3.1.7 \(\int \frac {\text {sech}^{-1}(a x)^2}{x^2} \, dx\) [7]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6420, 3377, 2718} \begin {gather*} -\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} \end {gather*}

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 2718

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

Rule 3377

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 6420

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])

Rubi steps

\begin {align*} \int \frac {\text {sech}^{-1}(a x)^2}{x^2} \, dx &=-\left (a \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.06, size = 42, normalized size = 0.86 \begin {gather*} -\frac {2-2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)+\text {sech}^{-1}(a x)^2}{x} \end {gather*}

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 = 0.16, size = 61, normalized size = 1.24

method	result	size
derivativedivides	\(a \left (-\frac {\mathrm {arcsech}\left (a x \right )^{2}}{a x}+2 \,\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}-\frac {2}{a x}\right )\)	\(61\)
default	\(a \left (-\frac {\mathrm {arcsech}\left (a x \right )^{2}}{a x}+2 \,\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}-\frac {2}{a x}\right )\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(a*x)^2/x^2,x,method=_RETURNVERBOSE)

[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 = 0.26, size = 35, normalized size = 0.71 \begin {gather*} 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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (47) = 94\).
time = 0.34, size = 97, normalized size = 1.98 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asech}^{2}{\left (a x \right )}}{x^{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(1/(a*x))^2/x^2,x)

[Out]

int(acosh(1/(a*x))^2/x^2, x)

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