3.1.98 \(\int \frac {e^{\tan ^{-1}(x)} x}{(1+x^2)^{3/2}} \, dx\) [98]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {5185} \begin {gather*} -\frac {(1-x) e^{\tan ^{-1}(x)}}{2 \sqrt {x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5185

Int[(E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(1 - a*n*x))*(E^(n*Ar
cTan[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] Result contains complex when optimal does not.
time = 0.01, size = 37, normalized size = 1.68 \begin {gather*} \frac {1}{2} (1-i x)^{-\frac {1}{2}+\frac {i}{2}} (1+i x)^{-\frac {1}{2}-\frac {i}{2}} (-1+x) \end {gather*}

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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Mathics [A]
time = 12.08, size = 16, normalized size = 0.73 \begin {gather*} \frac {\left (-1+x\right ) E^{\text {ArcTan}\left [x\right ]}}{2 \sqrt {1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.07, size = 16, normalized size = 0.73

method	result	size
gosper	\(\frac {\left (-1+x \right ) {\mathrm e}^{\arctan \left (x \right )}}{2 \sqrt {x^{2}+1}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.34, size = 15, normalized size = 0.68 \begin {gather*} \frac {{\left (x - 1\right )} e^{\arctan \left (x\right )}}{2 \, \sqrt {x^{2} + 1}} \end {gather*}

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 = 11.80, size = 31, normalized size = 1.41 \begin {gather*} \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 {gather*}

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 [A]
time = 0.00, size = 33, normalized size = 1.50 \begin {gather*} \mathrm {e}^{\arctan x} \left (-\frac {1}{2 \sqrt {x^{2}+1}}+\frac {x}{2 \sqrt {x^{2}+1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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