3.72 \(\int \frac{e^{2 \text{sech}^{-1}(a x)}}{x^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{a^2}{2 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{3 a^2}{2 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{2 a^2}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{a^2}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}+\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.45291, antiderivative size = 147, 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, 1612, 207} \[ \frac{a^2}{2 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{3 a^2}{2 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{2 a^2}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{a^2}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}+\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcSech[a*x])/x^3,x]

[Out]

-(a^2/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) + (2*a^2)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3 - (3*a^2)/(2*(1 - Sqrt[(1
- a*x)/(1 + a*x)])^2) + a^2/(2*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) + (a^2*ArcTanh[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 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{2 \text{sech}^{-1}(a x)}}{x^3} \, 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^3} \, dx\\ &=\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{x \left (1+x^2\right )}{(-1+x)^5 (1+x)} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=\left (4 a^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^5}+\frac{3}{2 (-1+x)^4}+\frac{3}{4 (-1+x)^3}+\frac{1}{8 (-1+x)^2}-\frac{1}{8 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{a^2}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^4}+\frac{2 a^2}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^3}-\frac{3 a^2}{2 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^2}{2 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{a^2}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^4}+\frac{2 a^2}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^3}-\frac{3 a^2}{2 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^2}{2 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}+\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.122758, size = 121, normalized size = 0.82 \[ \frac{\frac{(a x+1) \left (a^2 x^2 \sqrt{\frac{1-a x}{a x+1}}+2 a x-2 \sqrt{\frac{1-a x}{a x+1}}-2\right )}{x^4}+a^4 (-\log (x))+a^4 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )}{4 a^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcSech[a*x])/x^3,x]

[Out]

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

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Maple [A]  time = 0.192, size = 131, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{1}{4\,{x}^{4}}}+{\frac{{a}^{2}}{2\,{x}^{2}}} \right ) }+{\frac{1}{4\,{x}^{3}a}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ({\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){x}^{4}{a}^{4}+{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-2\,\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{4\,{a}^{2}{x}^{4}}} \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^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (\frac{1}{8} \, a^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1} a^{4} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}}{8 \, x^{2}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{4 \, x^{4}}\right )}}{a^{2}} - \frac{1}{2 \, a^{2} x^{4}} - \int \frac{1}{x^{3}}\,{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^3,x, algorithm="maxima")

[Out]

2*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^5, x)/a^2 - 1/2/(a^2*x^4) - integrate(x^(-3), x)

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Fricas [A]  time = 2.12267, size = 316, normalized size = 2.15 \begin{align*} \frac{a^{4} x^{4} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) + 4 \, a^{2} x^{2} + 2 \,{\left (a^{3} x^{3} - 2 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 4}{8 \, a^{2} x^{4}} \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^3,x, algorithm="fricas")

[Out]

1/8*(a^4*x^4*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - a^4*x^4*log(a*x*sqrt((a*x + 1)/(a*x))
*sqrt(-(a*x - 1)/(a*x)) - 1) + 4*a^2*x^2 + 2*(a^3*x^3 - 2*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) -
4)/(a^2*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{2}{x^{5}}\, dx + \int - \frac{a^{2}}{x^{3}}\, dx + \int \frac{2 a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{4}}\, 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**3,x)

[Out]

(Integral(2/x**5, x) + Integral(-a**2/x**3, x) + Integral(2*a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**4, 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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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