Optimal. Leaf size=22 \[ -\log \left (x^2+1\right )+\log (x)+x \tan ^{-1}(x)-\frac{\tan ^{-1}(x)}{x} \]
[Out]
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Rubi [A] time = 0.0600282, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ -\log \left (x^2+1\right )+\log (x)+x \tan ^{-1}(x)-\frac{\tan ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
[In] Int[((1 + x^2)*ArcTan[x])/x^2,x]
[Out]
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Rubi in Sympy [A] time = 4.00374, size = 22, normalized size = 1. \[ x \operatorname{atan}{\left (x \right )} + \frac{\log{\left (x^{2} \right )}}{2} - \log{\left (x^{2} + 1 \right )} - \frac{\operatorname{atan}{\left (x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+1)*atan(x)/x**2,x)
[Out]
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Mathematica [A] time = 0.00665661, size = 22, normalized size = 1. \[ -\log \left (x^2+1\right )+\log (x)+x \tan ^{-1}(x)-\frac{\tan ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
[In] Integrate[((1 + x^2)*ArcTan[x])/x^2,x]
[Out]
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Maple [A] time = 0.007, size = 23, normalized size = 1.1 \[ -{\frac{\arctan \left ( x \right ) }{x}}+x\arctan \left ( x \right ) +\ln \left ( x \right ) -\ln \left ({x}^{2}+1 \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+1)*arctan(x)/x^2,x)
[Out]
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Maxima [A] time = 1.53725, size = 28, normalized size = 1.27 \[{\left (x - \frac{1}{x}\right )} \arctan \left (x\right ) - \log \left (x^{2} + 1\right ) + \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)*arctan(x)/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231375, size = 35, normalized size = 1.59 \[ \frac{{\left (x^{2} - 1\right )} \arctan \left (x\right ) - x \log \left (x^{2} + 1\right ) + x \log \left (x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)*arctan(x)/x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.629659, size = 19, normalized size = 0.86 \[ x \operatorname{atan}{\left (x \right )} + \log{\left (x \right )} - \log{\left (x^{2} + 1 \right )} - \frac{\operatorname{atan}{\left (x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+1)*atan(x)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.206921, size = 34, normalized size = 1.55 \[{\left (x - \frac{1}{x}\right )} \arctan \left (x\right ) -{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)*arctan(x)/x^2,x, algorithm="giac")
[Out]