Integrand size = 27, antiderivative size = 114 \[ \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}-\frac {3 a \sin (c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}+\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^4(c+d x)}{4 d}-\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d} \]
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Time = 0.07 (sec) , antiderivative size = 114, 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} \[ \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^6(c+d x)}{6 d}-\frac {a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}-\frac {3 a \sin (c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^2 (a-x)^3 (a+x)^4}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \left (-3 a^5+\frac {a^7}{x^2}+\frac {a^6}{x}-3 a^4 x+3 a^3 x^2+3 a^2 x^3-a x^4-x^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = -\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}-\frac {3 a \sin (c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}+\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^4(c+d x)}{4 d}-\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}-\frac {3 a \sin (c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}+\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^4(c+d x)}{4 d}-\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d} \]
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Time = 0.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(105\) |
default | \(\frac {a \left (\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(105\) |
parallelrisch | \(\frac {\left (-100-192 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {12 \left (-206+222 \cos \left (d x +c \right )-32 \cos \left (2 d x +2 c \right )+17 \cos \left (3 d x +3 c \right )-2 \cos \left (4 d x +4 c \right )+\cos \left (5 d x +5 c \right )\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}-96 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (6 d x +6 c \right )+192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+87 \cos \left (2 d x +2 c \right )+12 \cos \left (4 d x +4 c \right )\right ) a}{192 d}\) | \(149\) |
risch | \(\frac {19 i a \,{\mathrm e}^{i \left (d x +c \right )}}{16 d}-i a x -\frac {2 i a c}{d}+\frac {29 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 d}-\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {19 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {29 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {a \cos \left (6 d x +6 c \right )}{192 d}-\frac {a \sin \left (5 d x +5 c \right )}{80 d}+\frac {a \cos \left (4 d x +4 c \right )}{16 d}-\frac {3 a \sin \left (3 d x +3 c \right )}{16 d}\) | \(183\) |
norman | \(\frac {-\frac {a}{2 d}-\frac {19 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {65 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {599 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 d}-\frac {599 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 d}-\frac {65 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {19 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {68 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {12 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {12 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(275\) |
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99 \[ \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {48 \, a \cos \left (d x + c\right )^{6} + 96 \, a \cos \left (d x + c\right )^{4} + 384 \, a \cos \left (d x + c\right )^{2} + 240 \, a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 5 \, {\left (8 \, a \cos \left (d x + c\right )^{6} + 12 \, a \cos \left (d x + c\right )^{4} + 24 \, a \cos \left (d x + c\right )^{2} - 19 \, a\right )} \sin \left (d x + c\right ) - 768 \, a}{240 \, d \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.80 \[ \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {10 \, a \sin \left (d x + c\right )^{6} + 12 \, a \sin \left (d x + c\right )^{5} - 45 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} + 90 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left (\sin \left (d x + c\right )\right ) + 180 \, a \sin \left (d x + c\right ) + \frac {60 \, a}{\sin \left (d x + c\right )}}{60 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {10 \, a \sin \left (d x + c\right )^{6} + 12 \, a \sin \left (d x + c\right )^{5} - 45 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} + 90 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 180 \, a \sin \left (d x + c\right ) + \frac {60 \, {\left (a \sin \left (d x + c\right ) + a\right )}}{\sin \left (d x + c\right )}}{60 \, d} \]
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Time = 11.17 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.98 \[ \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}-\frac {6\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}+\frac {18\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {104\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3\,d}+\frac {44\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d}-\frac {32\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d}+\frac {32\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3\,d}-\frac {13\,a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {14\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {112\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {136\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {96\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {32\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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