3.74 \(\int \frac{e^{2 \text{sech}^{-1}(a x)}}{x^5} \, dx\)

Optimal. Leaf size=267 \[ \frac{3 a^4}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}+\frac{a^4}{8 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}-\frac{11 a^4}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{a^4}{8 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}+\frac{8 a^4}{3 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{3 a^4}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}+\frac{2 a^4}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^5}-\frac{2 a^4}{3 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^6}+\frac{1}{4} a^4 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]

[Out]

(-2*a^4)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^6) + (2*a^4)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5 - (3*a^4)/(1 - Sqrt
[(1 - a*x)/(1 + a*x)])^4 + (8*a^4)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) - (11*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1
+ a*x)])^2) + (3*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^4/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2) + a^4/(8
*(1 + Sqrt[(1 - a*x)/(1 + a*x)])) + (a^4*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/4

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Rubi [A]  time = 0.542506, antiderivative size = 267, 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} \[ \frac{3 a^4}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}+\frac{a^4}{8 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}-\frac{11 a^4}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{a^4}{8 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}+\frac{8 a^4}{3 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{3 a^4}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}+\frac{2 a^4}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^5}-\frac{2 a^4}{3 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^6}+\frac{1}{4} a^4 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcSech[a*x])/x^5,x]

[Out]

(-2*a^4)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^6) + (2*a^4)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5 - (3*a^4)/(1 - Sqrt
[(1 - a*x)/(1 + a*x)])^4 + (8*a^4)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) - (11*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1
+ a*x)])^2) + (3*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^4/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2) + a^4/(8
*(1 + Sqrt[(1 - a*x)/(1 + a*x)])) + (a^4*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/4

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^5} \, 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^5} \, dx\\ &=(4 a) \operatorname{Subst}\left (\int \frac{x \left (a+a x^2\right )^3}{(-1+x)^7 (1+x)^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=(4 a) \operatorname{Subst}\left (\int \left (\frac{a^3}{(-1+x)^7}+\frac{5 a^3}{2 (-1+x)^6}+\frac{3 a^3}{(-1+x)^5}+\frac{2 a^3}{(-1+x)^4}+\frac{11 a^3}{16 (-1+x)^3}+\frac{3 a^3}{32 (-1+x)^2}+\frac{a^3}{16 (1+x)^3}-\frac{a^3}{32 (1+x)^2}-\frac{a^3}{16 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{2 a^4}{3 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^6}+\frac{2 a^4}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^5}-\frac{3 a^4}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^4}+\frac{8 a^4}{3 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^3}-\frac{11 a^4}{8 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{3 a^4}{8 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{a^4}{8 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^4}{8 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{1}{4} a^4 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{2 a^4}{3 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^6}+\frac{2 a^4}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^5}-\frac{3 a^4}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^4}+\frac{8 a^4}{3 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^3}-\frac{11 a^4}{8 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{3 a^4}{8 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{a^4}{8 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^4}{8 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}+\frac{1}{4} a^4 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.142864, size = 137, normalized size = 0.51 \[ \frac{6 a^2 x^2+\sqrt{\frac{1-a x}{a x+1}} \left (3 a^5 x^5+3 a^4 x^4+2 a^3 x^3+2 a^2 x^2-8 a x-8\right )-3 a^6 x^6 \log (x)+3 a^6 x^6 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-8}{24 a^2 x^6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcSech[a*x])/x^5,x]

[Out]

(-8 + 6*a^2*x^2 + Sqrt[(1 - a*x)/(1 + a*x)]*(-8 - 8*a*x + 2*a^2*x^2 + 2*a^3*x^3 + 3*a^4*x^4 + 3*a^5*x^5) - 3*a
^6*x^6*Log[x] + 3*a^6*x^6*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/(24*a^2*x^6)

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Maple [A]  time = 0.227, size = 153, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}}{4\,{x}^{4}}}-{\frac{1}{6\,{x}^{6}}} \right ) }+{\frac{1}{24\,a{x}^{5}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){x}^{6}{a}^{6}+3\,\sqrt{-{a}^{2}{x}^{2}+1}{x}^{4}{a}^{4}+2\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-8\,\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{6\,{x}^{6}{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^5,x)

[Out]

1/a^2*(1/4*a^2/x^4-1/6/x^6)+1/24/a*(-(a*x-1)/a/x)^(1/2)/x^5*((a*x+1)/a/x)^(1/2)*(3*arctanh(1/(-a^2*x^2+1)^(1/2
))*x^6*a^6+3*(-a^2*x^2+1)^(1/2)*x^4*a^4+2*a^2*x^2*(-a^2*x^2+1)^(1/2)-8*(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^(1/2)-
1/6/a^2/x^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (\frac{1}{16} \, a^{6} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{1}{16} \, \sqrt{-a^{2} x^{2} + 1} a^{6} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{4}}{16 \, x^{2}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}}{8 \, x^{4}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}}\right )}}{a^{2}} - \frac{1}{3 \, a^{2} x^{6}} - \int \frac{1}{x^{5}}\,{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^5,x, algorithm="maxima")

[Out]

2*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^7, x)/a^2 - 1/3/(a^2*x^6) - integrate(x^(-5), x)

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Fricas [A]  time = 2.11585, size = 344, normalized size = 1.29 \begin{align*} \frac{3 \, a^{6} x^{6} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) + 12 \, a^{2} x^{2} + 2 \,{\left (3 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 16}{48 \, a^{2} x^{6}} \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^5,x, algorithm="fricas")

[Out]

1/48*(3*a^6*x^6*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - 3*a^6*x^6*log(a*x*sqrt((a*x + 1)/(
a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) + 12*a^2*x^2 + 2*(3*a^5*x^5 + 2*a^3*x^3 - 8*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(
-(a*x - 1)/(a*x)) - 16)/(a^2*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{2}{x^{7}}\, dx + \int - \frac{a^{2}}{x^{5}}\, dx + \int \frac{2 a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{6}}\, 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**5,x)

[Out]

(Integral(2/x**7, x) + Integral(-a**2/x**5, x) + Integral(2*a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**6, 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^{5}}\,{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^5,x, algorithm="giac")

[Out]

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