Integrand size = 27, antiderivative size = 99 \[ \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{a d}+\frac {2 \sin ^3(c+d x)}{3 a d}+\frac {\sin ^4(c+d x)}{4 a d}-\frac {\sin ^5(c+d x)}{5 a d} \]
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Time = 0.08 (sec) , antiderivative size = 99, 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} \[ \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d}+\frac {2 \sin ^3(c+d x)}{3 a d}-\frac {\sin ^2(c+d x)}{a d}-\frac {\sin (c+d x)}{a d}+\frac {\log (\sin (c+d x))}{a d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a (a-x)^3 (a+x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^6 d} \\ & = \frac {\text {Subst}\left (\int \left (-a^4+\frac {a^5}{x}-2 a^3 x+2 a^2 x^2+a x^3-x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d} \\ & = \frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{a d}+\frac {2 \sin ^3(c+d x)}{3 a d}+\frac {\sin ^4(c+d x)}{4 a d}-\frac {\sin ^5(c+d x)}{5 a d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {60 \log (\sin (c+d x))-60 \sin (c+d x)-60 \sin ^2(c+d x)+40 \sin ^3(c+d x)+15 \sin ^4(c+d x)-12 \sin ^5(c+d x)}{60 a d} \]
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Time = 0.35 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\left (\sin ^{2}\left (d x +c \right )\right )-\sin \left (d x +c \right )+\ln \left (\sin \left (d x +c \right )\right )}{d a}\) | \(64\) |
default | \(\frac {-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\left (\sin ^{2}\left (d x +c \right )\right )-\sin \left (d x +c \right )+\ln \left (\sin \left (d x +c \right )\right )}{d a}\) | \(64\) |
parallelrisch | \(\frac {480 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-480 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-195-6 \sin \left (5 d x +5 c \right )-50 \sin \left (3 d x +3 c \right )-300 \sin \left (d x +c \right )+180 \cos \left (2 d x +2 c \right )+15 \cos \left (4 d x +4 c \right )}{480 a d}\) | \(89\) |
risch | \(-\frac {i x}{a}+\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}}{16 a d}+\frac {3 \,{\mathrm e}^{-2 i \left (d x +c \right )}}{16 a d}-\frac {2 i c}{a d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}-\frac {5 \sin \left (d x +c \right )}{8 a d}-\frac {\sin \left (5 d x +5 c \right )}{80 d a}+\frac {\cos \left (4 d x +4 c \right )}{32 a d}-\frac {5 \sin \left (3 d x +3 c \right )}{48 d a}\) | \(137\) |
norman | \(\frac {\frac {2}{a d}+\frac {2 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {6 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {10 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {40 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {40 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {68 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {68 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {38 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {38 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(288\) |
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Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 60 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{60 \, a d} \]
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Timed out. \[ \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 40 \, \sin \left (d x + c\right )^{3} + 60 \, \sin \left (d x + c\right )^{2} + 60 \, \sin \left (d x + c\right )}{a} - \frac {60 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{60 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {12 \, a^{4} \sin \left (d x + c\right )^{5} - 15 \, a^{4} \sin \left (d x + c\right )^{4} - 40 \, a^{4} \sin \left (d x + c\right )^{3} + 60 \, a^{4} \sin \left (d x + c\right )^{2} + 60 \, a^{4} \sin \left (d x + c\right )}{a^{5}}}{60 \, d} \]
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Time = 10.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a\,d}-\frac {\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{a\,d}-\frac {8\,\sin \left (c+d\,x\right )}{15\,a\,d}+\frac {{\cos \left (c+d\,x\right )}^2}{2\,a\,d}+\frac {{\cos \left (c+d\,x\right )}^4}{4\,a\,d}-\frac {4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{15\,a\,d}-\frac {{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{5\,a\,d} \]
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