\(\int e^{\text {sech}^{-1}(a x)} \, dx\) [36]

Optimal result

Integrand size = 6, antiderivative size = 24 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=e^{\text {sech}^{-1}(a x)} x-\frac {\text {sech}^{-1}(a x)}{a}+\frac {\log (x)}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6464, 1984, 214} \[ \int e^{\text {sech}^{-1}(a x)} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {\frac {1-a x}{a x+1}}\right )}{a}+\frac {\log (x)}{a}+x e^{\text {sech}^{-1}(a x)} \]

[In]

Int[E^ArcSech[a*x],x]

[Out]

E^ArcSech[a*x]*x - (2*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/a + Log[x]/a

Rule 214

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

Rule 1984

Int[(u_)^(r_.)*(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Wi
th[{q = Denominator[p]}, Dist[q*e*((b*c - a*d)/n), Subst[Int[SimplifyIntegrand[x^(q*(p + 1) - 1)*(((-a)*e + c*
x^q)^((m + 1)/n - 1)/(b*e - d*x^q)^((m + 1)/n + 1))*(u /. x -> ((-a)*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r,
x], x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && Frac
tionQ[p] && IntegerQ[1/n] && IntegersQ[m, r]

Rule 6464

Int[E^ArcSech[(a_.)*(x_)], x_Symbol] :> Simp[x*E^ArcSech[a*x], x] + (Dist[1/a, Int[(1/(x*(1 - a*x)))*Sqrt[(1 -
a*x)/(1 + a*x)], x], x] + Simp[Log[x]/a, x]) /; FreeQ[a, x]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(24)=48\).

Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.29 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)+2 \log (a x)-\log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{a} \]

[In]

Integrate[E^ArcSech[a*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33

method	result	size
default	\(\frac {\ln \left (x \right )}{a}-\frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (-\sqrt {-a^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{\sqrt {-a^{2} x^{2}+1}}\)	\(80\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (49) = 98\).

Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.79 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) + \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, \log \left (x\right )}{2 \, a} \]

[In]

integrate(1/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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 7.58 \[ \int e^{\text {sech}^{-1}(a x)} \, dx=\frac {\ln \left (x\right )}{a}-\frac {4\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{a}+\frac {\frac {5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1}{\frac {4\,a\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}+\frac {4\,a\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}}+\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{4\,a\,\left (\sqrt {\frac {1}{a\,x}+1}-1\right )} \]

[In]

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

[Out]

log(x)/a - (4*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/a + ((5*((1/(a*x) - 1)^(1/2) - 1i)^
2)/((1/(a*x) + 1)^(1/2) - 1)^2 + 1)/((4*a*((1/(a*x) - 1)^(1/2) - 1i))/((1/(a*x) + 1)^(1/2) - 1) + (4*a*((1/(a*
x) - 1)^(1/2) - 1i)^3)/((1/(a*x) + 1)^(1/2) - 1)^3) + ((1/(a*x) - 1)^(1/2) - 1i)/(4*a*((1/(a*x) + 1)^(1/2) - 1
))