Integrand size = 17, antiderivative size = 69 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=-\frac {2 \cot ^2(a+b x)}{b}-\frac {\cot ^4(a+b x)}{4 b}+\frac {6 \log (\tan (a+b x))}{b}+\frac {2 \tan ^2(a+b x)}{b}+\frac {\tan ^4(a+b x)}{4 b} \]
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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 = {2700, 272, 45} \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\frac {\tan ^4(a+b x)}{4 b}+\frac {2 \tan ^2(a+b x)}{b}-\frac {\cot ^4(a+b x)}{4 b}-\frac {2 \cot ^2(a+b x)}{b}+\frac {6 \log (\tan (a+b x))}{b} \]
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Rule 45
Rule 272
Rule 2700
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^4}{x^5} \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {(1+x)^4}{x^3} \, dx,x,\tan ^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,\tan ^2(a+b x)\right )}{2 b} \\ & = -\frac {2 \cot ^2(a+b x)}{b}-\frac {\cot ^4(a+b x)}{4 b}+\frac {6 \log (\tan (a+b x))}{b}+\frac {2 \tan ^2(a+b x)}{b}+\frac {\tan ^4(a+b x)}{4 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.32 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=32 \left (-\frac {3 \csc ^2(a+b x)}{64 b}-\frac {\csc ^4(a+b x)}{128 b}-\frac {3 \log (\cos (a+b x))}{16 b}+\frac {3 \log (\sin (a+b x))}{16 b}+\frac {3 \sec ^2(a+b x)}{64 b}+\frac {\sec ^4(a+b x)}{128 b}\right ) \]
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Time = 0.47 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {1}{4 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{4}}-\frac {1}{2 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{4}}+\frac {3}{2 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2}}-\frac {3}{\sin \left (b x +a \right )^{2}}+6 \ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(79\) |
default | \(\frac {\frac {1}{4 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{4}}-\frac {1}{2 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{4}}+\frac {3}{2 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2}}-\frac {3}{\sin \left (b x +a \right )^{2}}+6 \ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(79\) |
risch | \(\frac {12 \,{\mathrm e}^{14 i \left (b x +a \right )}-44 \,{\mathrm e}^{10 i \left (b x +a \right )}-44 \,{\mathrm e}^{6 i \left (b x +a \right )}+12 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{4}}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}-\frac {6 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}\) | \(112\) |
parallelrisch | \(\frac {\left (-384 \cos \left (2 b x +2 a \right )-96 \cos \left (4 b x +4 a \right )-288\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-384 \cos \left (2 b x +2 a \right )-96 \cos \left (4 b x +4 a \right )-288\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+\left (384 \cos \left (2 b x +2 a \right )+96 \cos \left (4 b x +4 a \right )+288\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\left (3384 \cos \left (b x +a \right )-1284 \cos \left (2 b x +2 a \right )+328 \cos \left (3 b x +3 a \right )-41 \cos \left (4 b x +4 a \right )-3227\right ) \left (\cot ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+592 \left (\cot ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\csc ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-2 \left (\csc ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\sec ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )+16 \left (\sec ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-140\right )}{16 b \left (\cos \left (4 b x +4 a \right )+4 \cos \left (2 b x +2 a \right )+3\right )}\) | \(258\) |
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Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (65) = 130\).
Time = 0.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.14 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\frac {12 \, \cos \left (b x + a\right )^{6} - 18 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} - 12 \, {\left (\cos \left (b x + a\right )^{8} - 2 \, \cos \left (b x + a\right )^{6} + \cos \left (b x + a\right )^{4}\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 12 \, {\left (\cos \left (b x + a\right )^{8} - 2 \, \cos \left (b x + a\right )^{6} + \cos \left (b x + a\right )^{4}\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) + 1}{4 \, {\left (b \cos \left (b x + a\right )^{8} - 2 \, b \cos \left (b x + a\right )^{6} + b \cos \left (b x + a\right )^{4}\right )}} \]
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\[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\int \frac {\sec ^{5}{\left (a + b x \right )}}{\sin ^{5}{\left (a + b x \right )}}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.33 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=-\frac {\frac {12 \, \sin \left (b x + a\right )^{6} - 18 \, \sin \left (b x + a\right )^{4} + 4 \, \sin \left (b x + a\right )^{2} + 1}{\sin \left (b x + a\right )^{8} - 2 \, \sin \left (b x + a\right )^{6} + \sin \left (b x + a\right )^{4}} + 12 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 12 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{4 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (65) = 130\).
Time = 0.32 (sec) , antiderivative size = 278, normalized size of antiderivative = 4.03 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\frac {\frac {{\left (\frac {28 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {288 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} + \frac {28 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {32 \, {\left (\frac {84 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {126 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {84 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {25 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 25\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{4}} + 192 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 384 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{64 \, b} \]
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Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\frac {2\,{\mathrm {tan}\left (a+b\,x\right )}^2}{b}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^4}{4\,b}+\frac {6\,\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^4\,\left (2\,{\mathrm {tan}\left (a+b\,x\right )}^2+\frac {1}{4}\right )}{b} \]
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