3.70 \(\int \frac{e^{2 \text{sech}^{-1}(a x)}}{x} \, dx\)

Optimal. Leaf size=86 \[ \frac{2}{1-\sqrt{\frac{1-a x}{a x+1}}}-\frac{2}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\log (a x+1)-2 \log \left (1-\sqrt{\frac{1-a x}{a x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.445547, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6337, 1629, 260} \[ \frac{2}{1-\sqrt{\frac{1-a x}{a x+1}}}-\frac{2}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\log (a x+1)-2 \log \left (1-\sqrt{\frac{1-a x}{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcSech[a*x])/x,x]

[Out]

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

Rule 6337

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

Rule 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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 \frac{e^{2 \text{sech}^{-1}(a x)}}{x} \, dx &=\int \frac{\left (\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}\right )^2}{x} \, dx\\ &=4 \operatorname{Subst}\left (\int \frac{x (1+x)}{(-1+x)^3 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=4 \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^3}+\frac{1}{2 (-1+x)^2}-\frac{1}{2 (-1+x)}+\frac{x}{2 \left (1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{2}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{2}{1-\sqrt{\frac{1-a x}{1+a x}}}-2 \log \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )+2 \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{2}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{2}{1-\sqrt{\frac{1-a x}{1+a x}}}-\log (1+a x)-2 \log \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0658266, size = 86, normalized size = 1. \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a^2 x^2}-\frac{1}{a^2 x^2}+\log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-2 \log (x) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcSech[a*x])/x,x]

[Out]

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

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Maple [A]  time = 0.187, size = 97, normalized size = 1.1 \begin{align*} -\ln \left ( x \right ) -{\frac{1}{{a}^{2}{x}^{2}}}-{\frac{1}{ax}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( -{a}^{2}{x}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +\sqrt{-{a}^{2}{x}^{2}+1} \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))^2/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (\frac{1}{2} \, a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} a^{2} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{2 \, x^{2}}\right )}}{a^{2}} - \frac{1}{a^{2} x^{2}} - \int \frac{1}{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))^2/x,x, algorithm="maxima")

[Out]

2*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^3, x)/a^2 - 1/(a^2*x^2) - integrate(1/x, x)

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Fricas [A]  time = 2.14903, size = 306, normalized size = 3.56 \begin{align*} \frac{a^{2} x^{2} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 2}{2 \, a^{2} x^{2}} \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))^2/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{2}{x^{3}}\, dx + \int - \frac{a^{2}}{x}\, dx + \int \frac{2 a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{2}}\, dx}{a^{2}} \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))**2/x,x)

[Out]

(Integral(2/x**3, x) + Integral(-a**2/x, x) + Integral(2*a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**2, x))/a**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}^{2}}{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))^2/x,x, algorithm="giac")

[Out]

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