Integrand size = 27, antiderivative size = 74 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {3 b \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {3 b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot ^3(c+d x) \csc (c+d x)}{4 d} \]
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Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2917, 2687, 30, 2691, 3855} \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot ^5(c+d x)}{5 d}-\frac {3 b \text {arctanh}(\cos (c+d x))}{8 d}-\frac {b \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 b \cot (c+d x) \csc (c+d x)}{8 d} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2917
Rule 3855
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx+b \int \cot ^4(c+d x) \csc (c+d x) \, dx \\ & = -\frac {b \cot ^3(c+d x) \csc (c+d x)}{4 d}-\frac {1}{4} (3 b) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\frac {a \text {Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \cot ^5(c+d x)}{5 d}+\frac {3 b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {1}{8} (3 b) \int \csc (c+d x) \, dx \\ & = -\frac {3 b \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {3 b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot ^3(c+d x) \csc (c+d x)}{4 d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.82 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot ^5(c+d x)}{5 d}+\frac {5 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {b \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {3 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {5 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {b \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \]
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Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+b \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(100\) |
default | \(\frac {-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+b \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(100\) |
risch | \(-\frac {40 i a \,{\mathrm e}^{8 i \left (d x +c \right )}+25 b \,{\mathrm e}^{9 i \left (d x +c \right )}-10 b \,{\mathrm e}^{7 i \left (d x +c \right )}+80 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+10 b \,{\mathrm e}^{3 i \left (d x +c \right )}+8 i a -25 b \,{\mathrm e}^{i \left (d x +c \right )}}{20 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\) | \(135\) |
parallelrisch | \(\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 a \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -5 b \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 a \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +40 b \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -20 a \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{320 d}\) | \(156\) |
norman | \(\frac {-\frac {a}{160 d}+\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d}-\frac {a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d}+\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {7 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {7 b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(220\) |
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (66) = 132\).
Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.16 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {16 \, a \cos \left (d x + c\right )^{5} + 15 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10 \, {\left (5 \, b \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
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none
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {5 \, b {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {16 \, a}{\tan \left (d x + c\right )^{5}}}{80 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (66) = 132\).
Time = 0.37 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.34 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 20 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {274 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{320 \, d} \]
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Time = 10.40 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.35 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {3\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a}{5}\right )}{32\,d} \]
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