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

Optimal. Leaf size=132 \[ \frac{1}{8} a^3 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{a x+1}}}+\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3} \]

[Out]

1/(12*a*x^4) - E^ArcSech[a*x]/(3*x^3) + Sqrt[1 - a*x]/(12*a*x^4*Sqrt[(1 + a*x)^(-1)]) + (a*Sqrt[1 - a*x])/(8*x
^2*Sqrt[(1 + a*x)^(-1)]) + (a^3*Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/8

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Rubi [A]  time = 0.058869, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{1}{8} a^3 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{a x+1}}}+\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(12*a*x^4) - E^ArcSech[a*x]/(3*x^3) + Sqrt[1 - a*x]/(12*a*x^4*Sqrt[(1 + a*x)^(-1)]) + (a*Sqrt[1 - a*x])/(8*x
^2*Sqrt[(1 + a*x)^(-1)]) + (a^3*Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/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^4} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}-\frac{\int \frac{1}{x^5} \, dx}{3 a}-\frac{\left (\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}{3 a}\\ &=\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\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}{12 a}\\ &=\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}-\frac{1}{4} \left (a \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}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{8} \left (a \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}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{8} \left (a^3 \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}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{8} \left (a^4 \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}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{8} a^3 \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 = 0.0733636, size = 110, normalized size = 0.83 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} \left (a^3 x^3+a^2 x^2-2 a x-2\right )-a^4 x^4 \log (x)+a^4 x^4 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-2}{8 a x^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[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 [A]  time = 0.184, size = 110, normalized size = 0.8 \begin{align*}{\frac{1}{8\,{x}^{3}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ({\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){x}^{4}{a}^{4}+{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-2\,\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{4\,{x}^{4}a}} \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^4,x)

[Out]

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

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

[Out]

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

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Fricas [A]  time = 1.80857, size = 298, normalized size = 2.26 \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/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*} \frac{\int \frac{1}{x^{5}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{4}}\, 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**4,x)

[Out]

(Integral(x**(-5), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**4, 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^{4}}\,{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^4,x, algorithm="giac")

[Out]

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