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

Optimal. Leaf size=65 \[ \frac{a \text{Unintegrable}\left (\frac{x}{(a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)},x\right )}{c}+\frac{1}{a c \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)} \]

[Out]

1/(a*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]) + (a*Unintegrable[x/((-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCos
h[a*x]), x])/c

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Rubi [A]  time = 0.245898, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

1/(a*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]) + (a*Defer[Int][x/((-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[
a*x]), x])/c

Rubi steps

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

Mathematica [A]  time = 2.97771, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( -{a}^{2}c{x}^{2}+c \right ) \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a x + \sqrt{a x + 1} \sqrt{a x - 1}}{{\left (a^{3} c x^{2} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} c x - a c\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )} + \int \frac{a^{4} x^{4} +{\left (a^{2} x^{2} - 1\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} +{\left (2 \, a^{3} x^{3} - a x\right )} \sqrt{a x + 1} \sqrt{a x - 1} - 1}{{\left (a^{6} c x^{6} - 3 \, a^{4} c x^{4} + 3 \, a^{2} c x^{2} +{\left (a^{4} c x^{4} - a^{2} c x^{2}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} + 2 \,{\left (a^{5} c x^{5} - 2 \, a^{3} c x^{3} + a c x\right )} \sqrt{a x + 1} \sqrt{a x - 1} - c\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*c*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*c*x - a*c)*log(a*x + sqrt(a*
x + 1)*sqrt(a*x - 1))) + integrate((a^4*x^4 + (a^2*x^2 - 1)*(a*x + 1)*(a*x - 1) + (2*a^3*x^3 - a*x)*sqrt(a*x +
1)*sqrt(a*x - 1) - 1)/((a^6*c*x^6 - 3*a^4*c*x^4 + 3*a^2*c*x^2 + (a^4*c*x^4 - a^2*c*x^2)*(a*x + 1)*(a*x - 1) +
2*(a^5*c*x^5 - 2*a^3*c*x^3 + a*c*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - c)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))),
x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{{\left (a^{2} c x^{2} - c\right )} \operatorname{arcosh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{1}{a^{2} x^{2} \operatorname{acosh}^{2}{\left (a x \right )} - \operatorname{acosh}^{2}{\left (a x \right )}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (a^{2} c x^{2} - c\right )} \operatorname{arcosh}\left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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