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

Optimal. Leaf size=114 \[ -\frac {8 a \sqrt {a x-1} \sqrt {a x+1} \text {Int}\left (\frac {x}{\left (a^2 x^2-1\right )^3 \cosh ^{-1}(a x)^{3/2}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{3/2}} \]

[Out]

-2/3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/(-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(3/2)-8/3*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)*U
nintegrable(x/(a^2*x^2-1)^3/arccosh(a*x)^(3/2),x)/c^2/(-a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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 = 2.36, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx \]

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]

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

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]

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

________________________________________________________________________________________

maple [A]  time = 0.75, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}\, dx \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________