Integrand size = 17, antiderivative size = 49 \[ \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\frac {3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 \csc (a+b x)}{2 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2701, 294, 327, 213} \[ \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\frac {3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 \csc (a+b x)}{2 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{2 b} \]
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Rule 213
Rule 294
Rule 327
Rule 2701
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{b} \\ & = \frac {\csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {3 \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b} \\ & = -\frac {3 \csc (a+b x)}{2 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b} \\ & = \frac {3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 \csc (a+b x)}{2 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{2 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.55 \[ \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {\csc (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},\sin ^2(a+b x)\right )}{b} \]
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Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )}-\frac {3}{2 \sin \left (b x +a \right )}+\frac {3 \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2}}{b}\) | \(50\) |
default | \(\frac {\frac {1}{2 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )}-\frac {3}{2 \sin \left (b x +a \right )}+\frac {3 \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2}}{b}\) | \(50\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{5 i \left (b x +a \right )}+2 \,{\mathrm e}^{3 i \left (b x +a \right )}+3 \,{\mathrm e}^{i \left (b x +a \right )}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{2 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{2 b}\) | \(104\) |
parallelrisch | \(\frac {\left (-3 \cos \left (2 b x +2 a \right )-3\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (3 \cos \left (2 b x +2 a \right )+3\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+\left (-6 \cos \left (b x +a \right )+6\right ) \cot \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \sec \left (\frac {b x}{2}+\frac {a}{2}\right ) \csc \left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b \left (1+\cos \left (2 b x +2 a \right )\right )}\) | \(112\) |
norman | \(\frac {-\frac {1}{2 b}+\frac {3 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b}+\frac {3 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b}-\frac {\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}-\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{2 b}+\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{2 b}\) | \(117\) |
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Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.73 \[ \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\frac {3 \, \cos \left (b x + a\right )^{2} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, \cos \left (b x + a\right )^{2} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 2}{4 \, b \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )} \]
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\[ \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\int \frac {\sec ^{3}{\left (a + b x \right )}}{\sin ^{2}{\left (a + b x \right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.24 \[ \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} - 2\right )}}{\sin \left (b x + a\right )^{3} - \sin \left (b x + a\right )} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\sin \left (b x + a\right ) - 1\right )}{4 \, b} \]
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Time = 0.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.29 \[ \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} - 2\right )}}{\sin \left (b x + a\right )^{3} - \sin \left (b x + a\right )} - 3 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 3 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{4 \, b} \]
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Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98 \[ \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\frac {3\,\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{2\,b}+\frac {\frac {3\,{\sin \left (a+b\,x\right )}^2}{2}-1}{b\,\left (\sin \left (a+b\,x\right )-{\sin \left (a+b\,x\right )}^3\right )} \]
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