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

Optimal. Leaf size=24 \[ \frac{\log (x)}{a}+x e^{\text{sech}^{-1}(a x)}-\frac{\text{sech}^{-1}(a x)}{a} \]

[Out]

E^ArcSech[a*x]*x - ArcSech[a*x]/a + Log[x]/a

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Rubi [A]  time = 0.136978, antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6329, 1962, 208} \[ \frac{\log (x)}{a}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{a}+x e^{\text{sech}^{-1}(a x)} \]

Warning: Unable to verify antiderivative.

[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 6329

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

Rule 1962

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)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^((m + 1)/n + 1),
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 208

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

Rubi steps

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

Mathematica [B]  time = 0.0402629, size = 79, normalized size = 3.29 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)+2 \log (a x)-\log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )}{a} \]

Warning: Unable to verify antiderivative.

[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

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Maple [A]  time = 0.184, size = 79, normalized size = 3.3 \begin{align*}{\frac{\ln \left ( x \right ) }{a}}+{x\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( \sqrt{-{a}^{2}{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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/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)

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Fricas [B]  time = 1.83216, size = 258, normalized size = 10.75 \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/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*} \frac{\int \frac{1}{x}\, dx + \int a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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/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)