3.642 \(\int \frac{x}{\sqrt{-1+e^{2 x^2}}} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{e^{2 x^2}-1}\right ) \]

[Out]

ArcTan[Sqrt[-1 + E^(2*x^2)]]/2

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Rubi [A]  time = 0.0947207, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{e^{2 x^2}-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[-1 + E^(2*x^2)],x]

[Out]

ArcTan[Sqrt[-1 + E^(2*x^2)]]/2

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Rubi in Sympy [A]  time = 6.8302, size = 14, normalized size = 0.78 \[ \frac{\operatorname{atan}{\left (\sqrt{e^{2 x^{2}} - 1} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(exp(2*x**2) - 1))/2

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Mathematica [A]  time = 0.01407, size = 18, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{e^{2 x^2}-1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[-1 + E^(2*x^2)],x]

[Out]

ArcTan[Sqrt[-1 + E^(2*x^2)]]/2

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Maple [A]  time = 0.016, size = 14, normalized size = 0.8 \[{\frac{1}{2}\arctan \left ( \sqrt{-1+{{\rm e}^{2\,{x}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-1+exp(2*x^2))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.881927, size = 18, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{e^{\left (2 \, x^{2}\right )} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(e^(2*x^2) - 1),x, algorithm="maxima")

[Out]

1/2*arctan(sqrt(e^(2*x^2) - 1))

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Fricas [A]  time = 0.249721, size = 18, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{e^{\left (2 \, x^{2}\right )} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(e^(2*x^2) - 1),x, algorithm="fricas")

[Out]

1/2*arctan(sqrt(e^(2*x^2) - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (e^{x^{2}} - 1\right ) \left (e^{x^{2}} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt((exp(x**2) - 1)*(exp(x**2) + 1)), x)

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GIAC/XCAS [A]  time = 0.323234, size = 18, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{e^{\left (2 \, x^{2}\right )} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(e^(2*x^2) - 1),x, algorithm="giac")

[Out]

1/2*arctan(sqrt(e^(2*x^2) - 1))