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]

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

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.49, size = 66, normalized size = 1.65 \[ \frac {b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - b \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - a}{x} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.06, size = 58, normalized size = 1.45 \[ c \left (-\frac {a}{c x}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right ) \]

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 = 0.31, size = 32, normalized size = 0.80 \[ {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b - \frac {a}{x} \]

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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mupad [B]  time = 1.48, size = 46, normalized size = 1.15 \[ b\,c\,\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}-\frac {b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x}-\frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))/x^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{2}}\, dx \]

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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