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]

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

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Rubi [A]  time = 0.50, antiderivative size = 233, 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, 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 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 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^{-\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*}

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Mathematica [A]  time = 0.07, 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]

-1/15*(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))/(a*x^5)

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fricas [A]  time = 0.74, size = 60, normalized size = 0.26 \[ -\frac {{\left (2 \, a^{5} x^{5} + a^{3} x^{3} - 3 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 3}{15 \, a x^{5}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{5} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]

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)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right ) x^{5}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{5} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]

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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mupad [B]  time = 2.37, size = 75, normalized size = 0.32 \[ -\frac {1}{5\,a\,x^5}-\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {a\,x^2}{15}-\frac {x}{5}-\frac {1}{5\,a}+\frac {a^2\,x^3}{15}+\frac {2\,a^3\,x^4}{15}+\frac {2\,a^4\,x^5}{15}\right )}{x^5\,\sqrt {\frac {1}{a\,x}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 1/(5*a*x^5) - ((1/(a*x) - 1)^(1/2)*((a*x^2)/15 - x/5 - 1/(5*a) + (a^2*x^3)/15 + (2*a^3*x^4)/15 + (2*a^4*x^5)
/15))/(x^5*(1/(a*x) + 1)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \int \frac {1}{a x^{5} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + x^{4}}\, dx \]

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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