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

Optimal. Leaf size=40 \[ \frac{b \sqrt{1-c x}}{x \sqrt{\frac{1}{c x+1}}}-\frac{a+b \text{sech}^{-1}(c x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0201413, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6283, 95} \[ \frac{b \sqrt{1-c x}}{x \sqrt{\frac{1}{c x+1}}}-\frac{a+b \text{sech}^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])/x^2,x]

[Out]

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

Rule 6283

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSech[c*
x]))/(d*(m + 1)), x] + Dist[(b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)])/(m + 1), Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c
*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rule 95

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] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0540285, size = 42, normalized size = 1.05 \[ -\frac{a}{x}+b \left (c+\frac{1}{x}\right ) \sqrt{\frac{1-c x}{c x+1}}-\frac{b \text{sech}^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSech[c*x])/x^2,x]

[Out]

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

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Maple [A]  time = 0.178, size = 58, normalized size = 1.5 \begin{align*} c \left ( -{\frac{a}{cx}}+b \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))/x^2,x)

[Out]

c*(-a/c/x+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 = 0.981928, size = 43, normalized size = 1.08 \begin{align*}{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} - \frac{\operatorname{arsech}\left (c x\right )}{x}\right )} b - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.96489, size = 138, normalized size = 3.45 \begin{align*} \frac{b c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - b \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asech(c*x))/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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