3.5.18 \(\int \frac {1}{(c-a^2 c x^2)^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx\) [418]

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

[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)

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Rubi [A]
time = 0.13, 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 = {} \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx \end {gather*}

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]*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]*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*}

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Mathematica [A]
time = 1.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx \end {gather*}

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 = 4.47, 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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

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)

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