3.642 \(\int e^x \sqrt{1-e^{2 x}} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{2} e^x \sqrt{1-e^{2 x}}+\frac{1}{2} \sin ^{-1}\left (e^x\right ) \]

[Out]

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

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Rubi [A]  time = 0.026527, antiderivative size = 29, 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, Rules used = {2249, 195, 216} \[ \frac{1}{2} e^x \sqrt{1-e^{2 x}}+\frac{1}{2} \sin ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Sqrt[1 - E^(2*x)],x]

[Out]

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

Rule 2249

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int e^x \sqrt{1-e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \sqrt{1-x^2} \, dx,x,e^x\right )\\ &=\frac{1}{2} e^x \sqrt{1-e^{2 x}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,e^x\right )\\ &=\frac{1}{2} e^x \sqrt{1-e^{2 x}}+\frac{1}{2} \sin ^{-1}\left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0115649, size = 26, normalized size = 0.9 \[ \frac{1}{2} \left (e^x \sqrt{1-e^{2 x}}+\sin ^{-1}\left (e^x\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sqrt[1 - E^(2*x)],x]

[Out]

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

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Maple [A]  time = 0.054, size = 21, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{x}}}{2}\sqrt{1- \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{\arcsin \left ({{\rm e}^{x}} \right ) }{2}} \end{align*}

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*arcsin(exp(x))

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Maxima [A]  time = 1.45959, size = 27, normalized size = 0.93 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 1} e^{x} + \frac{1}{2} \, \arcsin \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1-exp(2*x))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.892407, size = 95, normalized size = 3.28 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 1} e^{x} - \arctan \left ({\left (\sqrt{-e^{\left (2 \, x\right )} + 1} - 1\right )} e^{\left (-x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1-exp(2*x))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.27658, size = 24, normalized size = 0.83 \begin{align*} \begin{cases} \frac{\sqrt{1 - e^{2 x}} e^{x}}{2} + \frac{\operatorname{asin}{\left (e^{x} \right )}}{2} & \text{for}\: e^{x} < 0 \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1-exp(2*x))**(1/2),x)

[Out]

Piecewise((sqrt(1 - exp(2*x))*exp(x)/2 + asin(exp(x))/2, exp(x) < 0))

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Giac [A]  time = 1.2158, size = 27, normalized size = 0.93 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 1} e^{x} + \frac{1}{2} \, \arcsin \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1-exp(2*x))^(1/2),x, algorithm="giac")

[Out]

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