3.297 \(\int \frac {1}{x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x^{3} - x\right )} \operatorname {arcosh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)/((a^2*x^3 - x)*arccosh(a*x)), x)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} x \operatorname {arcosh}\left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \,\mathrm {arccosh}\left (a x \right ) \sqrt {-a^{2} x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} x \operatorname {arcosh}\left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x\,\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*acosh(a*x)*(1 - a^2*x^2)^(1/2)),x)

[Out]

int(1/(x*acosh(a*x)*(1 - a^2*x^2)^(1/2)), x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/acosh(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(1/(x*sqrt(-(a*x - 1)*(a*x + 1))*acosh(a*x)), x)

________________________________________________________________________________________