Integrand size = 15, antiderivative size = 57 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\log (\tan (a+b x))}{b}+\frac {3 \tan ^2(a+b x)}{2 b}+\frac {3 \tan ^4(a+b x)}{4 b}+\frac {\tan ^6(a+b x)}{6 b} \]
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Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2700, 272, 45} \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\tan ^6(a+b x)}{6 b}+\frac {3 \tan ^4(a+b x)}{4 b}+\frac {3 \tan ^2(a+b x)}{2 b}+\frac {\log (\tan (a+b x))}{b} \]
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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 )^3}{x} \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {(1+x)^3}{x} \, dx,x,\tan ^2(a+b x)\right )}{2 b} \\ & = \frac {\text {Subst}\left (\int \left (3+\frac {1}{x}+3 x+x^2\right ) \, dx,x,\tan ^2(a+b x)\right )}{2 b} \\ & = \frac {\log (\tan (a+b x))}{b}+\frac {3 \tan ^2(a+b x)}{2 b}+\frac {3 \tan ^4(a+b x)}{4 b}+\frac {\tan ^6(a+b x)}{6 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {\log (\cos (a+b x))}{b}+\frac {\log (\sin (a+b x))}{b}+\frac {\sec ^2(a+b x)}{2 b}+\frac {\sec ^4(a+b x)}{4 b}+\frac {\sec ^6(a+b x)}{6 b} \]
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Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {1}{6 \cos \left (b x +a \right )^{6}}+\frac {1}{4 \cos \left (b x +a \right )^{4}}+\frac {1}{2 \cos \left (b x +a \right )^{2}}+\ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(43\) |
default | \(\frac {\frac {1}{6 \cos \left (b x +a \right )^{6}}+\frac {1}{4 \cos \left (b x +a \right )^{4}}+\frac {1}{2 \cos \left (b x +a \right )^{2}}+\ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(43\) |
risch | \(\frac {2 \,{\mathrm e}^{10 i \left (b x +a \right )}+12 \,{\mathrm e}^{8 i \left (b x +a \right )}+\frac {92 \,{\mathrm e}^{6 i \left (b x +a \right )}}{3}+12 \,{\mathrm e}^{4 i \left (b x +a \right )}+2 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(109\) |
norman | \(\frac {\frac {6 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {6 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {12 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {12 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {68 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{6}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}\) | \(147\) |
parallelrisch | \(\frac {\left (-180 \cos \left (2 b x +2 a \right )-72 \cos \left (4 b x +4 a \right )-12 \cos \left (6 b x +6 a \right )-120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-180 \cos \left (2 b x +2 a \right )-72 \cos \left (4 b x +4 a \right )-12 \cos \left (6 b x +6 a \right )-120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+\left (180 \cos \left (2 b x +2 a \right )+72 \cos \left (4 b x +4 a \right )+12 \cos \left (6 b x +6 a \right )+120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-21 \cos \left (2 b x +2 a \right )-42 \cos \left (4 b x +4 a \right )-11 \cos \left (6 b x +6 a \right )+74}{12 b \left (\cos \left (6 b x +6 a \right )+6 \cos \left (4 b x +4 a \right )+15 \cos \left (2 b x +2 a \right )+10\right )}\) | \(218\) |
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Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.35 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {6 \, \cos \left (b x + a\right )^{6} \log \left (\cos \left (b x + a\right )^{2}\right ) - 6 \, \cos \left (b x + a\right )^{6} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 6 \, \cos \left (b x + a\right )^{4} - 3 \, \cos \left (b x + a\right )^{2} - 2}{12 \, b \cos \left (b x + a\right )^{6}} \]
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\[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\int \frac {\sec ^{7}{\left (a + b x \right )}}{\sin {\left (a + b x \right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {\frac {6 \, \sin \left (b x + a\right )^{4} - 15 \, \sin \left (b x + a\right )^{2} + 11}{\sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4} + 3 \, \sin \left (b x + a\right )^{2} - 1} + 6 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{12 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (51) = 102\).
Time = 0.34 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.75 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\frac {\frac {522 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {1485 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {1580 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {1485 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {522 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {147 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + 147}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{6}} + 30 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 60 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{60 \, b} \]
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Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{2}-\ln \left (\cos \left (a+b\,x\right )\right )+\frac {\frac {{\cos \left (a+b\,x\right )}^4}{2}+\frac {{\cos \left (a+b\,x\right )}^2}{4}+\frac {1}{6}}{{\cos \left (a+b\,x\right )}^6}}{b} \]
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