3.19 \(\int \tan ^{-1}(x) \, dx\)

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

[Out]

x*arctan(x)-1/2*ln(x^2+1)

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4846, 260} \[ x \tan ^{-1}(x)-\frac {1}{2} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[x],x]

[Out]

x*ArcTan[x] - Log[1 + x^2]/2

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ x \tan ^{-1}(x)-\frac {1}{2} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[x],x]

[Out]

x*ArcTan[x] - Log[1 + x^2]/2

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fricas [A]  time = 0.42, size = 13, normalized size = 0.87 \[ x \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x),x, algorithm="fricas")

[Out]

x*arctan(x) - 1/2*log(x^2 + 1)

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giac [A]  time = 0.92, size = 13, normalized size = 0.87 \[ x \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x),x, algorithm="giac")

[Out]

x*arctan(x) - 1/2*log(x^2 + 1)

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ x \arctan \relax (x )-\frac {\ln \left (x^{2}+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x),x)

[Out]

x*arctan(x)-1/2*ln(x^2+1)

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maxima [A]  time = 0.43, size = 13, normalized size = 0.87 \[ x \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x),x, algorithm="maxima")

[Out]

x*arctan(x) - 1/2*log(x^2 + 1)

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mupad [B]  time = 0.16, size = 13, normalized size = 0.87 \[ x\,\mathrm {atan}\relax (x)-\frac {\ln \left (x^2+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(x),x)

[Out]

x*atan(x) - log(x^2 + 1)/2

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sympy [A]  time = 0.20, size = 12, normalized size = 0.80 \[ x \operatorname {atan}{\relax (x )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(x),x)

[Out]

x*atan(x) - log(x**2 + 1)/2

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