3.87 \(\int \frac{e^{-\text{sech}^{-1}(a x)}}{x^7} \, dx\)

Optimal. Leaf size=353 \[ -\frac{5 a^6}{16 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{5 a^6}{16 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{3 a^6}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{a^6}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{5 a^6}{12 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{11 a^6}{6 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}+\frac{a^6}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}+\frac{9 a^6}{4 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{a^6}{10 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^5}-\frac{19 a^6}{10 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^5}+\frac{a^6}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^6}-\frac{2 a^6}{7 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^7} \]

[Out]

-a^6/(10*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5) + a^6/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) - (5*a^6)/(12*(1 - Sqr
t[(1 - a*x)/(1 + a*x)])^3) + (3*a^6)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - (5*a^6)/(16*(1 - Sqrt[(1 - a*x)/(
1 + a*x)])) - (2*a^6)/(7*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^7) + a^6/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^6 - (19*a^6)
/(10*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^5) + (9*a^6)/(4*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4) - (11*a^6)/(6*(1 + Sqr
t[(1 - a*x)/(1 + a*x)])^3) + a^6/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 - (5*a^6)/(16*(1 + Sqrt[(1 - a*x)/(1 + a*x)
]))

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Rubi [A]  time = 0.604905, antiderivative size = 353, 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} \[ -\frac{5 a^6}{16 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{5 a^6}{16 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{3 a^6}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{a^6}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{5 a^6}{12 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{11 a^6}{6 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}+\frac{a^6}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}+\frac{9 a^6}{4 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{a^6}{10 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^5}-\frac{19 a^6}{10 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^5}+\frac{a^6}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^6}-\frac{2 a^6}{7 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcSech[a*x]*x^7),x]

[Out]

-a^6/(10*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5) + a^6/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) - (5*a^6)/(12*(1 - Sqr
t[(1 - a*x)/(1 + a*x)])^3) + (3*a^6)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - (5*a^6)/(16*(1 - Sqrt[(1 - a*x)/(
1 + a*x)])) - (2*a^6)/(7*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^7) + a^6/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^6 - (19*a^6)
/(10*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^5) + (9*a^6)/(4*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4) - (11*a^6)/(6*(1 + Sqr
t[(1 - a*x)/(1 + a*x)])^3) + a^6/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 - (5*a^6)/(16*(1 + Sqrt[(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^7} \, dx &=\int \frac{1}{x^7 \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 )^5}{(-1+x)^6 (1+x)^8} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\left ((4 a) \operatorname{Subst}\left (\int \left (\frac{a^5}{8 (-1+x)^6}+\frac{a^5}{4 (-1+x)^5}+\frac{5 a^5}{16 (-1+x)^4}+\frac{3 a^5}{16 (-1+x)^3}+\frac{5 a^5}{64 (-1+x)^2}-\frac{a^5}{2 (1+x)^8}+\frac{3 a^5}{2 (1+x)^7}-\frac{19 a^5}{8 (1+x)^6}+\frac{9 a^5}{4 (1+x)^5}-\frac{11 a^5}{8 (1+x)^4}+\frac{a^5}{2 (1+x)^3}-\frac{5 a^5}{64 (1+x)^2}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\frac{a^6}{10 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^5}+\frac{a^6}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^4}-\frac{5 a^6}{12 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^3}+\frac{3 a^6}{8 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}-\frac{5 a^6}{16 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{2 a^6}{7 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^7}+\frac{a^6}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^6}-\frac{19 a^6}{10 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^5}+\frac{9 a^6}{4 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}-\frac{11 a^6}{6 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}+\frac{a^6}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}-\frac{5 a^6}{16 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}\\ \end{align*}

Mathematica [A]  time = 0.0969453, size = 76, normalized size = 0.22 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} \left (8 a^5 x^5-8 a^4 x^4+12 a^3 x^3-12 a^2 x^2+15 a x-15\right ) (a x+1)^2+15}{105 a x^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcSech[a*x]*x^7),x]

[Out]

-(15 + Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2*(-15 + 15*a*x - 12*a^2*x^2 + 12*a^3*x^3 - 8*a^4*x^4 + 8*a^5*x^5))
/(105*a*x^7)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7}} \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^7,x)

[Out]

int(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^7,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{7}{\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^7,x, algorithm="maxima")

[Out]

integrate(1/(x^7*(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))), x)

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Fricas [A]  time = 2.09432, size = 153, normalized size = 0.43 \begin{align*} -\frac{{\left (8 \, a^{7} x^{7} + 4 \, a^{5} x^{5} + 3 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 15}{105 \, a x^{7}} \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^7,x, algorithm="fricas")

[Out]

-1/105*((8*a^7*x^7 + 4*a^5*x^5 + 3*a^3*x^3 - 15*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 15)/(a*x^7
)

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

[Out]

a*Integral(1/(a*x**7*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + x**6), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{7}{\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^7,x, algorithm="giac")

[Out]

integrate(1/(x^7*(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))), x)