Integrand size = 8, antiderivative size = 30 \[ \int x \sec (x) \tan ^3(x) \, dx=\frac {5}{6} \text {arctanh}(\sin (x))-x \sec (x)+\frac {1}{3} x \sec ^3(x)-\frac {1}{6} \sec (x) \tan (x) \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2686, 4502, 3855, 3853} \[ \int x \sec (x) \tan ^3(x) \, dx=\frac {5}{6} \text {arctanh}(\sin (x))+\frac {1}{3} x \sec ^3(x)-x \sec (x)-\frac {1}{6} \tan (x) \sec (x) \]
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Rule 2686
Rule 3853
Rule 3855
Rule 4502
Rubi steps \begin{align*} \text {integral}& = -x \sec (x)+\frac {1}{3} x \sec ^3(x)-\int \left (-\sec (x)+\frac {\sec ^3(x)}{3}\right ) \, dx \\ & = -x \sec (x)+\frac {1}{3} x \sec ^3(x)-\frac {1}{3} \int \sec ^3(x) \, dx+\int \sec (x) \, dx \\ & = \text {arctanh}(\sin (x))-x \sec (x)+\frac {1}{3} x \sec ^3(x)-\frac {1}{6} \sec (x) \tan (x)-\frac {1}{6} \int \sec (x) \, dx \\ & = \frac {5}{6} \text {arctanh}(\sin (x))-x \sec (x)+\frac {1}{3} x \sec ^3(x)-\frac {1}{6} \sec (x) \tan (x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(30)=60\).
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.47 \[ \int x \sec (x) \tan ^3(x) \, dx=-\frac {1}{24} \sec ^3(x) \left (4 x+12 x \cos (2 x)+5 \cos (3 x) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+15 \cos (x) \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )\right )-5 \cos (3 x) \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+2 \sin (2 x)\right ) \]
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Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43
method | result | size |
parallelrisch | \(\ln \left (\left (-\cot \left (x \right )+1+\csc \left (x \right )\right )^{\frac {5}{6}}\right )+\ln \left (\frac {1}{\left (-\cot \left (x \right )+\csc \left (x \right )-1\right )^{\frac {5}{6}}}\right )+\frac {x \left (\sec ^{3}\left (x \right )\right )}{3}+\frac {\left (-6 x -\tan \left (x \right )\right ) \sec \left (x \right )}{6}\) | \(43\) |
norman | \(\frac {\frac {2 x}{3}-\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3}-2 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\frac {2 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3}+\frac {\tan \left (\frac {x}{2}\right )}{3}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}-\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{6}+\frac {5 \ln \left (1+\tan \left (\frac {x}{2}\right )\right )}{6}\) | \(76\) |
risch | \(-\frac {{\mathrm e}^{i x} \left (6 x \,{\mathrm e}^{4 i x}+4 x \,{\mathrm e}^{2 i x}-i {\mathrm e}^{4 i x}+6 x +i\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3}}+\frac {5 \ln \left (i+{\mathrm e}^{i x}\right )}{6}-\frac {5 \ln \left ({\mathrm e}^{i x}-i\right )}{6}\) | \(78\) |
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Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int x \sec (x) \tan ^3(x) \, dx=\frac {5 \, \cos \left (x\right )^{3} \log \left (\sin \left (x\right ) + 1\right ) - 5 \, \cos \left (x\right )^{3} \log \left (-\sin \left (x\right ) + 1\right ) - 12 \, x \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) \sin \left (x\right ) + 4 \, x}{12 \, \cos \left (x\right )^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (29) = 58\).
Time = 0.64 (sec) , antiderivative size = 551, normalized size of antiderivative = 18.37 \[ \int x \sec (x) \tan ^3(x) \, dx=\frac {4 x \tan ^{6}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} - \frac {12 x \tan ^{4}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} - \frac {12 x \tan ^{2}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} + \frac {4 x}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} - \frac {5 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} \tan ^{6}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} + \frac {15 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} \tan ^{4}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} - \frac {15 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} + \frac {5 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} + \frac {5 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{6}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} - \frac {15 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{4}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} + \frac {15 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} - \frac {5 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} - \frac {2 \tan ^{5}{\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} + \frac {2 \tan {\left (\frac {x}{2} \right )}}{6 \tan ^{6}{\left (\frac {x}{2} \right )} - 18 \tan ^{4}{\left (\frac {x}{2} \right )} + 18 \tan ^{2}{\left (\frac {x}{2} \right )} - 6} \]
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Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 619, normalized size of antiderivative = 20.63 \[ \int x \sec (x) \tan ^3(x) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (24) = 48\).
Time = 0.43 (sec) , antiderivative size = 341, normalized size of antiderivative = 11.37 \[ \int x \sec (x) \tan ^3(x) \, dx=\frac {8 \, x \tan \left (\frac {1}{2} \, x\right )^{6} + 5 \, \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{6} - 5 \, \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{6} - 24 \, x \tan \left (\frac {1}{2} \, x\right )^{4} - 15 \, \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{4} + 15 \, \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{4} - 4 \, \tan \left (\frac {1}{2} \, x\right )^{5} - 24 \, x \tan \left (\frac {1}{2} \, x\right )^{2} + 15 \, \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{2} - 15 \, \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + 8 \, x - 5 \, \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) + 5 \, \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) + 4 \, \tan \left (\frac {1}{2} \, x\right )}{12 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{6} - 3 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1\right )}} \]
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Time = 0.56 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int x \sec (x) \tan ^3(x) \, dx=-\frac {x\,{\cos \left (x\right )}^2-\frac {x}{3}+\frac {\sin \left (2\,x\right )}{12}}{{\cos \left (x\right )}^3}-\frac {\mathrm {atan}\left (\cos \left (x\right )+\sin \left (x\right )\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{3} \]
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