Integrand size = 19, antiderivative size = 46 \[ \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^3(x)} \, dx=\frac {2 x}{\sqrt {3}}+\frac {2 \arctan \left (\frac {1-2 \cos ^2(x)}{2+\sqrt {3}-2 \cos (x) \sin (x)}\right )}{\sqrt {3}}+\log (1+\tan (x)) \]
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Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4427, 1877, 31, 632, 210} \[ \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^3(x)} \, dx=\frac {2 \arctan \left (\frac {1-2 \cos ^2(x)}{-2 \sin (x) \cos (x)+\sqrt {3}+2}\right )}{\sqrt {3}}+\frac {2 x}{\sqrt {3}}+\log (\tan (x)+1) \]
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Rule 31
Rule 210
Rule 632
Rule 1877
Rule 4427
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {2+x^2}{1+x^3} \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan (x)\right )+\text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tan (x)\right ) \\ & = \log (1+\tan (x))-2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tan (x)\right ) \\ & = \frac {2 x}{\sqrt {3}}+\frac {2 \arctan \left (\frac {1-2 \cos ^2(x)}{2+\sqrt {3}-2 \cos (x) \sin (x)}\right )}{\sqrt {3}}+\log (1+\tan (x)) \\ \end{align*}
Time = 6.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^3(x)} \, dx=-\frac {2 \arctan \left (\frac {1-2 \tan (x)}{\sqrt {3}}\right )}{\sqrt {3}}-\log (\cos (x))+\log (\cos (x)+\sin (x)) \]
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Time = 1.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.52
method | result | size |
derivativedivides | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )-1\right ) \sqrt {3}}{3}\right )}{3}+\ln \left (1+\tan \left (x \right )\right )\) | \(24\) |
default | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )-1\right ) \sqrt {3}}{3}\right )}{3}+\ln \left (1+\tan \left (x \right )\right )\) | \(24\) |
risch | \(\ln \left (i+{\mathrm e}^{2 i x}\right )-\ln \left ({\mathrm e}^{2 i x}+1\right )+\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-2 i-i \sqrt {3}\right )}{3}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-2 i+i \sqrt {3}\right )}{3}\) | \(63\) |
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Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^3(x)} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {3}}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) - \frac {1}{2} \, \log \left (\cos \left (x\right )^{2}\right ) + \frac {1}{2} \, \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]
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Time = 2.50 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^3(x)} \, dx=\frac {2 \sqrt {3} \left (\operatorname {atan}{\left (\frac {2 \sqrt {3} \left (\tan {\left (x \right )} - \frac {1}{2}\right )}{3} \right )} + \pi \left \lfloor {\frac {x - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{3} + \log {\left (\tan {\left (x \right )} + 1 \right )} \]
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Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.50 \[ \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^3(x)} \, dx=\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right ) - 1\right )}\right ) + \log \left (\tan \left (x\right ) + 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.52 \[ \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^3(x)} \, dx=\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right ) - 1\right )}\right ) + \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) \]
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Time = 26.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.65 \[ \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^3(x)} \, dx=\ln \left (\mathrm {tan}\left (x\right )+1\right )-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}-\sqrt {3}\,\mathrm {tan}\left (x\right )}{\mathrm {tan}\left (x\right )+1}\right )}{3} \]
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