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

Optimal. Leaf size=113 \[ -\frac{8 a \sqrt{a x-1} \sqrt{a x+1} \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2-1\right )^3 \cosh ^{-1}(a x)^{3/2}},x\right )}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{3/2}} \]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*(c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^(3/2)) - (8*a*Sqrt[-1 + a*x]*Sqrt[1
+ a*x]*Unintegrable[x/((-1 + a^2*x^2)^3*ArcCosh[a*x]^(3/2)), x])/(3*c^2*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.225575, 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 = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*Sqrt[-1 + a*x])/(3*a*c^2*(1 - a*x)^2*(1 + a*x)^(3/2)*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2)) - (8*a*Sqrt[-
1 + a*x]*Sqrt[1 + a*x]*Defer[Int][x/((-1 + a^2*x^2)^3*ArcCosh[a*x]^(3/2)), x])/(3*c^2*Sqrt[c - a^2*c*x^2])

Rubi steps

\begin{align*} \int \frac{1}{\left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{(-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{2 \sqrt{-1+a x}}{3 a c^2 (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}-\frac{\left (8 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (-1+a^2 x^2\right )^3 \cosh ^{-1}(a x)^{3/2}} \, dx}{3 c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 2.26372, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.377, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{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)^(5/2)/arccosh(a*x)^(5/2),x)

[Out]

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

[Out]

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

[Out]

sage0*x