3.5.7 \(\int \frac {1}{(c-a^2 c x^2)^{5/2} \sqrt {\cosh ^{-1}(a x)}} \, dx\) [407]

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\cosh ^{-1}(a x)}},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, 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} \sqrt {\cosh ^{-1}(a x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\cosh ^{-1}(a x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 4.12, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\mathrm {arccosh}\left (a x \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \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)**(1/2),x)

[Out]

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

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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)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {\mathrm {acosh}\left (a\,x\right )}\,{\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)^(1/2)*(c - a^2*c*x^2)^(5/2)),x)

[Out]

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

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