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

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

[Out]

-ln(arccot(x))

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Rubi [A]  time = 0.02, antiderivative size = 5, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 5, normalized size = 1.00 \[ -\log \left (\cot ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^2)*ArcCot[x]),x]

[Out]

-Log[ArcCot[x]]

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fricas [A]  time = 0.46, size = 5, normalized size = 1.00 \[ -\log \left (\operatorname {arccot}\relax (x)\right ) \]

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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giac [A]  time = 0.12, size = 8, normalized size = 1.60 \[ -\log \left ({\left | \arctan \left (\frac {1}{x}\right ) \right |}\right ) \]

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)))

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maple [A]  time = 0.04, size = 6, normalized size = 1.20 \[ -\ln \left (\mathrm {arccot}\relax (x )\right ) \]

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 = 0.32, size = 5, normalized size = 1.00 \[ -\log \left (\operatorname {arccot}\relax (x)\right ) \]

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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mupad [B]  time = 0.59, size = 5, normalized size = 1.00 \[ -\ln \left (\mathrm {acot}\relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(acot(x)*(x^2 + 1)),x)

[Out]

-log(acot(x))

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sympy [A]  time = 0.24, size = 5, normalized size = 1.00 \[ - \log {\left (\operatorname {acot}{\relax (x )} \right )} \]

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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