3.44 \(\int \frac{e^{\text{sech}^{-1}(a x)}}{x^8} \, dx\)

Optimal. Leaf size=194 \[ \frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{a x+1}}}+\frac{5}{128} a^7 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{a x+1}}}+\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7} \]

[Out]

1/(56*a*x^8) - E^ArcSech[a*x]/(7*x^7) + Sqrt[1 - a*x]/(56*a*x^8*Sqrt[(1 + a*x)^(-1)]) + (a*Sqrt[1 - a*x])/(48*
x^6*Sqrt[(1 + a*x)^(-1)]) + (5*a^3*Sqrt[1 - a*x])/(192*x^4*Sqrt[(1 + a*x)^(-1)]) + (5*a^5*Sqrt[1 - a*x])/(128*
x^2*Sqrt[(1 + a*x)^(-1)]) + (5*a^7*Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/12
8

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Rubi [A]  time = 0.0984763, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6335, 30, 103, 12, 92, 208} \[ \frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{a x+1}}}+\frac{5}{128} a^7 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{a x+1}}}+\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSech[a*x]/x^8,x]

[Out]

1/(56*a*x^8) - E^ArcSech[a*x]/(7*x^7) + Sqrt[1 - a*x]/(56*a*x^8*Sqrt[(1 + a*x)^(-1)]) + (a*Sqrt[1 - a*x])/(48*
x^6*Sqrt[(1 + a*x)^(-1)]) + (5*a^3*Sqrt[1 - a*x])/(192*x^4*Sqrt[(1 + a*x)^(-1)]) + (5*a^5*Sqrt[1 - a*x])/(128*
x^2*Sqrt[(1 + a*x)^(-1)]) + (5*a^7*Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/12
8

Rule 6335

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*E^ArcSech[a*x^p])/(m + 1), x] + (Dist
[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[(p*Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/(a*(m + 1)), Int[x^(m - p
)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e^{\text{sech}^{-1}(a x)}}{x^8} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}-\frac{\int \frac{1}{x^9} \, dx}{7 a}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^9 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{7 a}\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{7 a^2}{x^7 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{56 a}\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}-\frac{1}{8} \left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^7 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{1}{48} \left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{5 a^2}{x^5 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}-\frac{1}{48} \left (5 a^3 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^5 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{1}{192} \left (5 a^3 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{3 a^2}{x^3 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}-\frac{1}{64} \left (5 a^5 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^3 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{128} \left (5 a^5 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{a^2}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{128} \left (5 a^7 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{128} \left (5 a^8 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{5}{128} a^7 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.132619, size = 145, normalized size = 0.75 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} \left (15 a^7 x^7+15 a^6 x^6+10 a^5 x^5+10 a^4 x^4+8 a^3 x^3+8 a^2 x^2-48 a x-48\right )-15 a^8 x^8 \log (x)+15 a^8 x^8 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-48}{384 a x^8} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x]/x^8,x]

[Out]

(-48 + Sqrt[(1 - a*x)/(1 + a*x)]*(-48 - 48*a*x + 8*a^2*x^2 + 8*a^3*x^3 + 10*a^4*x^4 + 10*a^5*x^5 + 15*a^6*x^6
+ 15*a^7*x^7) - 15*a^8*x^8*Log[x] + 15*a^8*x^8*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x
)]])/(384*a*x^8)

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Maple [A]  time = 0.203, size = 152, normalized size = 0.8 \begin{align*}{\frac{1}{384\,{x}^{7}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( 15\,{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){x}^{8}{a}^{8}+15\,\sqrt{-{a}^{2}{x}^{2}+1}{x}^{6}{a}^{6}+10\,\sqrt{-{a}^{2}{x}^{2}+1}{x}^{4}{a}^{4}+8\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-48\,\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{8\,a{x}^{8}}} \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))/x^8,x)

[Out]

1/384*(-(a*x-1)/a/x)^(1/2)/x^7*((a*x+1)/a/x)^(1/2)*(15*arctanh(1/(-a^2*x^2+1)^(1/2))*x^8*a^8+15*(-a^2*x^2+1)^(
1/2)*x^6*a^6+10*(-a^2*x^2+1)^(1/2)*x^4*a^4+8*a^2*x^2*(-a^2*x^2+1)^(1/2)-48*(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^(1
/2)-1/8/a/x^8

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{5}{128} \, a^{8} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{5}{128} \, \sqrt{-a^{2} x^{2} + 1} a^{8} - \frac{5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{6}}{128 \, x^{2}} - \frac{5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{4}}{64 \, x^{4}} - \frac{5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}}{48 \, x^{6}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{8 \, x^{8}}}{a} - \frac{1}{8 \, a x^{8}} \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))/x^8,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^9, x)/a - 1/8/(a*x^8)

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Fricas [A]  time = 2.13341, size = 348, normalized size = 1.79 \begin{align*} \frac{15 \, a^{8} x^{8} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - 15 \, a^{8} x^{8} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) + 2 \,{\left (15 \, a^{7} x^{7} + 10 \, a^{5} x^{5} + 8 \, a^{3} x^{3} - 48 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 96}{768 \, a x^{8}} \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))/x^8,x, algorithm="fricas")

[Out]

1/768*(15*a^8*x^8*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - 15*a^8*x^8*log(a*x*sqrt((a*x + 1
)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) + 2*(15*a^7*x^7 + 10*a^5*x^5 + 8*a^3*x^3 - 48*a*x)*sqrt((a*x + 1)/(a*x))*
sqrt(-(a*x - 1)/(a*x)) - 96)/(a*x^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{9}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{8}}\, dx}{a} \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))/x**8,x)

[Out]

(Integral(x**(-9), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**8, x))/a

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}{x^{8}}\,{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))/x^8,x, algorithm="giac")

[Out]

integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))/x^8, x)