3.406 \(\int \frac{1}{(c-a^2 c x^2)^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.198037, 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 = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.7483, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.3, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\rm arccosh} \left (ax\right )}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x)

[Out]

int(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \sqrt{\operatorname{acosh}{\left (a x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a**2*c*x**2+c)**(3/2)/acosh(a*x)**(1/2),x)

[Out]

Integral(1/((-c*(a*x - 1)*(a*x + 1))**(3/2)*sqrt(acosh(a*x))), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x, algorithm="giac")

[Out]

sage0*x