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

Optimal. Leaf size=27 \[ \frac{1}{2} e^x \sqrt{e^{2 x}+1}+\frac{1}{2} \sinh ^{-1}\left (e^x\right ) \]

[Out]

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

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Rubi [A]  time = 0.037997, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{2} e^x \sqrt{e^{2 x}+1}+\frac{1}{2} \sinh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x*Sqrt[1 + E^(2*x)],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.42649, size = 20, normalized size = 0.74 \[ \frac{\sqrt{e^{2 x} + 1} e^{x}}{2} + \frac{\operatorname{asinh}{\left (e^{x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0154689, size = 24, normalized size = 0.89 \[ \frac{1}{2} \left (e^x \sqrt{e^{2 x}+1}+\sinh ^{-1}\left (e^x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 19, normalized size = 0.7 \[{\frac{{{\rm e}^{x}}}{2}\sqrt{1+ \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{{\it Arcsinh} \left ({{\rm e}^{x}} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.846697, size = 24, normalized size = 0.89 \[ \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 1} e^{x} + \frac{1}{2} \, \operatorname{arsinh}\left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(e^(2*x) + 1)*e^x + 1/2*arcsinh(e^x)

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Fricas [A]  time = 0.254123, size = 122, normalized size = 4.52 \[ -\frac{{\left (2 \, \sqrt{e^{\left (2 \, x\right )} + 1} e^{x} - 2 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x}\right ) +{\left (2 \, e^{\left (3 \, x\right )} + e^{x}\right )} \sqrt{e^{\left (2 \, x\right )} + 1} - 2 \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )}}{2 \,{\left (2 \, \sqrt{e^{\left (2 \, x\right )} + 1} e^{x} - 2 \, e^{\left (2 \, x\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*((2*sqrt(e^(2*x) + 1)*e^x - 2*e^(2*x) - 1)*log(sqrt(e^(2*x) + 1) - e^x) + (
2*e^(3*x) + e^x)*sqrt(e^(2*x) + 1) - 2*e^(4*x) - 2*e^(2*x))/(2*sqrt(e^(2*x) + 1)
*e^x - 2*e^(2*x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{e^{2 x} + 1} e^{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.233985, size = 39, normalized size = 1.44 \[ \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 1} e^{x} - \frac{1}{2} \,{\rm ln}\left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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