3.643 \(\int e^x \sqrt{9+e^{2 x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{2} e^x \sqrt{e^{2 x}+9}+\frac{9}{2} \sinh ^{-1}\left (\frac{e^x}{3}\right ) \]

[Out]

(E^x*Sqrt[9 + E^(2*x)])/2 + (9*ArcSinh[E^x/3])/2

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Rubi [A]  time = 0.0384741, antiderivative size = 31, 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 \[ \frac{1}{2} e^x \sqrt{e^{2 x}+9}+\frac{9}{2} \sinh ^{-1}\left (\frac{e^x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x*Sqrt[9 + E^(2*x)],x]

[Out]

(E^x*Sqrt[9 + E^(2*x)])/2 + (9*ArcSinh[E^x/3])/2

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Rubi in Sympy [A]  time = 5.41781, size = 24, normalized size = 0.77 \[ \frac{\sqrt{e^{2 x} + 9} e^{x}}{2} + \frac{9 \operatorname{asinh}{\left (\frac{e^{x}}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)*(9+exp(2*x))**(1/2),x)

[Out]

sqrt(exp(2*x) + 9)*exp(x)/2 + 9*asinh(exp(x)/3)/2

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Mathematica [A]  time = 0.0187318, size = 31, normalized size = 1. \[ \frac{1}{2} e^x \sqrt{e^{2 x}+9}+\frac{9}{2} \sinh ^{-1}\left (\frac{e^x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x*Sqrt[9 + E^(2*x)],x]

[Out]

(E^x*Sqrt[9 + E^(2*x)])/2 + (9*ArcSinh[E^x/3])/2

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Maple [A]  time = 0.009, size = 21, normalized size = 0.7 \[{\frac{{{\rm e}^{x}}}{2}\sqrt{9+ \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{9}{2}{\it Arcsinh} \left ({\frac{{{\rm e}^{x}}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)*(9+exp(2*x))^(1/2),x)

[Out]

1/2*exp(x)*(9+exp(x)^2)^(1/2)+9/2*arcsinh(1/3*exp(x))

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Maxima [A]  time = 0.834955, size = 27, normalized size = 0.87 \[ \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 9} e^{x} + \frac{9}{2} \, \operatorname{arsinh}\left (\frac{1}{3} \, e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^(2*x) + 9)*e^x,x, algorithm="maxima")

[Out]

1/2*sqrt(e^(2*x) + 9)*e^x + 9/2*arcsinh(1/3*e^x)

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Fricas [A]  time = 0.238123, size = 126, normalized size = 4.06 \[ -\frac{9 \,{\left (2 \, \sqrt{e^{\left (2 \, x\right )} + 9} e^{x} - 2 \, e^{\left (2 \, x\right )} - 9\right )} \log \left (\sqrt{e^{\left (2 \, x\right )} + 9} - e^{x}\right ) +{\left (2 \, e^{\left (3 \, x\right )} + 9 \, e^{x}\right )} \sqrt{e^{\left (2 \, x\right )} + 9} - 2 \, e^{\left (4 \, x\right )} - 18 \, e^{\left (2 \, x\right )}}{2 \,{\left (2 \, \sqrt{e^{\left (2 \, x\right )} + 9} e^{x} - 2 \, e^{\left (2 \, x\right )} - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^(2*x) + 9)*e^x,x, algorithm="fricas")

[Out]

-1/2*(9*(2*sqrt(e^(2*x) + 9)*e^x - 2*e^(2*x) - 9)*log(sqrt(e^(2*x) + 9) - e^x) +
(2*e^(3*x) + 9*e^x)*sqrt(e^(2*x) + 9) - 2*e^(4*x) - 18*e^(2*x))/(2*sqrt(e^(2*x)
+ 9)*e^x - 2*e^(2*x) - 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{e^{2 x} + 9} e^{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)*(9+exp(2*x))**(1/2),x)

[Out]

Integral(sqrt(exp(2*x) + 9)*exp(x), x)

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GIAC/XCAS [A]  time = 0.238761, size = 39, normalized size = 1.26 \[ \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 9} e^{x} - \frac{9}{2} \,{\rm ln}\left (\sqrt{e^{\left (2 \, x\right )} + 9} - e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^(2*x) + 9)*e^x,x, algorithm="giac")

[Out]

1/2*sqrt(e^(2*x) + 9)*e^x - 9/2*ln(sqrt(e^(2*x) + 9) - e^x)