3.8 \(\int \frac {\cosh ^{-1}(a x)}{x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\cosh ^{-1}(a x)}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5662, 95} \[ \frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\cosh ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/x^3,x]

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*x) - ArcCosh[a*x]/(2*x^2)

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 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.92 \[ \frac {a x \sqrt {a x-1} \sqrt {a x+1}-\cosh ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x]/x^3,x]

[Out]

(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] - ArcCosh[a*x])/(2*x^2)

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fricas [A]  time = 0.53, size = 38, normalized size = 1.00 \[ \frac {\sqrt {a^{2} x^{2} - 1} a x - \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.42, size = 50, normalized size = 1.32 \[ \frac {a {\left | a \right |}}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 40, normalized size = 1.05 \[ a^{2} \left (-\frac {\mathrm {arccosh}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{2 a x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)/x^3,x)

[Out]

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

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maxima [A]  time = 0.87, size = 27, normalized size = 0.71 \[ \frac {\sqrt {a^{2} x^{2} - 1} a}{2 \, x} - \frac {\operatorname {arcosh}\left (a x\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/x^3,x, algorithm="maxima")

[Out]

1/2*sqrt(a^2*x^2 - 1)*a/x - 1/2*arccosh(a*x)/x^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)/x^3,x)

[Out]

int(acosh(a*x)/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)/x**3,x)

[Out]

Integral(acosh(a*x)/x**3, x)

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