Optimal. Leaf size=10 \[ x \sec (x)-\tanh ^{-1}(\sin (x)) \]
[Out]
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Rubi [A] time = 0.0168106, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ x \sec (x)-\tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
[In] Int[x*Sec[x]*Tan[x],x]
[Out]
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Rubi in Sympy [A] time = 1.37318, size = 8, normalized size = 0.8 \[ \frac{x}{\cos{\left (x \right )}} - \operatorname{atanh}{\left (\sin{\left (x \right )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*sec(x)*tan(x),x)
[Out]
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Mathematica [B] time = 0.0086549, size = 37, normalized size = 3.7 \[ x \sec (x)+\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*Sec[x]*Tan[x],x]
[Out]
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Maple [A] time = 0.009, size = 16, normalized size = 1.6 \[{\frac{x}{\cos \left ( x \right ) }}-\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*sec(x)*tan(x),x)
[Out]
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Maxima [A] time = 1.511, size = 163, normalized size = 16.3 \[ \frac{4 \, x \cos \left (2 \, x\right ) \cos \left (x\right ) + 4 \, x \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, x \cos \left (x\right ) -{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) +{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right )}{2 \,{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sec(x)*tan(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.244844, size = 39, normalized size = 3.9 \[ -\frac{\cos \left (x\right ) \log \left (\sin \left (x\right ) + 1\right ) - \cos \left (x\right ) \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, x}{2 \, \cos \left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sec(x)*tan(x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \tan{\left (x \right )} \sec{\left (x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sec(x)*tan(x),x)
[Out]
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GIAC/XCAS [A] time = 0.247866, size = 203, normalized size = 20.3 \[ -\frac{2 \, x \tan \left (\frac{1}{2} \, x\right )^{2} +{\rm ln}\left (\frac{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac{1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac{1}{2} \, x\right )^{2} -{\rm ln}\left (\frac{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac{1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac{1}{2} \, x\right )^{2} + 2 \, x -{\rm ln}\left (\frac{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac{1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) +{\rm ln}\left (\frac{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac{1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )}{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sec(x)*tan(x),x, algorithm="giac")
[Out]