Optimal. Leaf size=30 \[ \frac {1}{3} x \sec ^3(x)-x \sec (x)+\frac {5}{6} \tanh ^{-1}(\sin (x))-\frac {1}{6} \tan (x) \sec (x) \]
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Rubi [A] time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2606, 4417, 3770, 3768} \[ \frac {1}{3} x \sec ^3(x)-x \sec (x)+\frac {5}{6} \tanh ^{-1}(\sin (x))-\frac {1}{6} \tan (x) \sec (x) \]
Antiderivative was successfully verified.
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Rule 2606
Rule 3768
Rule 3770
Rule 4417
Rubi steps
\begin {align*} \int x \sec (x) \tan ^3(x) \, dx &=-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\\ &=\tanh ^{-1}(\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} \tanh ^{-1}(\sin (x))-x \sec (x)+\frac {1}{3} x \sec ^3(x)-\frac {1}{6} \sec (x) \tan (x)\\ \end {align*}
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Mathematica [B] time = 0.13, size = 104, normalized size = 3.47 \[ -\frac {1}{24} \sec ^3(x) \left (4 x+2 \sin (2 x)+12 x \cos (2 x)+5 \cos (3 x) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+15 \cos (x) \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )-5 \cos (3 x) \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sec (x) \tan ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.20, size = 47, normalized size = 1.57 \[ \frac {5 \, \cos \relax (x)^{3} \log \left (\sin \relax (x) + 1\right ) - 5 \, \cos \relax (x)^{3} \log \left (-\sin \relax (x) + 1\right ) - 12 \, x \cos \relax (x)^{2} - 2 \, \cos \relax (x) \sin \relax (x) + 4 \, x}{12 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.22, size = 341, normalized size = 11.37 \[ \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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 76, normalized size = 2.53
method | result | size |
norman | \(\frac {\frac {2 x}{3}-\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3}-2 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\frac {2 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3}+\frac {\tan \left (\frac {x}{2}\right )}{3}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}-\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{6}+\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{6}\) | \(76\) |
risch | \(-\frac {6 x \,{\mathrm e}^{5 i x}+4 x \,{\mathrm e}^{3 i x}-i {\mathrm e}^{5 i x}+6 x \,{\mathrm e}^{i x}+i {\mathrm e}^{i x}}{3 \left (1+{\mathrm e}^{2 i x}\right )^{3}}-\frac {5 \ln \left ({\mathrm e}^{i x}-i\right )}{6}+\frac {5 \ln \left ({\mathrm e}^{i x}+i\right )}{6}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.47, size = 619, normalized size = 20.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 35, normalized size = 1.17 \[ -\frac {x\,{\cos \relax (x)}^2-\frac {x}{3}+\frac {\sin \left (2\,x\right )}{12}}{{\cos \relax (x)}^3}-\frac {\mathrm {atan}\left (\cos \relax (x)+\sin \relax (x)\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.69, size = 551, normalized size = 18.37 \[ \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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