Integrand size = 9, antiderivative size = 20 \[ \int \cos (5 x) \csc ^5(x) \, dx=6 \csc ^2(x)-\frac {\csc ^4(x)}{4}+16 \log (\sin (x)) \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4451, 1261, 712} \[ \int \cos (5 x) \csc ^5(x) \, dx=-\frac {1}{4} \csc ^4(x)+6 \csc ^2(x)+16 \log (\sin (x)) \]
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Rule 712
Rule 1261
Rule 4451
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x \left (5-20 x^2+16 x^4\right )}{\left (1-x^2\right )^3} \, dx,x,\cos (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {5-20 x+16 x^2}{(1-x)^3} \, dx,x,\cos ^2(x)\right )\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{(-1+x)^3}-\frac {12}{(-1+x)^2}-\frac {16}{-1+x}\right ) \, dx,x,\cos ^2(x)\right )\right ) \\ & = 6 \csc ^2(x)-\frac {\csc ^4(x)}{4}+16 \log (\sin (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \cos (5 x) \csc ^5(x) \, dx=6 \csc ^2(x)-\frac {\csc ^4(x)}{4}+16 \log (\sin (x)) \]
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Time = 25.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75
method | result | size |
default | \(-\frac {5}{4 \sin \left (x \right )^{4}}+\frac {5 \left (\cos ^{4}\left (x \right )\right )}{\sin \left (x \right )^{4}}-4 \left (\cot ^{4}\left (x \right )\right )+8 \left (\cot ^{2}\left (x \right )\right )+16 \ln \left (\sin \left (x \right )\right )\) | \(35\) |
risch | \(-16 i x -\frac {4 \left (6 \,{\mathrm e}^{6 i x}-11 \,{\mathrm e}^{4 i x}+6 \,{\mathrm e}^{2 i x}\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}+16 \ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(49\) |
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.15 \[ \int \cos (5 x) \csc ^5(x) \, dx=-\frac {24 \, \cos \left (x\right )^{2} - 64 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) - 23}{4 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]
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Time = 8.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \cos (5 x) \csc ^5(x) \, dx=8 \log {\left (\sin ^{2}{\left (x \right )} \right )} + \frac {6}{\sin ^{2}{\left (x \right )}} - \frac {1}{4 \sin ^{4}{\left (x \right )}} \]
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none
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \cos (5 x) \csc ^5(x) \, dx=\frac {5}{\sin \left (x\right )^{2}} + \frac {4 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} + \frac {11}{2} \, \log \left (\sin \left (x\right )^{2}\right ) + 5 \, \log \left (\sin \left (x\right )\right ) \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \cos (5 x) \csc ^5(x) \, dx=\frac {24 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} + 16 \, \log \left ({\left | \sin \left (x\right ) \right |}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \cos (5 x) \csc ^5(x) \, dx=8\,\ln \left ({\sin \left (x\right )}^2\right )+\frac {6\,{\sin \left (x\right )}^2-\frac {1}{4}}{{\sin \left (x\right )}^4} \]
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