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