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]

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

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Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.01, 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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fricas [A]  time = 0.43, size = 15, normalized size = 0.68 \[ \frac {{\left (x - 1\right )} e^{\arctan \relax (x)}}{2 \, \sqrt {x^{2} + 1}} \]

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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giac [A]  time = 0.01, size = 24, normalized size = 1.09 \[ \frac {1}{2} \, {\left (\frac {x}{\sqrt {x^{2} + 1}} - \frac {1}{\sqrt {x^{2} + 1}}\right )} e^{\arctan \relax (x)} \]

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]

1/2*(x/sqrt(x^2 + 1) - 1/sqrt(x^2 + 1))*e^arctan(x)

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maple [A]  time = 0.01, size = 16, normalized size = 0.73 \[ \frac {\left (x -1\right ) {\mathrm e}^{\arctan \relax (x )}}{2 \sqrt {x^{2}+1}} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x e^{\arctan \relax (x)}}{{\left (x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {x\,{\mathrm {e}}^{\mathrm {atan}\relax (x)}}{{\left (x^2+1\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 18.84, size = 31, normalized size = 1.41 \[ \frac {x e^{\operatorname {atan}{\relax (x )}}}{2 \sqrt {x^{2} + 1}} - \frac {e^{\operatorname {atan}{\relax (x )}}}{2 \sqrt {x^{2} + 1}} \]

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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