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

Optimal. Leaf size=192 \[ \frac{a \sqrt{a x-1} \sqrt{a x+1} \text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right ) \sqrt{\cosh ^{-1}(a x)}},x\right )}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2-1\right )^2 \sqrt{\cosh ^{-1}(a x)}},x\right )}{6 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{\cosh ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

(x*Sqrt[ArcCosh[a*x]])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcCosh[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) + (a
*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((1 - a^2*x^2)*Sqrt[ArcCosh[a*x]]), x])/(3*c^2*Sqrt[c - a^2*c*x^2
]) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((-1 + a^2*x^2)^2*Sqrt[ArcCosh[a*x]]), x])/(6*c^2*Sqrt[c -
a^2*c*x^2])

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Rubi [A]  time = 0.432165, 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 = {} \[ \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(2*x*Sqrt[ArcCosh[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) + (x*Sqrt[ArcCosh[a*x]])/(3*c^2*(1 - a*x)*(1 + a*x)*Sqrt[
c - a^2*c*x^2]) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Defer[Int][x/((1 - a^2*x^2)*Sqrt[ArcCosh[a*x]]), x])/(3*c^2*
Sqrt[c - a^2*c*x^2]) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Defer[Int][x/((-1 + a^2*x^2)^2*Sqrt[ArcCosh[a*x]]), x])
/(6*c^2*Sqrt[c - a^2*c*x^2])

Rubi steps

\begin{align*} \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\sqrt{\cosh ^{-1}(a x)}}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sqrt{\cosh ^{-1}(a x)}}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\sqrt{\cosh ^{-1}(a x)}}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (-1+a^2 x^2\right )^2 \sqrt{\cosh ^{-1}(a x)}} \, dx}{6 c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{\cosh ^{-1}(a x)}}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (-1+a^2 x^2\right )^2 \sqrt{\cosh ^{-1}(a x)}} \, dx}{6 c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (1-a^2 x^2\right ) \sqrt{\cosh ^{-1}(a x)}} \, dx}{3 c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 2.12787, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.376, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{{\rm arccosh} \left (ax\right )} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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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(acosh(a*x)**(1/2)/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Timed out

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

[Out]

sage0*x