3.80 \(\int e^{-\text{sech}^{-1}(a x)} \, dx\)

Optimal. Leaf size=65 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a}+\frac{\log (a x+1)}{a}+\frac{2 \log \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}{a} \]

[Out]

-((Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/a) + Log[1 + a*x]/a + (2*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)]])/a

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Rubi [A]  time = 0.165642, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6332, 1647, 801, 260} \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a}+\frac{\log (a x+1)}{a}+\frac{2 \log \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(-ArcSech[a*x]),x]

[Out]

-((Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/a) + Log[1 + a*x]/a + (2*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)]])/a

Rule 6332

Int[E^(ArcSech[u_]*(n_.)), x_Symbol] :> Int[(1/u + Sqrt[(1 - u)/(1 + u)] + (1*Sqrt[(1 - u)/(1 + u)])/u)^n, x]
/; IntegerQ[n]

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int e^{-\text{sech}^{-1}(a x)} \, dx &=\int \frac{1}{\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}} \, dx\\ &=\frac{4 \operatorname{Subst}\left (\int \frac{(-1+x) x}{(1+x) \left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)}{a}-\frac{2 \operatorname{Subst}\left (\int \frac{-1+x}{(1+x) \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)}{a}-\frac{2 \operatorname{Subst}\left (\int \left (\frac{1}{-1-x}+\frac{x}{1+x^2}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)}{a}+\frac{2 \log \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}{a}-\frac{2 \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)}{a}+\frac{\log (1+a x)}{a}+\frac{2 \log \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0305943, size = 72, normalized size = 1.11 \[ \frac{\log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(-ArcSech[a*x]),x]

[Out]

(-(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)) + Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/
a

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Maple [B]  time = 0.475, size = 2632, normalized size = 40.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*x-1)/x^3*(x^2*ln(a^2*x^2)*((a*x+1)/a/x)^(3/2)*(-(a*x-1)/a/x)^(1/2)*(-a^2*x^2+1)^(1/2)*a^2-ln(2*((-a^2*
(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2
*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)*x-(-a^2*x^2+1)^(1/2)-
1)*a^2/(-a^2*x+(-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((
a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2))
)*x^3*a^3-ln(2*((-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*((
(a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)
*x+(-a^2*x^2+1)^(1/2)+1)*a^2/(a^2*x+(-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-
(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1
)/x^2/(a*x-1))^(1/2)))*x^3*a^3+2*(-a^2*x^2+1)^(1/2)*x^3*a^3-ln(2*((-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x
)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1
/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)*x-(-a^2*x^2+1)^(1/2)-1)*a^2/(-a^2*x+(-a^2*(((a*x+1)/a/x)^(1
/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(
1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)))*x^2*a^2-ln(2*((-a^2*(((a*x+1)/a/x)^(
1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^
(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)*x+(-a^2*x^2+1)^(1/2)+1)*a^2/(a^2*x+(-
a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)
*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)))*x^2*a^2+2*a^2*x
^2*(-a^2*x^2+1)^(1/2)+ln(2*((-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*
x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a
*x-1))^(1/2)*x-(-a^2*x^2+1)^(1/2)-1)*a^2/(-a^2*x+(-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1
)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^
(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)))*x*a+ln(2*((-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x
)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/
2))/(a*x+1)/x^2/(a*x-1))^(1/2)*x+(-a^2*x^2+1)^(1/2)+1)*a^2/(a^2*x+(-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x
)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1
/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)))*x*a-2*(-a^2*x^2+1)^(1/2)*a*x+ln(2*((-a^2*(((a*x+1)/a/x)^(
1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^
(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)*x-(-a^2*x^2+1)^(1/2)-1)*a^2/(-a^2*x+(
-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2
)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)))+ln(2*((-a^2*((
(a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x+1)/a/x)^(1/2)*x^2*(
-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)*x+(-a^2*x^2+1)^(1/2)+1)
*a^2/(a^2*x+(-a^2*(((a*x+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a-((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))*(((a*x
+1)/a/x)^(1/2)*x^2*(-(a*x-1)/a/x)^(1/2)*a+((a*x+1)*x)^(1/2)*(-(a*x-1)*x)^(1/2))/(a*x+1)/x^2/(a*x-1))^(1/2)))-2
*(-a^2*x^2+1)^(1/2))/a^4/((a*x+1)/a/x)^(3/2)/(-(a*x-1)/a/x)^(3/2)/(-a^2*x^2+1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x)), x)

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Fricas [A]  time = 2.08143, size = 259, normalized size = 3.98 \begin{align*} -\frac{2 \, a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) + \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) - 2 \, \log \left (x\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)
) + 1) + log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) - 2*log(x))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a \int \frac{x}{a x \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)**(1/2)*(1+1/a/x)**(1/2)),x)

[Out]

a*Integral(x/(a*x*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x)), x)