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

Optimal. Leaf size=5 \[ \log \left (\tan ^{-1}(x)+2\right ) \]

[Out]

ln(2+arctan(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 = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4882} \[ \log \left (\tan ^{-1}(x)+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^2)*(2 + ArcTan[x])),x]

[Out]

Log[2 + ArcTan[x]]

Rule 4882

Int[1/(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[Log[RemoveContent[a + b*Ar
cTan[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 ) \left (2+\tan ^{-1}(x)\right )} \, dx &=\log \left (2+\tan ^{-1}(x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 5, normalized size = 1.00 \[ \log \left (\tan ^{-1}(x)+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^2)*(2 + ArcTan[x])),x]

[Out]

Log[2 + ArcTan[x]]

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fricas [A]  time = 0.50, size = 5, normalized size = 1.00 \[ \log \left (\arctan \relax (x) + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(arctan(x) + 2)

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giac [A]  time = 0.12, size = 5, normalized size = 1.00 \[ \log \left (\arctan \relax (x) + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(arctan(x) + 2)

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maple [A]  time = 0.04, size = 6, normalized size = 1.20 \[ \ln \left (2+\arctan \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(2+arctan(x))

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maxima [A]  time = 0.31, size = 5, normalized size = 1.00 \[ \log \left (\arctan \relax (x) + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(arctan(x) + 2)

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mupad [B]  time = 0.22, size = 5, normalized size = 1.00 \[ \ln \left (\mathrm {atan}\relax (x)+2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^2 + 1)*(atan(x) + 2)),x)

[Out]

log(atan(x) + 2)

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sympy [A]  time = 0.26, size = 5, normalized size = 1.00 \[ \log {\left (\operatorname {atan}{\relax (x )} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(atan(x) + 2)

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