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]

Log[2 + ArcTan[x]]

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Rubi [A]  time = 0.0245949, antiderivative size = 5, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0293372, size = 5, normalized size = 1. \[ \log \left (\tan ^{-1}(x)+2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[2 + ArcTan[x]]

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Maple [A]  time = 0.036, size = 6, normalized size = 1.2 \begin{align*} \ln \left ( 2+\arctan \left ( x \right ) \right ) \end{align*}

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 = 0.953525, size = 7, normalized size = 1.4 \begin{align*} \log \left (\arctan \left (x\right ) + 2\right ) \end{align*}

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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Fricas [A]  time = 1.9855, size = 27, normalized size = 5.4 \begin{align*} \log \left (\arctan \left (x\right ) + 2\right ) \end{align*}

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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Sympy [A]  time = 0.349717, size = 5, normalized size = 1. \begin{align*} \log{\left (\operatorname{atan}{\left (x \right )} + 2 \right )} \end{align*}

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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Giac [A]  time = 1.10889, size = 7, normalized size = 1.4 \begin{align*} \log \left (\arctan \left (x\right ) + 2\right ) \end{align*}

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)