3.38 \(\int \frac{e^{\text{sech}^{-1}(a x)}}{x^2} \, dx\)

Optimal. Leaf size=35 \[ a \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )-\frac{e^{\text{sech}^{-1}(a x)}}{2 x} \]

[Out]

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

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Rubi [B]  time = 0.0412326, antiderivative size = 99, normalized size of antiderivative = 2.83, number of steps used = 6, 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} \[ \frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{a x+1}}}+\frac{1}{2 a x^2}+\frac{1}{2} a \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )-\frac{e^{\text{sech}^{-1}(a x)}}{x} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcSech[a*x]/x^2,x]

[Out]

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

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^2} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{x}-\frac{\int \frac{1}{x^3} \, dx}{a}-\frac{\left (\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}{a}\\ &=\frac{1}{2 a x^2}-\frac{e^{\text{sech}^{-1}(a x)}}{x}+\frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{1+a x}}}-\frac{\left (\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}{2 a}\\ &=\frac{1}{2 a x^2}-\frac{e^{\text{sech}^{-1}(a x)}}{x}+\frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{2} \left (a \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}{2 a x^2}-\frac{e^{\text{sech}^{-1}(a x)}}{x}+\frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{2} \left (a^2 \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}{2 a x^2}-\frac{e^{\text{sech}^{-1}(a x)}}{x}+\frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{2} a \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 [B]  time = 0.0598851, size = 93, normalized size = 2.66 \[ \frac{1}{2} \left (-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a x^2}-\frac{1}{a x^2}-a \log (x)+a \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x]/x^2,x]

[Out]

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

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Maple [A]  time = 0.204, size = 90, normalized size = 2.6 \begin{align*} -{\frac{1}{2\,x}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( -{a}^{2}{x}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{2\,a{x}^{2}}} \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^2,x)

[Out]

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

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

[Out]

integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^3, x)/a - 1/2/(a*x^2)

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Fricas [B]  time = 1.81015, size = 278, normalized size = 7.94 \begin{align*} \frac{a^{2} x^{2} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) - 2 \, a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 2}{4 \, a x^{2}} \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{3}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{2}}\, 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**2,x)

[Out]

(Integral(x**(-3), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**2, 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^{2}}\,{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^2,x, algorithm="giac")

[Out]

integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))/x^2, x)