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

Optimal. Leaf size=4 \[ \sinh ^{-1}\left (e^x\right ) \]

[Out]

ArcSinh[E^x]

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Rubi [A]  time = 0.0319986, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \sinh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/Sqrt[1 + E^(2*x)],x]

[Out]

ArcSinh[E^x]

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Rubi in Sympy [A]  time = 5.193, size = 3, normalized size = 0.75 \[ \operatorname{asinh}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(exp(x))

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Mathematica [A]  time = 0.0117559, size = 4, normalized size = 1. \[ \sinh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcSinh[E^x]

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Maple [A]  time = 0.013, size = 4, normalized size = 1. \[{\it Arcsinh} \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsinh(exp(x))

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Maxima [A]  time = 0.862088, size = 4, normalized size = 1. \[ \operatorname{arsinh}\left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsinh(e^x)

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Fricas [A]  time = 0.251439, size = 22, normalized size = 5.5 \[ -\log \left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(sqrt(e^(2*x) + 1) - e^x)

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Sympy [A]  time = 0.508696, size = 3, normalized size = 0.75 \[ \operatorname{asinh}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(exp(x))

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GIAC/XCAS [A]  time = 0.22804, size = 22, normalized size = 5.5 \[ -{\rm ln}\left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(sqrt(e^(2*x) + 1) - e^x)