Integrand size = 27, antiderivative size = 179 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^3 d}+\frac {b \csc ^4(c+d x)}{4 a^2 d}-\frac {\csc ^5(c+d x)}{5 a d}-\frac {b \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^6 d}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^6 d} \]
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Time = 0.15 (sec) , antiderivative size = 179, 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 = {2916, 12, 908} \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \csc ^4(c+d x)}{4 a^2 d}-\frac {b \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^6 d}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^6 d}-\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^3 d}-\frac {\csc ^5(c+d x)}{5 a d} \]
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Rule 12
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^6 \left (b^2-x^2\right )^2}{x^6 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {b \text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x^6 (a+x)} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b \text {Subst}\left (\int \left (\frac {b^4}{a x^6}-\frac {b^4}{a^2 x^5}+\frac {-2 a^2 b^2+b^4}{a^3 x^4}+\frac {2 a^2 b^2-b^4}{a^4 x^3}+\frac {\left (a^2-b^2\right )^2}{a^5 x^2}-\frac {\left (a^2-b^2\right )^2}{a^6 x}+\frac {\left (a^2-b^2\right )^2}{a^6 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d} \\ & = -\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^3 d}+\frac {b \csc ^4(c+d x)}{4 a^2 d}-\frac {\csc ^5(c+d x)}{5 a d}-\frac {b \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^6 d}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^6 d} \\ \end{align*}
Time = 6.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^3 d}+\frac {b \csc ^4(c+d x)}{4 a^2 d}-\frac {\csc ^5(c+d x)}{5 a d}-\frac {b \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^6 d}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^6 d} \]
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Time = 0.59 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {\left (\csc ^{5}\left (d x +c \right )\right ) a^{4}}{5}-\frac {b \left (\csc ^{4}\left (d x +c \right )\right ) a^{3}}{4}-\frac {2 a^{4} \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{2} b^{2} \left (\csc ^{3}\left (d x +c \right )\right )}{3}+a^{3} b \left (\csc ^{2}\left (d x +c \right )\right )-\frac {a \,b^{3} \left (\csc ^{2}\left (d x +c \right )\right )}{2}+\csc \left (d x +c \right ) a^{4}-2 \csc \left (d x +c \right ) a^{2} b^{2}+\csc \left (d x +c \right ) b^{4}}{a^{5}}+\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a \csc \left (d x +c \right )+b \right )}{a^{6}}}{d}\) | \(160\) |
default | \(\frac {-\frac {\frac {\left (\csc ^{5}\left (d x +c \right )\right ) a^{4}}{5}-\frac {b \left (\csc ^{4}\left (d x +c \right )\right ) a^{3}}{4}-\frac {2 a^{4} \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{2} b^{2} \left (\csc ^{3}\left (d x +c \right )\right )}{3}+a^{3} b \left (\csc ^{2}\left (d x +c \right )\right )-\frac {a \,b^{3} \left (\csc ^{2}\left (d x +c \right )\right )}{2}+\csc \left (d x +c \right ) a^{4}-2 \csc \left (d x +c \right ) a^{2} b^{2}+\csc \left (d x +c \right ) b^{4}}{a^{5}}+\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a \csc \left (d x +c \right )+b \right )}{a^{6}}}{d}\) | \(160\) |
parallelrisch | \(\frac {960 b \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-960 b \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}-\frac {5 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b}{2}+\left (-\frac {25}{3} a^{4}+\frac {20}{3} a^{2} b^{2}\right ) \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -20 a \,b^{3}\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (50 a^{4}-140 a^{2} b^{2}+80 b^{4}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b}{2}+\left (-\frac {25}{3} a^{4}+\frac {20}{3} a^{2} b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -20 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+50 a^{4}-140 a^{2} b^{2}+80 b^{4}\right )\right ) a}{960 a^{6} d}\) | \(303\) |
norman | \(\frac {-\frac {1}{160 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}+\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a^{2} d}+\frac {b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2} d}+\frac {\left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a^{3} d}+\frac {\left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a^{3} d}-\frac {\left (5 a^{4}-14 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5} d}-\frac {\left (5 a^{4}-14 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5} d}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{4} d}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{4} d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{a^{6} d}-\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6} d}\) | \(363\) |
risch | \(-\frac {2 i \left (15 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}-30 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+15 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}-20 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}+100 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-15 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+45 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+58 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}-140 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+90 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+15 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-45 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-20 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+100 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+60 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+30 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+15 a^{4} {\mathrm e}^{i \left (d x +c \right )}-30 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+15 b^{4} {\mathrm e}^{i \left (d x +c \right )}-30 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-60 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{15 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}-\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{6} d}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{2} d}-\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{6} d}\) | \(558\) |
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Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (171) = 342\).
Time = 0.42 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.93 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {32 \, a^{5} - 100 \, a^{3} b^{2} + 60 \, a b^{4} + 60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{4} - 20 \, {\left (4 \, a^{5} - 11 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - 60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) + 60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 15 \, {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - 2 \, {\left (2 \, a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{6} d \cos \left (d x + c\right )^{4} - 2 \, a^{6} d \cos \left (d x + c\right )^{2} + a^{6} d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6}} - \frac {60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{6}} + \frac {15 \, a^{3} b \sin \left (d x + c\right ) - 60 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4} - 12 \, a^{4} - 30 \, {\left (2 \, a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{3} + 20 \, {\left (2 \, a^{4} - a^{2} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{5} \sin \left (d x + c\right )^{5}}}{60 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.40 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{6}} - \frac {60 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b} - \frac {137 \, a^{4} b \sin \left (d x + c\right )^{5} - 274 \, a^{2} b^{3} \sin \left (d x + c\right )^{5} + 137 \, b^{5} \sin \left (d x + c\right )^{5} - 60 \, a^{5} \sin \left (d x + c\right )^{4} + 120 \, a^{3} b^{2} \sin \left (d x + c\right )^{4} - 60 \, a b^{4} \sin \left (d x + c\right )^{4} - 60 \, a^{4} b \sin \left (d x + c\right )^{3} + 30 \, a^{2} b^{3} \sin \left (d x + c\right )^{3} + 40 \, a^{5} \sin \left (d x + c\right )^{2} - 20 \, a^{3} b^{2} \sin \left (d x + c\right )^{2} + 15 \, a^{4} b \sin \left (d x + c\right ) - 12 \, a^{5}}{a^{6} \sin \left (d x + c\right )^{5}}}{60 \, d} \]
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Time = 12.05 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.13 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {b^2}{8\,a^3}-\frac {5}{16\,a}+\frac {2\,b\,\left (\frac {b}{16\,a^2}+\frac {2\,b\,\left (\frac {5}{32\,a}-\frac {b^2}{8\,a^3}\right )}{a}\right )}{a}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {b}{32\,a^2}+\frac {b\,\left (\frac {5}{32\,a}-\frac {b^2}{8\,a^3}\right )}{a}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5}{96\,a}-\frac {b^2}{24\,a^3}\right )}{d}+\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {5\,a^4}{3}-\frac {4\,a^2\,b^2}{3}\right )-\frac {a^4}{5}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (10\,a^4-28\,a^2\,b^2+16\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a\,b^3-6\,a^3\,b\right )+\frac {a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{32\,a^5\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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