Optimal. Leaf size=12 \[ \frac{\tan ^2(x)}{2}+\log (\cos (x)) \]
[Out]
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Rubi [A] time = 0.0105905, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{\tan ^2(x)}{2}+\log (\cos (x)) \]
Antiderivative was successfully verified.
[In] Int[Tan[x]^3,x]
[Out]
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Rubi in Sympy [A] time = 0.493162, size = 10, normalized size = 0.83 \[ \log{\left (\cos{\left (x \right )} \right )} + \frac{\tan ^{2}{\left (x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(tan(x)**3,x)
[Out]
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Mathematica [A] time = 0.00503877, size = 12, normalized size = 1. \[ \frac{\sec ^2(x)}{2}+\log (\cos (x)) \]
Antiderivative was successfully verified.
[In] Integrate[Tan[x]^3,x]
[Out]
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Maple [A] time = 0., size = 17, normalized size = 1.4 \[{\frac{ \left ( \tan \left ( x \right ) \right ) ^{2}}{2}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(tan(x)^3,x)
[Out]
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Maxima [A] time = 1.35687, size = 27, normalized size = 2.25 \[ -\frac{1}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )}} + \frac{1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238815, size = 24, normalized size = 2. \[ \frac{1}{2} \, \tan \left (x\right )^{2} + \frac{1}{2} \, \log \left (\frac{1}{\tan \left (x\right )^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.087913, size = 12, normalized size = 1. \[ \log{\left (\cos{\left (x \right )} \right )} + \frac{1}{2 \cos ^{2}{\left (x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.201146, size = 22, normalized size = 1.83 \[ \frac{1}{2} \, \tan \left (x\right )^{2} - \frac{1}{2} \,{\rm ln}\left (\tan \left (x\right )^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)^3,x, algorithm="giac")
[Out]