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

Optimal. Leaf size=61 \[ \frac{2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{2 b^2}{x} \]

[Out]

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

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Rubi [A]  time = 0.0699774, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6285, 3296, 2638} \[ \frac{2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{2 b^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])^2/x^2,x]

[Out]

(-2*b^2)/x + (2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/x - (a + b*ArcSech[c*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{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x^2} \, dx &=-\left (c \operatorname{Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}+(2 b c) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=\frac{2 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=-\frac{2 b^2}{x}+\frac{2 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}\\ \end{align*}

Mathematica [A]  time = 0.228817, size = 87, normalized size = 1.43 \[ -\frac{a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 b \text{sech}^{-1}(c x) \left (b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-a\right )+b^2 \text{sech}^{-1}(c x)^2+2 b^2}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSech[c*x])^2/x^2,x]

[Out]

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

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Maple [B]  time = 0.217, size = 124, normalized size = 2. \begin{align*} c \left ( -{\frac{{a}^{2}}{cx}}+{b}^{2} \left ( -{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{cx}}+2\,{\rm arcsech} \left (cx\right )\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}-2\,{\frac{1}{cx}} \right ) +2\,ab \left ( -{\frac{{\rm arcsech} \left (cx\right )}{cx}}+\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(c*x))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.02864, size = 105, normalized size = 1.72 \begin{align*} 2 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} - \frac{\operatorname{arsech}\left (c x\right )}{x}\right )} a b + 2 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} \operatorname{arsech}\left (c x\right ) - \frac{1}{x}\right )} b^{2} - \frac{b^{2} \operatorname{arsech}\left (c x\right )^{2}}{x} - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c*sqrt(1/(c^2*x^2) - 1) - arcsech(c*x)/x)*a*b + 2*(c*sqrt(1/(c^2*x^2) - 1)*arcsech(c*x) - 1/x)*b^2 - b^2*ar
csech(c*x)^2/x - a^2/x

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Fricas [B]  time = 1.65977, size = 301, normalized size = 4.93 \begin{align*} \frac{2 \, a b c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - b^{2} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - a^{2} - 2 \, b^{2} + 2 \,{\left (b^{2} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))**2/x**2,x)

[Out]

Integral((a + b*asech(c*x))**2/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsech(c*x) + a)^2/x^2, x)