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]

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

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Rubi [A]  time = 0.39, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 1.06, size = 61, normalized size = 1.07 \[ \frac {3 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2}{3 \, a^{2} x^{3}} \]

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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giac [B]  time = 0.21, size = 122, normalized size = 2.14 \[ \frac {3 \, {\left (a^{2} + \frac {a}{x}\right )} a^{2} - {\left (9 \, a^{2} + {\left (a^{2} + \frac {a}{x}\right )} {\left (\frac {2 \, {\left (a^{2} + \frac {a}{x}\right )}}{a^{2}} - 7\right )}\right )} \sqrt {a^{2} + \frac {a}{x}} \sqrt {-a^{2} + \frac {a}{x}} + 3 \, {\left (2 \, a^{2} - \frac {a}{x}\right )} \sqrt {a^{2} + \frac {a}{x}} \sqrt {-a^{2} + \frac {a}{x}} - \frac {2 \, a}{x^{3}}}{3 \, a^{3}} \]

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*(3*(a^2 + a/x)*a^2 - (9*a^2 + (a^2 + a/x)*(2*(a^2 + a/x)/a^2 - 7))*sqrt(a^2 + a/x)*sqrt(-a^2 + a/x) + 3*(2
*a^2 - a/x)*sqrt(a^2 + a/x)*sqrt(-a^2 + a/x) - 2*a/x^3)/a^3

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maple [A]  time = 0.06, size = 73, normalized size = 1.28 \[ \frac {\frac {a^{2}}{x}-\frac {1}{3 x^{3}}}{a^{2}}+\frac {2 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2}-1\right )}{3 a \,x^{2}}-\frac {1}{3 a^{2} x^{3}} \]

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 = 0.35, size = 46, normalized size = 0.81 \[ \frac {1}{x} + \frac {2 \, {\left (a^{2} x^{3} - x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{3 \, a^{2} x^{4}} - \frac {2}{3 \, a^{2} x^{3}} \]

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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mupad [B]  time = 1.80, size = 67, normalized size = 1.18 \[ \frac {a^2\,x^2-\frac {2}{3}}{a^2\,x^3}-\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,\sqrt {\frac {1}{a\,x}+1}}{3\,a}-\frac {2\,a\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{3}\right )}{x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {2}{x^{4}}\, dx + \int \left (- \frac {a^{2}}{x^{2}}\right )\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{3}}\, dx}{a^{2}} \]

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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