3.98 \(\int \frac{e^{\tan ^{-1}(x)} x}{(1+x^2)^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{(1-x) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]

[Out]

-(E^ArcTan[x]*(1 - x))/(2*Sqrt[1 + x^2])

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Rubi [A]  time = 0.0412641, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {5077} \[ -\frac{(1-x) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTan[x]*x)/(1 + x^2)^(3/2),x]

[Out]

-(E^ArcTan[x]*(1 - x))/(2*Sqrt[1 + x^2])

Rule 5077

Int[(E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[((1 - a*n*x)*E^(n*ArcT
an[a*x]))/(d*(n^2 + 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] &&  !IntegerQ[I*n]

Rubi steps

\begin{align*} \int \frac{e^{\tan ^{-1}(x)} x}{\left (1+x^2\right )^{3/2}} \, dx &=-\frac{e^{\tan ^{-1}(x)} (1-x)}{2 \sqrt{1+x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0099006, size = 37, normalized size = 1.68 \[ \frac{1}{2} (1-i x)^{-\frac{1}{2}+\frac{i}{2}} (1+i x)^{-\frac{1}{2}-\frac{i}{2}} (x-1) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcTan[x]*x)/(1 + x^2)^(3/2),x]

[Out]

(-1 + x)/(2*(1 - I*x)^(1/2 - I/2)*(1 + I*x)^(1/2 + I/2))

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Maple [A]  time = 0.003, size = 16, normalized size = 0.7 \begin{align*}{\frac{ \left ( -1+x \right ){{\rm e}^{\arctan \left ( x \right ) }}}{2}{\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arctan(x))*x/(x^2+1)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{\arctan \left (x\right )}}{{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(x))*x/(x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x*e^arctan(x)/(x^2 + 1)^(3/2), x)

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Fricas [A]  time = 0.664137, size = 53, normalized size = 2.41 \begin{align*} \frac{{\left (x - 1\right )} e^{\arctan \left (x\right )}}{2 \, \sqrt{x^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(x))*x/(x^2+1)^(3/2),x, algorithm="fricas")

[Out]

1/2*(x - 1)*e^arctan(x)/sqrt(x^2 + 1)

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Sympy [A]  time = 119.363, size = 31, normalized size = 1.41 \begin{align*} \frac{x e^{\operatorname{atan}{\left (x \right )}}}{2 \sqrt{x^{2} + 1}} - \frac{e^{\operatorname{atan}{\left (x \right )}}}{2 \sqrt{x^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(atan(x))*x/(x**2+1)**(3/2),x)

[Out]

x*exp(atan(x))/(2*sqrt(x**2 + 1)) - exp(atan(x))/(2*sqrt(x**2 + 1))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{\arctan \left (x\right )}}{{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(x))*x/(x^2+1)^(3/2),x, algorithm="giac")

[Out]

integrate(x*e^arctan(x)/(x^2 + 1)^(3/2), x)