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

Optimal. Leaf size=30 \[ b c \sqrt {\frac {1}{c^2 x^2}+1}-\frac {a+b \text {csch}^{-1}(c x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 30, 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 = {6284, 261} \[ b c \sqrt {\frac {1}{c^2 x^2}+1}-\frac {a+b \text {csch}^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])/x^2,x]

[Out]

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 6284

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsch[c*
x]))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^2} \, dx &=-\frac {a+b \text {csch}^{-1}(c x)}{x}-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^3} \, dx}{c}\\ &=b c \sqrt {1+\frac {1}{c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{x}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 40, normalized size = 1.33 \[ -\frac {a}{x}+b c \sqrt {\frac {c^2 x^2+1}{c^2 x^2}}-\frac {b \text {csch}^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsch[c*x])/x^2,x]

[Out]

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

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fricas [B]  time = 0.78, size = 64, normalized size = 2.13 \[ \frac {b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - b \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 62, normalized size = 2.07 \[ c \left (-\frac {a}{c x}+b \left (-\frac {\mathrm {arccsch}\left (c x \right )}{c x}+\frac {c^{2} x^{2}+1}{\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 32, normalized size = 1.07 \[ {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b - \frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.32, size = 35, normalized size = 1.17 \[ b\,c\,\sqrt {\frac {1}{c^2\,x^2}+1}-\frac {a}{x}-\frac {b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(1/(c*x)))/x^2,x)

[Out]

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

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sympy [A]  time = 1.82, size = 36, normalized size = 1.20 \[ \begin {cases} - \frac {a}{x} + b c \sqrt {1 + \frac {1}{c^{2} x^{2}}} - \frac {b \operatorname {acsch}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\- \frac {a + \tilde {\infty } b}{x} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/x + b*c*sqrt(1 + 1/(c**2*x**2)) - b*acsch(c*x)/x, Ne(c, 0)), (-(a + zoo*b)/x, True))

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