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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{\left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)},x\right ) \]

[Out]

Unintegrable(1/(-a^2*c*x^2+c)^2/arccosh(a*x),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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \operatorname {arcosh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*arccosh(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{2} \mathrm {arccosh}\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\mathrm {acosh}\left (a\,x\right )\,{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{4} x^{4} \operatorname {acosh}{\left (a x \right )} - 2 a^{2} x^{2} \operatorname {acosh}{\left (a x \right )} + \operatorname {acosh}{\left (a x \right )}}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**4*x**4*acosh(a*x) - 2*a**2*x**2*acosh(a*x) + acosh(a*x)), x)/c**2

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