}}\) | \(62\) |
default | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \sin \left (d x +c \right )+6 \ln \left (1+\sin \left (d x +c \right )\right )+\frac {4}{1+\sin \left (d x +c \right )}-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{3}}\) | \(62\) |
parallelrisch | \(\frac {\left (-48 \cos \left (2 d x +2 c \right )+192 \sin \left (d x +c \right )+144\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (96 \cos \left (2 d x +2 c \right )-384 \sin \left (d x +c \right )-288\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-68 \cos \left (2 d x +2 c \right )-\cos \left (4 d x +4 c \right )+120 \sin \left (d x +c \right )-8 \sin \left (3 d x +3 c \right )+69}{8 d \,a^{3} \left (-3+\cos \left (2 d x +2 c \right )-4 \sin \left (d x +c \right )\right )}\) | \(141\) |
risch | \(-\frac {6 i x}{a^{3}}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {12 i c}{d \,a^{3}}+\frac {2 i \left (7 i {\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{3 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4}}+\frac {12 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}\) | \(168\) |
norman | \(\frac {-\frac {12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {12 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {124 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {124 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {416 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {416 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {672 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {672 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {260 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {260 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {48 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {48 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {580 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {580 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{3}}-\frac {6 \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(341\) |
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Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.15 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, \cos \left (d x + c\right )^{4} + 19 \, \cos \left (d x + c\right )^{2} - 24 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) - 8}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (80) = 160\).
Time = 0.94 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.73 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\begin {cases} \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {24 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {\sin ^{4}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {4 \sin ^{3}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {24 \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {18}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\left (c \right )} \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {8 \, \sin \left (d x + c\right ) + 7}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} + \frac {\sin \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right )}{a^{3}} + \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{2 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} + \frac {8 \, \sin \left (d x + c\right ) + 7}{a^{3} {\left (\sin \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{2} - 6 \, a^{3} \sin \left (d x + c\right )}{a^{6}}}{2 \, d} \]
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Time = 9.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^3\,d}+\frac {4\,\sin \left (c+d\,x\right )+\frac {7}{2}}{d\,\left (a^3\,{\sin \left (c+d\,x\right )}^2+2\,a^3\,\sin \left (c+d\,x\right )+a^3\right )}-\frac {3\,\sin \left (c+d\,x\right )}{a^3\,d}+\frac {{\sin \left (c+d\,x\right )}^2}{2\,a^3\,d} \]
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