Integrand size = 17, antiderivative size = 70 \[ \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx=-\frac {15 \text {arctanh}(\cos (a+b x))}{8 b}+\frac {15 \sec (a+b x)}{8 b}-\frac {5 \csc ^2(a+b x) \sec (a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec (a+b x)}{4 b} \]
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Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2702, 294, 327, 213} \[ \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx=-\frac {15 \text {arctanh}(\cos (a+b x))}{8 b}+\frac {15 \sec (a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec (a+b x)}{4 b}-\frac {5 \csc ^2(a+b x) \sec (a+b x)}{8 b} \]
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Rule 213
Rule 294
Rule 327
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^3} \, dx,x,\sec (a+b x)\right )}{b} \\ & = -\frac {\csc ^4(a+b x) \sec (a+b x)}{4 b}+\frac {5 \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{4 b} \\ & = -\frac {5 \csc ^2(a+b x) \sec (a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec (a+b x)}{4 b}+\frac {15 \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{8 b} \\ & = \frac {15 \sec (a+b x)}{8 b}-\frac {5 \csc ^2(a+b x) \sec (a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec (a+b x)}{4 b}+\frac {15 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{8 b} \\ & = -\frac {15 \text {arctanh}(\cos (a+b x))}{8 b}+\frac {15 \sec (a+b x)}{8 b}-\frac {5 \csc ^2(a+b x) \sec (a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec (a+b x)}{4 b} \\ \end{align*}
Time = 2.88 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.84 \[ \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx=-\frac {14 \csc ^2\left (\frac {1}{2} (a+b x)\right )+\csc ^4\left (\frac {1}{2} (a+b x)\right )+\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right ) \left (78+\cos (a+b x) \left (-8 \left (8+15 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-15 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )+\sec ^4\left (\frac {1}{2} (a+b x)\right )\right )-14 \tan ^2\left (\frac {1}{2} (a+b x)\right )\right )}{-1+\tan ^2\left (\frac {1}{2} (a+b x)\right )}}{64 b} \]
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Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {1}{4 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{4}}-\frac {5}{8 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2}}+\frac {15}{8 \cos \left (b x +a \right )}+\frac {15 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{b}\) | \(70\) |
default | \(\frac {-\frac {1}{4 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{4}}-\frac {5}{8 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2}}+\frac {15}{8 \cos \left (b x +a \right )}+\frac {15 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{b}\) | \(70\) |
parallelrisch | \(\frac {\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )+\cot ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )+15 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+120 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+15 \left (\cot ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-120 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-160}{64 b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}\) | \(106\) |
norman | \(\frac {\frac {1}{64 b}+\frac {15 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64 b}+\frac {15 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64 b}+\frac {\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}-\frac {5 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}+\frac {15 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}\) | \(114\) |
risch | \(\frac {15 \,{\mathrm e}^{9 i \left (b x +a \right )}-40 \,{\mathrm e}^{7 i \left (b x +a \right )}+18 \,{\mathrm e}^{5 i \left (b x +a \right )}-40 \,{\mathrm e}^{3 i \left (b x +a \right )}+15 \,{\mathrm e}^{i \left (b x +a \right )}}{4 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}-\frac {15 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{8 b}+\frac {15 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{8 b}\) | \(123\) |
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (62) = 124\).
Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.89 \[ \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx=\frac {30 \, \cos \left (b x + a\right )^{4} - 50 \, \cos \left (b x + a\right )^{2} - 15 \, {\left (\cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} + \cos \left (b x + a\right )\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} + \cos \left (b x + a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 16}{16 \, {\left (b \cos \left (b x + a\right )^{5} - 2 \, b \cos \left (b x + a\right )^{3} + b \cos \left (b x + a\right )\right )}} \]
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\[ \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx=\int \frac {\sec ^{2}{\left (a + b x \right )}}{\sin ^{5}{\left (a + b x \right )}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.13 \[ \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx=\frac {\frac {2 \, {\left (15 \, \cos \left (b x + a\right )^{4} - 25 \, \cos \left (b x + a\right )^{2} + 8\right )}}{\cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} + \cos \left (b x + a\right )} - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (62) = 124\).
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.33 \[ \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx=\frac {\frac {{\left (\frac {16 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {90 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} - \frac {16 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {128}{\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1} + 60 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx=\frac {\frac {15\,{\cos \left (a+b\,x\right )}^4}{8}-\frac {25\,{\cos \left (a+b\,x\right )}^2}{8}+1}{b\,\left ({\cos \left (a+b\,x\right )}^5-2\,{\cos \left (a+b\,x\right )}^3+\cos \left (a+b\,x\right )\right )}-\frac {15\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{8\,b} \]
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