3.417 \(\int \frac{1}{(c-a^2 c x^2)^{3/2} \cosh ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{4 a \sqrt{a x-1} \sqrt{a x+1} \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2-1\right )^2 \cosh ^{-1}(a x)^{3/2}},x\right )}{3 c \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{3/2}} \]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)) + (4*a*Sqrt[-1 + a*x]*Sqrt[1
+ a*x]*Unintegrable[x/((-1 + a^2*x^2)^2*ArcCosh[a*x]^(3/2)), x])/(3*c*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.228131, 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 = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2)),x]

[Out]

(-2*Sqrt[-1 + a*x])/(3*a*c*(1 - a*x)*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2)) + (4*a*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]*Defer[Int][x/((-1 + a^2*x^2)^2*ArcCosh[a*x]^(3/2)), x])/(3*c*Sqrt[c - a^2*c*x^2])

Rubi steps

\begin{align*} \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2}} \, 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} \cosh ^{-1}(a x)^{5/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=-\frac{2 \sqrt{-1+a x}}{3 a c (1-a x) \sqrt{1+a x} \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}+\frac{\left (4 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (-1+a^2 x^2\right )^2 \cosh ^{-1}(a x)^{3/2}} \, dx}{3 c \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 1.73978, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2)),x]

[Out]

Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2)), x]

________________________________________________________________________________________

Maple [A]  time = 0.3, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{5}{2}}}}\, 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)^(5/2),x)

[Out]

int(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(5/2),x)

________________________________________________________________________________________

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}} \operatorname{arcosh}\left (a x\right )^{\frac{5}{2}}}\,{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)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((-a^2*c*x^2 + c)^(3/2)*arccosh(a*x)^(5/2)), x)

________________________________________________________________________________________

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)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

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)^(5/2),x, algorithm="giac")

[Out]

sage0*x