3.692 \(\int e^x \sqrt{9-e^{2 x}} \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{2} e^x \sqrt{9-e^{2 x}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^x}{3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0439058, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{2} e^x \sqrt{9-e^{2 x}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x*Sqrt[9 - E^(2*x)],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.97625, size = 24, normalized size = 0.73 \[ \frac{\sqrt{- e^{2 x} + 9} e^{x}}{2} + \frac{9 \operatorname{asin}{\left (\frac{e^{x}}{3} \right )}}{2} \]

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*asin(exp(x)/3)/2

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Mathematica [A]  time = 0.0219208, size = 33, normalized size = 1. \[ \frac{1}{2} e^x \sqrt{9-e^{2 x}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^x}{3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 23, normalized size = 0.7 \[{\frac{{{\rm e}^{x}}}{2}\sqrt{9- \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{9}{2}\arcsin \left ({\frac{{{\rm e}^{x}}}{3}} \right ) } \]

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*arcsin(1/3*exp(x))

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Maxima [A]  time = 0.859182, size = 30, normalized size = 0.91 \[ \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 9} e^{x} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, e^{x}\right ) \]

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*arcsin(1/3*e^x)

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Fricas [A]  time = 0.238799, size = 124, normalized size = 3.76 \[ -\frac{18 \,{\left (6 \, \sqrt{-e^{\left (2 \, x\right )} + 9} + e^{\left (2 \, x\right )} - 18\right )} \arctan \left ({\left (\sqrt{-e^{\left (2 \, x\right )} + 9} - 3\right )} e^{\left (-x\right )}\right ) -{\left (e^{\left (3 \, x\right )} - 18 \, e^{x}\right )} \sqrt{-e^{\left (2 \, x\right )} + 9} + 6 \, e^{\left (3 \, x\right )} - 54 \, e^{x}}{2 \,{\left (6 \, \sqrt{-e^{\left (2 \, x\right )} + 9} + e^{\left (2 \, x\right )} - 18\right )}} \]

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*(18*(6*sqrt(-e^(2*x) + 9) + e^(2*x) - 18)*arctan((sqrt(-e^(2*x) + 9) - 3)*e
^(-x)) - (e^(3*x) - 18*e^x)*sqrt(-e^(2*x) + 9) + 6*e^(3*x) - 54*e^x)/(6*sqrt(-e^
(2*x) + 9) + e^(2*x) - 18)

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Sympy [A]  time = 1.58338, size = 29, normalized size = 0.88 \[ \begin{cases} \frac{\sqrt{- e^{2 x} + 9} e^{x}}{2} + \frac{9 \operatorname{asin}{\left (\frac{e^{x}}{3} \right )}}{2} & \text{for}\: e^{x} < \log{\left (3 \right )} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((sqrt(-exp(2*x) + 9)*exp(x)/2 + 9*asin(exp(x)/3)/2, exp(x) < log(3)))

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GIAC/XCAS [A]  time = 0.241794, size = 30, normalized size = 0.91 \[ \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 9} e^{x} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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