3.51 \(\int \frac{1}{(1+x^2) \cot ^{-1}(x)} \, dx\)

Optimal. Leaf size=5 \[ -\log \left (\cot ^{-1}(x)\right ) \]

[Out]

-Log[ArcCot[x]]

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Rubi [A]  time = 0.0199193, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4883} \[ -\log \left (\cot ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^2)*ArcCot[x]),x]

[Out]

-Log[ArcCot[x]]

Rule 4883

Int[1/(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[Log[RemoveContent[a + b*A
rcCot[c*x], x]]/(b*c*d), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+x^2\right ) \cot ^{-1}(x)} \, dx &=-\log \left (\cot ^{-1}(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0226783, size = 5, normalized size = 1. \[ -\log \left (\cot ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^2)*ArcCot[x]),x]

[Out]

-Log[ArcCot[x]]

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Maple [A]  time = 0.019, size = 6, normalized size = 1.2 \begin{align*} -\ln \left ({\rm arccot} \left (x\right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)/arccot(x),x)

[Out]

-ln(arccot(x))

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Maxima [A]  time = 0.946703, size = 7, normalized size = 1.4 \begin{align*} -\log \left (\operatorname{arccot}\left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/arccot(x),x, algorithm="maxima")

[Out]

-log(arccot(x))

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Fricas [A]  time = 1.76909, size = 23, normalized size = 4.6 \begin{align*} -\log \left (\operatorname{arccot}\left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/arccot(x),x, algorithm="fricas")

[Out]

-log(arccot(x))

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Sympy [A]  time = 0.344778, size = 5, normalized size = 1. \begin{align*} - \log{\left (\operatorname{acot}{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+1)/acot(x),x)

[Out]

-log(acot(x))

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Giac [A]  time = 1.09545, size = 11, normalized size = 2.2 \begin{align*} -\log \left ({\left | \arctan \left (\frac{1}{x}\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/arccot(x),x, algorithm="giac")

[Out]

-log(abs(arctan(1/x)))