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

Optimal. Leaf size=49 \[ -\frac{\text{sech}^{-1}(a x)^2}{x}+\frac{2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{x}-\frac{2}{x} \]

[Out]

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

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Rubi [A]  time = 0.0483221, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6285, 3296, 2638} \[ -\frac{\text{sech}^{-1}(a x)^2}{x}+\frac{2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{x}-\frac{2}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6285

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\text{sech}^{-1}(a x)^2}{x^2} \, dx &=-\left (a \operatorname{Subst}\left (\int x^2 \sinh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\right )\\ &=-\frac{\text{sech}^{-1}(a x)^2}{x}+(2 a) \operatorname{Subst}\left (\int x \cosh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=\frac{2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)}{x}-\frac{\text{sech}^{-1}(a x)^2}{x}-(2 a) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=-\frac{2}{x}+\frac{2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)}{x}-\frac{\text{sech}^{-1}(a x)^2}{x}\\ \end{align*}

Mathematica [A]  time = 0.0870271, size = 42, normalized size = 0.86 \[ -\frac{\text{sech}^{-1}(a x)^2-2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)+2}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.213, size = 61, normalized size = 1.2 \begin{align*} a \left ( -{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{ax}}+2\,{\rm arcsech} \left (ax\right )\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}-2\,{\frac{1}{ax}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(a*x)^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.01563, size = 47, normalized size = 0.96 \begin{align*} 2 \, a \sqrt{\frac{1}{a^{2} x^{2}} - 1} \operatorname{arsech}\left (a x\right ) - \frac{\operatorname{arsech}\left (a x\right )^{2}}{x} - \frac{2}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^2/x^2,x, algorithm="maxima")

[Out]

2*a*sqrt(1/(a^2*x^2) - 1)*arcsech(a*x) - arcsech(a*x)^2/x - 2/x

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Fricas [B]  time = 1.70892, size = 208, normalized size = 4.24 \begin{align*} \frac{2 \, a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) - \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - 2}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^2/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asech}^{2}{\left (a x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(a*x)**2/x**2,x)

[Out]

Integral(asech(a*x)**2/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (a x\right )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^2/x^2,x, algorithm="giac")

[Out]

integrate(arcsech(a*x)^2/x^2, x)