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

Optimal. Leaf size=4 \[ \sin ^{-1}\left (e^x\right ) \]

[Out]

arcsin(exp(x))

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Rubi [A]  time = 0.02, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2249, 216} \[ \sin ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x/Sqrt[1 - E^(2*x)],x]

[Out]

ArcSin[E^x]

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 4, normalized size = 1.00 \[ \sin ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/Sqrt[1 - E^(2*x)],x]

[Out]

ArcSin[E^x]

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fricas [B]  time = 0.41, size = 20, normalized size = 5.00 \[ -2 \, \arctan \left ({\left (\sqrt {-e^{\left (2 \, x\right )} + 1} - 1\right )} e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 3, normalized size = 0.75 \[ \arcsin \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(e^x)

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maple [A]  time = 0.04, size = 4, normalized size = 1.00 \[ \arcsin \left ({\mathrm e}^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(-exp(2*x)+1)^(1/2),x)

[Out]

arcsin(exp(x))

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maxima [A]  time = 2.07, size = 3, normalized size = 0.75 \[ \arcsin \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(e^x)

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mupad [B]  time = 3.53, size = 3, normalized size = 0.75 \[ \mathrm {asin}\left ({\mathrm {e}}^x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(1 - exp(2*x))^(1/2),x)

[Out]

asin(exp(x))

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sympy [A]  time = 0.71, size = 3, normalized size = 0.75 \[ \operatorname {asin}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asin(exp(x))

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