Integrand size = 17, antiderivative size = 69 \[ \int \cos ^4(a+b x) \cot ^5(a+b x) \, dx=\frac {2 \csc ^2(a+b x)}{b}-\frac {\csc ^4(a+b x)}{4 b}+\frac {6 \log (\sin (a+b x))}{b}-\frac {2 \sin ^2(a+b x)}{b}+\frac {\sin ^4(a+b x)}{4 b} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2670, 272, 45} \[ \int \cos ^4(a+b x) \cot ^5(a+b x) \, dx=\frac {\sin ^4(a+b x)}{4 b}-\frac {2 \sin ^2(a+b x)}{b}-\frac {\csc ^4(a+b x)}{4 b}+\frac {2 \csc ^2(a+b x)}{b}+\frac {6 \log (\sin (a+b x))}{b} \]
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Rule 45
Rule 272
Rule 2670
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^5} \, dx,x,-\sin (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {(1-x)^4}{x^3} \, dx,x,\sin ^2(a+b x)\right )}{2 b} \\ & = \frac {\text {Subst}\left (\int \left (-4+\frac {1}{x^3}-\frac {4}{x^2}+\frac {6}{x}+x\right ) \, dx,x,\sin ^2(a+b x)\right )}{2 b} \\ & = \frac {2 \csc ^2(a+b x)}{b}-\frac {\csc ^4(a+b x)}{4 b}+\frac {6 \log (\sin (a+b x))}{b}-\frac {2 \sin ^2(a+b x)}{b}+\frac {\sin ^4(a+b x)}{4 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \cos ^4(a+b x) \cot ^5(a+b x) \, dx=\frac {2 \csc ^2(a+b x)}{b}-\frac {\csc ^4(a+b x)}{4 b}+\frac {6 \log (\sin (a+b x))}{b}-\frac {2 \sin ^2(a+b x)}{b}+\frac {\sin ^4(a+b x)}{4 b} \]
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Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {-\frac {\cos ^{10}\left (b x +a \right )}{4 \sin \left (b x +a \right )^{4}}+\frac {3 \left (\cos ^{10}\left (b x +a \right )\right )}{4 \sin \left (b x +a \right )^{2}}+\frac {3 \left (\cos ^{8}\left (b x +a \right )\right )}{4}+\cos ^{6}\left (b x +a \right )+\frac {3 \left (\cos ^{4}\left (b x +a \right )\right )}{2}+3 \left (\cos ^{2}\left (b x +a \right )\right )+6 \ln \left (\sin \left (b x +a \right )\right )}{b}\) | \(89\) |
default | \(\frac {-\frac {\cos ^{10}\left (b x +a \right )}{4 \sin \left (b x +a \right )^{4}}+\frac {3 \left (\cos ^{10}\left (b x +a \right )\right )}{4 \sin \left (b x +a \right )^{2}}+\frac {3 \left (\cos ^{8}\left (b x +a \right )\right )}{4}+\cos ^{6}\left (b x +a \right )+\frac {3 \left (\cos ^{4}\left (b x +a \right )\right )}{2}+3 \left (\cos ^{2}\left (b x +a \right )\right )+6 \ln \left (\sin \left (b x +a \right )\right )}{b}\) | \(89\) |
risch | \(-6 i x +\frac {{\mathrm e}^{4 i \left (b x +a \right )}}{64 b}+\frac {7 \,{\mathrm e}^{2 i \left (b x +a \right )}}{16 b}+\frac {7 \,{\mathrm e}^{-2 i \left (b x +a \right )}}{16 b}+\frac {{\mathrm e}^{-4 i \left (b x +a \right )}}{64 b}-\frac {12 i a}{b}-\frac {4 \left (2 \,{\mathrm e}^{6 i \left (b x +a \right )}-3 \,{\mathrm e}^{4 i \left (b x +a \right )}+2 \,{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4}}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(138\) |
parallelrisch | \(-\frac {3 \left (\left (\frac {3}{4}-\cos \left (2 b x +2 a \right )+\frac {\cos \left (4 b x +4 a \right )}{4}\right ) \ln \left (\sec ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\left (\cos \left (2 b x +2 a \right )-\frac {\cos \left (4 b x +4 a \right )}{4}-\frac {3}{4}\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {71 \cos \left (2 b x +2 a \right )}{192}+\frac {41 \cos \left (4 b x +4 a \right )}{192}-\frac {\cos \left (6 b x +6 a \right )}{64}-\frac {\cos \left (8 b x +8 a \right )}{1536}+\frac {131}{512}\right ) \left (\sec ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\csc ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16 b}\) | \(144\) |
norman | \(\frac {-\frac {1}{64 b}+\frac {3 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}+\frac {3 \left (\tan ^{14}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {\tan ^{16}\left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}-\frac {93 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {93 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {591 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{32 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}+\frac {6 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {6 \ln \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\) | \(165\) |
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Time = 0.33 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.45 \[ \int \cos ^4(a+b x) \cot ^5(a+b x) \, dx=\frac {8 \, \cos \left (b x + a\right )^{8} + 32 \, \cos \left (b x + a\right )^{6} - 115 \, \cos \left (b x + a\right )^{4} + 38 \, \cos \left (b x + a\right )^{2} + 192 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right ) + 29}{32 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1664 vs. \(2 (58) = 116\).
Time = 7.68 (sec) , antiderivative size = 1664, normalized size of antiderivative = 24.12 \[ \int \cos ^4(a+b x) \cot ^5(a+b x) \, dx=\text {Too large to display} \]
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Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int \cos ^4(a+b x) \cot ^5(a+b x) \, dx=\frac {\sin \left (b x + a\right )^{4} - 8 \, \sin \left (b x + a\right )^{2} + \frac {8 \, \sin \left (b x + a\right )^{2} - 1}{\sin \left (b x + a\right )^{4}} + 12 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{4 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (65) = 130\).
Time = 0.32 (sec) , antiderivative size = 277, normalized size of antiderivative = 4.01 \[ \int \cos ^4(a+b x) \cot ^5(a+b x) \, dx=-\frac {\frac {{\left (\frac {28 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {288 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} + \frac {28 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {32 \, {\left (\frac {84 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {126 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {84 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {25 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - 25\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{4}} - 192 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 384 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{64 \, b} \]
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Time = 1.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.33 \[ \int \cos ^4(a+b x) \cot ^5(a+b x) \, dx=\frac {6\,\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b}-\frac {3\,\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )}{b}+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^6+\frac {9\,{\mathrm {tan}\left (a+b\,x\right )}^4}{2}+{\mathrm {tan}\left (a+b\,x\right )}^2-\frac {1}{4}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^8+2\,{\mathrm {tan}\left (a+b\,x\right )}^6+{\mathrm {tan}\left (a+b\,x\right )}^4\right )} \]
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