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

Optimal. Leaf size=48 \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.154337, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5713, 5676} \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rubi steps

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

Mathematica [A]  time = 0.0328619, size = 48, normalized size = 1. \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 41, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,a}\sqrt{ax-1}\sqrt{ax+1} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a^{2} c x^{2} + c} \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)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.96189, size = 134, normalized size = 2.79 \begin{align*} \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} - 1}}{3 \,{\left (a^{3} c x^{2} - a c\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="fricas")

[Out]

2/3*sqrt(-a^2*c*x^2 + c)*sqrt(a^2*x^2 - 1)/((a^3*c*x^2 - a*c)*log(a*x + sqrt(a^2*x^2 - 1))^(3/2))

________________________________________________________________________________________

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)**(1/2)/acosh(a*x)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  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)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x