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

Optimal. Leaf size=57 \[ \frac{2 a}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{4 a}{3 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3} \]

[Out]

(-4*a)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) + (2*a)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2

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Rubi [A]  time = 0.390062, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6337, 43} \[ \frac{2 a}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{4 a}{3 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) + (2*a)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0662674, size = 52, normalized size = 0.91 \[ \frac{3 a^2 x^2+2 (a x-1) \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2-2}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.194, size = 73, normalized size = 1.3 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}}{x}}-{\frac{1}{3\,{x}^{3}}} \right ) }+{\frac{2\,{a}^{2}{x}^{2}-2}{3\,a{x}^{2}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{1}{3\,{x}^{3}{a}^{2}}} \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^2,x)

[Out]

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

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Maxima [A]  time = 1.04345, size = 62, normalized size = 1.09 \begin{align*} \frac{1}{x} + \frac{2 \,{\left (a^{2} x^{3} - x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{3 \, a^{2} x^{4}} - \frac{2}{3 \, a^{2} x^{3}} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.05545, size = 130, normalized size = 2.28 \begin{align*} \frac{3 \, a^{2} x^{2} + 2 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 2}{3 \, a^{2} x^{3}} \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{2}{x^{4}}\, dx + \int - \frac{a^{2}}{x^{2}}\, dx + \int \frac{2 a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{3}}\, 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**2,x)

[Out]

(Integral(2/x**4, x) + Integral(-a**2/x**2, x) + Integral(2*a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**3, x))/a
**2

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Giac [A]  time = 1.15194, size = 126, normalized size = 2.21 \begin{align*} \frac{2 \,{\left (a^{2} + \frac{a}{x}\right )}^{\frac{3}{2}}{\left (a^{2} - \frac{a}{x}\right )} \sqrt{-a^{2} + \frac{a}{x}} a^{4} -{\left (a^{4} + \frac{a}{x^{3}}\right )} a^{6} +{\left (3 \,{\left (a^{2} + \frac{a}{x}\right )}^{2} a^{2} -{\left (a^{2} + \frac{a}{x}\right )}^{3}\right )} a^{4}}{3 \, a^{9}} \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^2,x, algorithm="giac")

[Out]

1/3*(2*(a^2 + a/x)^(3/2)*(a^2 - a/x)*sqrt(-a^2 + a/x)*a^4 - (a^4 + a/x^3)*a^6 + (3*(a^2 + a/x)^2*a^2 - (a^2 +
a/x)^3)*a^4)/a^9