3.84 \(\int \frac{e^{-\text{sech}^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=200 \[ \frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}+\frac{a^3}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}-\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{a^3}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}+\frac{a^3}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}-\frac{a^3}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{1}{4} a^3 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.500469, antiderivative size = 200, 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} \[ \frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}+\frac{a^3}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}-\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{a^3}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}+\frac{a^3}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}-\frac{a^3}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{1}{4} a^3 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcSech[a*x]*x^4),x]

[Out]

-a^3/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) + a^3/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^3/(2*(1 + Sqrt[(1 - a
*x)/(1 + a*x)])^4) + a^3/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3 - a^3/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 + a^3/(2*(1
+ Sqrt[(1 - a*x)/(1 + a*x)])) - (a^3*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^{-\text{sech}^{-1}(a x)}}{x^4} \, dx &=\int \frac{1}{x^4 \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\\ &=(4 a) \operatorname{Subst}\left (\int \frac{x \left (a+a x^2\right )^2}{(-1+x)^3 (1+x)^5} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=(4 a) \operatorname{Subst}\left (\int \left (\frac{a^2}{8 (-1+x)^3}+\frac{a^2}{16 (-1+x)^2}+\frac{a^2}{2 (1+x)^5}-\frac{3 a^2}{4 (1+x)^4}+\frac{a^2}{2 (1+x)^3}-\frac{a^2}{8 (1+x)^2}+\frac{a^2}{16 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{a^3}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}+\frac{a^3}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}-\frac{a^3}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^3}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}+\frac{1}{4} a^3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{a^3}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}+\frac{a^3}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}-\frac{a^3}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^3}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{1}{4} a^3 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0996863, size = 110, normalized size = 0.55 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} \left (a^3 x^3+a^2 x^2-2 a x-2\right )-a^4 x^4 \log (x)+a^4 x^4 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )+2}{8 a x^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcSech[a*x]*x^4),x]

[Out]

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

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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^4,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4}{\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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.07651, size = 300, normalized size = 1.5 \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 ) + 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}{16 \, a x^{4}} \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^4,x, algorithm="fricas")

[Out]

-1/16*(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) + 2*(a^3*x^3 - 2*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 4)/(a*x^4)

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4}{\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^4,x, algorithm="giac")

[Out]

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