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

Optimal. Leaf size=20 \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.0350961, antiderivative size = 20, 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 \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.39228, size = 17, normalized size = 0.85 \[ \frac{\left (x + 1\right ) e^{\operatorname{atan}{\left (x \right )}}}{2 \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(atan(x))/(x**2+1)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.064379, size = 20, normalized size = 1. \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 16, normalized size = 0.8 \[{\frac{{{\rm e}^{\arctan \left ( x \right ) }} \left ( 1+x \right ) }{2}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(arctan(x))/(x^2+1)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^arctan(x)/(x^2 + 1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.227675, size = 20, normalized size = 1. \[ \frac{{\left (x + 1\right )} e^{\arctan \left (x\right )}}{2 \, \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^arctan(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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\arctan \left (x\right )}}{{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^arctan(x)/(x^2 + 1)^(3/2),x, algorithm="giac")

[Out]

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