\(\int e^{-\text {sech}^{-1}(a x)} \, dx\) [80]

Optimal result

Integrand size = 8, antiderivative size = 65 \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a}+\frac {\log (1+a x)}{a}+\frac {2 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6467, 1661, 815, 266} \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a}+\frac {\log (a x+1)}{a}+\frac {2 \log \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}{a} \]

[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 266

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]

Rule 815

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 1661

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 6467

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

Rubi steps \begin{align*} \text {integral}& = \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 \text {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 \text {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 \text {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 \text {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] (verified)

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=\frac {-\sqrt {\frac {1-a x}{1+a x}} (1+a x)+\log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{a} \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2611\) vs. \(2(61)=122\).

Time = 0.28 (sec) , antiderivative size = 2612, normalized size of antiderivative = 40.18

method	result	size
default	\(\text {Expression too large to display}\)	\(2612\)

[In]

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

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.77 \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=-\frac {2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) + \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, \log \left (x\right )}{2 \, a} \]

[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

Sympy [F]

\[ \int e^{-\text {sech}^{-1}(a x)} \, dx=a \int \frac {x}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \]

[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)

Maxima [F]

\[ \int e^{-\text {sech}^{-1}(a x)} \, dx=\int { \frac {1}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}} \,d x } \]

[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)

Giac [F]

\[ \int e^{-\text {sech}^{-1}(a x)} \, dx=\int { \frac {1}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 7.36 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=\frac {\mathrm {acosh}\left (\frac {1}{a\,x}\right )}{a}-\frac {\ln \left (\frac {1}{x}\right )}{a}-x\,\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1} \]

[In]

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

[Out]

acosh(1/(a*x))/a - log(1/x)/a - x*(1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2)