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

Optimal. Leaf size=233 \[ -\frac{3 a^4}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{3 a^4}{8 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{a^4}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{a^4}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{a^4}{6 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{4 a^4}{3 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}+\frac{a^4}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{2 a^4}{5 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^5} \]

[Out]

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

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Rubi [A]  time = 0.499871, antiderivative size = 233, 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, 1612} \[ -\frac{3 a^4}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{3 a^4}{8 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{a^4}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{a^4}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{a^4}{6 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{4 a^4}{3 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}+\frac{a^4}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{2 a^4}{5 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcSech[a*x]*x^5),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0688688, size = 60, normalized size = 0.26 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} \left (2 a^3 x^3-2 a^2 x^2+3 a x-3\right ) (a x+1)^2+3}{15 a x^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcSech[a*x]*x^5),x]

[Out]

-(3 + Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2*(-3 + 3*a*x - 2*a^2*x^2 + 2*a^3*x^3))/(15*a*x^5)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5}{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.03202, size = 130, normalized size = 0.56 \begin{align*} -\frac{{\left (2 \, a^{5} x^{5} + a^{3} x^{3} - 3 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 3}{15 \, a x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="fricas")

[Out]

-1/15*((2*a^5*x^5 + a^3*x^3 - 3*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 3)/(a*x^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*Integral(1/(a*x**5*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + x**4), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5}{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="giac")

[Out]

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