3.676 \(\int \frac{\left (1+x^2\right ) \tan ^{-1}(x)}{x^2} \, dx\)

Optimal. Leaf size=22 \[ -\log \left (x^2+1\right )+\log (x)+x \tan ^{-1}(x)-\frac{\tan ^{-1}(x)}{x} \]

[Out]

-(ArcTan[x]/x) + x*ArcTan[x] + Log[x] - Log[1 + x^2]

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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]

-(ArcTan[x]/x) + x*ArcTan[x] + Log[x] - Log[1 + x^2]

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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]

x*atan(x) + log(x**2)/2 - log(x**2 + 1) - atan(x)/x

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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]

-(ArcTan[x]/x) + x*ArcTan[x] + Log[x] - Log[1 + x^2]

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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]

-arctan(x)/x+x*arctan(x)+ln(x)-ln(x^2+1)

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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]

(x - 1/x)*arctan(x) - log(x^2 + 1) + log(x)

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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]

((x^2 - 1)*arctan(x) - x*log(x^2 + 1) + x*log(x))/x

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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]

x*atan(x) + log(x) - log(x**2 + 1) - atan(x)/x

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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]

(x - 1/x)*arctan(x) - ln(x^2 + 1) + 1/2*ln(x^2)