Integrand size = 15, antiderivative size = 38 \[ \int \csc ^4(a+b x) \sec (a+b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b} \]
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Time = 0.02 (sec) , antiderivative size = 38, 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 = {2701, 308, 213} \[ \int \csc ^4(a+b x) \sec (a+b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\csc ^3(a+b x)}{3 b}-\frac {\csc (a+b x)}{b} \]
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Rule 213
Rule 308
Rule 2701
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{b} \\ & = -\frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}-\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{b} \\ & = \frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \csc ^4(a+b x) \sec (a+b x) \, dx=-\frac {\csc ^3(a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\sin ^2(a+b x)\right )}{3 b} \]
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Time = 0.13 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \sin \left (b x +a \right )^{3}}-\frac {1}{\sin \left (b x +a \right )}+\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b}\) | \(40\) |
default | \(\frac {-\frac {1}{3 \sin \left (b x +a \right )^{3}}-\frac {1}{\sin \left (b x +a \right )}+\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b}\) | \(40\) |
parallelrisch | \(\frac {-\left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\cot ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-15 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )-15 \cot \left (\frac {b x}{2}+\frac {a}{2}\right )+24 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )-24 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{24 b}\) | \(83\) |
risch | \(-\frac {2 i \left (3 \,{\mathrm e}^{5 i \left (b x +a \right )}-10 \,{\mathrm e}^{3 i \left (b x +a \right )}+3 \,{\mathrm e}^{i \left (b x +a \right )}\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{b}\) | \(90\) |
norman | \(\frac {-\frac {1}{24 b}-\frac {5 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {5 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}\) | \(101\) |
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (36) = 72\).
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.47 \[ \int \csc ^4(a+b x) \sec (a+b x) \, dx=\frac {3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 8}{6 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
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\[ \int \csc ^4(a+b x) \sec (a+b x) \, dx=\int \frac {\sec {\left (a + b x \right )}}{\sin ^{4}{\left (a + b x \right )}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \csc ^4(a+b x) \sec (a+b x) \, dx=-\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\sin \left (b x + a\right ) - 1\right )}{6 \, b} \]
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Time = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \csc ^4(a+b x) \sec (a+b x) \, dx=-\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 3 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 3 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{6 \, b} \]
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Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \csc ^4(a+b x) \sec (a+b x) \, dx=\frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )-\frac {{\sin \left (a+b\,x\right )}^2+\frac {1}{3}}{{\sin \left (a+b\,x\right )}^3}}{b} \]
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