Integrand size = 27, antiderivative size = 131 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc (c+d x)}{d}-\frac {3 a^3 \csc ^2(c+d x)}{2 d}-\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {5 a^3 \log (\sin (c+d x))}{d}-\frac {5 a^3 \sin (c+d x)}{d}+\frac {a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin ^3(c+d x)}{d}+\frac {a^3 \sin ^4(c+d x)}{4 d} \]
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Time = 0.08 (sec) , antiderivative size = 131, 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 = {2915, 12, 90} \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^4(c+d x)}{4 d}+\frac {a^3 \sin ^3(c+d x)}{d}+\frac {a^3 \sin ^2(c+d x)}{2 d}-\frac {5 a^3 \sin (c+d x)}{d}-\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc ^2(c+d x)}{2 d}-\frac {a^3 \csc (c+d x)}{d}-\frac {5 a^3 \log (\sin (c+d x))}{d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^4 (a-x)^2 (a+x)^5}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^2 (a+x)^5}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \left (-5 a^3+\frac {a^7}{x^4}+\frac {3 a^6}{x^3}+\frac {a^5}{x^2}-\frac {5 a^4}{x}+a^2 x+3 a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = -\frac {a^3 \csc (c+d x)}{d}-\frac {3 a^3 \csc ^2(c+d x)}{2 d}-\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {5 a^3 \log (\sin (c+d x))}{d}-\frac {5 a^3 \sin (c+d x)}{d}+\frac {a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin ^3(c+d x)}{d}+\frac {a^3 \sin ^4(c+d x)}{4 d} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \left (12 \csc (c+d x)+18 \csc ^2(c+d x)+4 \csc ^3(c+d x)+60 \log (\sin (c+d x))+60 \sin (c+d x)-6 \sin ^2(c+d x)-12 \sin ^3(c+d x)-3 \sin ^4(c+d x)\right )}{12 d} \]
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Time = 0.40 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.44
method | result | size |
parallelrisch | \(\frac {a^{3} \left (960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (3 d x +3 c \right )-960 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (3 d x +3 c \right )-2880 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+2880 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )-615 \sin \left (3 d x +3 c \right )+1944 \cos \left (2 d x +2 c \right )+489 \sin \left (d x +c \right )-3 \sin \left (7 d x +7 c \right )+45 \sin \left (5 d x +5 c \right )-24 \cos \left (6 d x +6 c \right )-336 \cos \left (4 d x +4 c \right )-1840\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6144 d}\) | \(189\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(210\) |
default | \(\frac {a^{3} \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(210\) |
risch | \(5 i a^{3} x +\frac {a^{3} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}+\frac {i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{8 d}-\frac {3 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {17 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {17 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}-\frac {i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{8 d}+\frac {a^{3} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {10 i a^{3} c}{d}-\frac {2 i a^{3} \left (9 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}-9 i {\mathrm e}^{2 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(259\) |
norman | \(\frac {-\frac {a^{3}}{24 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {19 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {107 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {683 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {683 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {107 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {19 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {3 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {14 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {43 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {43 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a^{3} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(304\) |
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Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.21 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {96 \, a^{3} \cos \left (d x + c\right )^{6} + 192 \, a^{3} \cos \left (d x + c\right )^{4} - 768 \, a^{3} \cos \left (d x + c\right )^{2} + 512 \, a^{3} - 480 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{6} - 40 \, a^{3} \cos \left (d x + c\right )^{4} + 45 \, a^{3} \cos \left (d x + c\right )^{2} + 35 \, a^{3}\right )} \sin \left (d x + c\right )}{96 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 \, a^{3} \sin \left (d x + c\right )^{4} + 12 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} - 60 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) - 60 \, a^{3} \sin \left (d x + c\right ) - \frac {2 \, {\left (6 \, a^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3}\right )}}{\sin \left (d x + c\right )^{3}}}{12 \, d} \]
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Time = 0.41 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.93 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 \, a^{3} \sin \left (d x + c\right )^{4} + 12 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} - 60 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 60 \, a^{3} \sin \left (d x + c\right ) + \frac {2 \, {\left (55 \, a^{3} \sin \left (d x + c\right )^{3} - 6 \, a^{3} \sin \left (d x + c\right )^{2} - 9 \, a^{3} \sin \left (d x + c\right ) - 2 \, a^{3}\right )}}{\sin \left (d x + c\right )^{3}}}{12 \, d} \]
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Time = 9.71 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.54 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5\,a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {5\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {5\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {85\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {589\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-52\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {622\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+102\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \]
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