3.64 \(\int \frac{1}{x^2 \cosh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{1}{x^2 \cosh ^{-1}(a x)^3},x\right ) \]

[Out]

Unintegrable[1/(x^2*ArcCosh[a*x]^3), x]

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Rubi [A]  time = 0.0149753, 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 = {} \[ \int \frac{1}{x^2 \cosh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*ArcCosh[a*x]^3),x]

[Out]

Defer[Int][1/(x^2*ArcCosh[a*x]^3), x]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \cosh ^{-1}(a x)^3} \, dx &=\int \frac{1}{x^2 \cosh ^{-1}(a x)^3} \, dx\\ \end{align*}

Mathematica [A]  time = 2.19435, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \cosh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*ArcCosh[a*x]^3),x]

[Out]

Integrate[1/(x^2*ArcCosh[a*x]^3), x]

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Maple [A]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/arccosh(a*x)^3,x)

[Out]

int(1/x^2/arccosh(a*x)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/arccosh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^8 - 3*a^6*x^6 + 3*a^4*x^4 + (a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - a^2*x^2 + (3*a^6
*x^6 - 5*a^4*x^4 + 2*a^2*x^2)*(a*x + 1)*(a*x - 1) + (3*a^7*x^7 - 7*a^5*x^5 + 5*a^3*x^3 - a*x)*sqrt(a*x + 1)*sq
rt(a*x - 1) - (a^8*x^8 - 3*a^6*x^6 + 3*a^4*x^4 + (a^5*x^5 - 4*a^3*x^3 + 3*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2)
- a^2*x^2 + (3*a^6*x^6 - 11*a^4*x^4 + 10*a^2*x^2 - 2)*(a*x + 1)*(a*x - 1) + (3*a^7*x^7 - 10*a^5*x^5 + 10*a^3*
x^3 - 3*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^8*x^9 + (a*x + 1)^(3/2)*
(a*x - 1)^(3/2)*a^5*x^6 - 3*a^6*x^7 + 3*a^4*x^5 - a^2*x^3 + 3*(a^6*x^7 - a^4*x^5)*(a*x + 1)*(a*x - 1) + 3*(a^7
*x^8 - 2*a^5*x^6 + a^3*x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2) + integrate
(1/2*(a^10*x^10 - 4*a^8*x^8 + 6*a^6*x^6 - 4*a^4*x^4 + (a^6*x^6 - 12*a^4*x^4 + 15*a^2*x^2)*(a*x + 1)^2*(a*x - 1
)^2 + (4*a^7*x^7 - 40*a^5*x^5 + 57*a^3*x^3 - 18*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + a^2*x^2 + 3*(2*a^8*x^8
- 16*a^6*x^6 + 25*a^4*x^4 - 13*a^2*x^2 + 2)*(a*x + 1)*(a*x - 1) + (4*a^9*x^9 - 24*a^7*x^7 + 39*a^5*x^5 - 25*a^
3*x^3 + 6*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^10*x^12 + (a*x + 1)^2*(a*x - 1)^2*a^6*x^8 - 4*a^8*x^10 + 6*a^6
*x^8 - 4*a^4*x^6 + a^2*x^4 + 4*(a^7*x^9 - a^5*x^7)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 6*(a^8*x^10 - 2*a^6*x^8 +
a^4*x^6)*(a*x + 1)*(a*x - 1) + 4*(a^9*x^11 - 3*a^7*x^9 + 3*a^5*x^7 - a^3*x^5)*sqrt(a*x + 1)*sqrt(a*x - 1))*lo
g(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x^{2} \operatorname{arcosh}\left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(1/(x^2*arccosh(a*x)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \operatorname{acosh}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/acosh(a*x)**3,x)

[Out]

Integral(1/(x**2*acosh(a*x)**3), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \operatorname{arcosh}\left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/arccosh(a*x)^3,x, algorithm="giac")

[Out]

integrate(1/(x^2*arccosh(a*x)^3), x)