3.82 \(\int \cot ^{-1}(x) \, dx\)

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

[Out]

x*ArcCot[x] + Log[1 + x^2]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.00722682, 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. \[ \frac{1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[ArcCot[x],x]

[Out]

x*ArcCot[x] + Log[1 + x^2]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.842846, size = 12, normalized size = 0.8 \[ x \operatorname{acot}{\left (x \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(acot(x),x)

[Out]

x*acot(x) + log(x**2 + 1)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00247219, size = 15, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcCot[x],x]

[Out]

x*ArcCot[x] + Log[1 + x^2]/2

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 14, normalized size = 0.9 \[ x{\rm arccot} \left (x\right )+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arccot(x),x)

[Out]

x*arccot(x)+1/2*ln(x^2+1)

_______________________________________________________________________________________

Maxima [A]  time = 1.38675, size = 18, normalized size = 1.2 \[ x \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arccot(x) + 1/2*log(x^2 + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.22433, size = 18, normalized size = 1.2 \[ x \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arccot(x) + 1/2*log(x^2 + 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.233644, size = 12, normalized size = 0.8 \[ x \operatorname{acot}{\left (x \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(acot(x),x)

[Out]

x*acot(x) + log(x**2 + 1)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218393, size = 20, normalized size = 1.33 \[ x \arctan \left (\frac{1}{x}\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arctan(1/x) + 1/2*ln(x^2 + 1)