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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{\left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.58, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{2} c x^{2} - c\right )} \operatorname {arcosh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-1/((a^2*c*x^2 - c)*arccosh(a*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} c \,x^{2}+c \right ) \mathrm {arccosh}\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(acosh(a*x)*(c - a^2*c*x^2)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{a^{2} x^{2} \operatorname {acosh}{\left (a x \right )} - \operatorname {acosh}{\left (a x \right )}}\, dx}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(a**2*x**2*acosh(a*x) - acosh(a*x)), x)/c

________________________________________________________________________________________