3.19 \(\int \tan ^{-1}(x) \, dx\)

Optimal. Leaf size=15 \[ x \tan ^{-1}(x)-\frac{1}{2} \log \left (x^2+1\right ) \]

[Out]

x*ArcTan[x] - Log[1 + x^2]/2

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Rubi [A]  time = 0.0067478, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1. \[ x \tan ^{-1}(x)-\frac{1}{2} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[x],x]

[Out]

x*ArcTan[x] - Log[1 + x^2]/2

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Rubi in Sympy [A]  time = 0.908175, size = 12, normalized size = 0.8 \[ x \operatorname{atan}{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(atan(x),x)

[Out]

x*atan(x) - log(x**2 + 1)/2

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Mathematica [A]  time = 0.00229204, size = 15, normalized size = 1. \[ x \tan ^{-1}(x)-\frac{1}{2} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcTan[x],x]

[Out]

x*ArcTan[x] - Log[1 + x^2]/2

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Maple [A]  time = 0.003, size = 14, normalized size = 0.9 \[ x\arctan \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x),x)

[Out]

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

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Maxima [A]  time = 1.35691, size = 18, normalized size = 1.2 \[ x \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x),x, algorithm="maxima")

[Out]

x*arctan(x) - 1/2*log(x^2 + 1)

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Fricas [A]  time = 0.232247, size = 18, normalized size = 1.2 \[ x \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x),x, algorithm="fricas")

[Out]

x*arctan(x) - 1/2*log(x^2 + 1)

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Sympy [A]  time = 0.235287, size = 12, normalized size = 0.8 \[ x \operatorname{atan}{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(atan(x),x)

[Out]

x*atan(x) - log(x**2 + 1)/2

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GIAC/XCAS [A]  time = 0.199097, size = 18, normalized size = 1.2 \[ x \arctan \left (x\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x),x, algorithm="giac")

[Out]

x*arctan(x) - 1/2*ln(x^2 + 1)