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

Optimal. Leaf size=13 \[ \text {Int}\left (\frac {1}{x^2 \cosh ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, 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 = {} \[ \int \frac {1}{x^2 \cosh ^{-1}(a x)^3} \, dx \]

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*}

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Mathematica [A]  time = 2.38, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \cosh ^{-1}(a x)^3} \, dx \]

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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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x^{2} \operatorname {arcosh}\left (a x\right )^{3}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \operatorname {arcosh}\left (a x\right )^{3}}\,{d x} \]

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)

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maple [A]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \mathrm {arccosh}\left (a x \right )^{3}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{8} x^{8} - 3 \, a^{6} x^{6} + 3 \, a^{4} x^{4} + {\left (a^{5} x^{5} - a^{3} x^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} - a^{2} x^{2} + {\left (3 \, a^{6} x^{6} - 5 \, a^{4} x^{4} + 2 \, a^{2} x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (3 \, a^{7} x^{7} - 7 \, a^{5} x^{5} + 5 \, a^{3} x^{3} - a x\right )} \sqrt {a x + 1} \sqrt {a x - 1} - {\left (a^{8} x^{8} - 3 \, a^{6} x^{6} + 3 \, a^{4} x^{4} + {\left (a^{5} x^{5} - 4 \, a^{3} x^{3} + 3 \, a x\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} - a^{2} x^{2} + {\left (3 \, a^{6} x^{6} - 11 \, a^{4} x^{4} + 10 \, a^{2} x^{2} - 2\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (3 \, a^{7} x^{7} - 10 \, a^{5} x^{5} + 10 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{2 \, {\left (a^{8} x^{9} + {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} a^{5} x^{6} - 3 \, a^{6} x^{7} + 3 \, a^{4} x^{5} - a^{2} x^{3} + 3 \, {\left (a^{6} x^{7} - a^{4} x^{5}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 3 \, {\left (a^{7} x^{8} - 2 \, a^{5} x^{6} + a^{3} x^{4}\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}} + \int \frac {a^{10} x^{10} - 4 \, a^{8} x^{8} + 6 \, a^{6} x^{6} - 4 \, a^{4} x^{4} + {\left (a^{6} x^{6} - 12 \, a^{4} x^{4} + 15 \, a^{2} x^{2}\right )} {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{2} + {\left (4 \, a^{7} x^{7} - 40 \, a^{5} x^{5} + 57 \, a^{3} x^{3} - 18 \, a x\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + a^{2} x^{2} + 3 \, {\left (2 \, a^{8} x^{8} - 16 \, a^{6} x^{6} + 25 \, a^{4} x^{4} - 13 \, a^{2} x^{2} + 2\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (4 \, a^{9} x^{9} - 24 \, a^{7} x^{7} + 39 \, a^{5} x^{5} - 25 \, a^{3} x^{3} + 6 \, a x\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{2 \, {\left (a^{10} x^{12} + {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{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 \, {\left (a^{7} x^{9} - a^{5} x^{7}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + 6 \, {\left (a^{8} x^{10} - 2 \, a^{6} x^{8} + a^{4} x^{6}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 4 \, {\left (a^{9} x^{11} - 3 \, a^{7} x^{9} + 3 \, a^{5} x^{7} - a^{3} x^{5}\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{x^2\,{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(x^2*acosh(a*x)^3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]

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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