\(\int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx\) [64]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx=\text {Int}\left (\frac {1}{x^2 \text {arccosh}(a x)^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx=\int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx \]

[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*} \text {integral}& = \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx=\int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx \]

[In]

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

[Out]

Integrate[1/(x^2*ArcCosh[a*x]^3), x]

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{2} \operatorname {arccosh}\left (a x \right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx=\int { \frac {1}{x^{2} \operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

integral(1/(x^2*arccosh(a*x)^3), x)

Sympy [N/A]

Not integrable

Time = 2.61 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx=\int \frac {1}{x^{2} \operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(x**2*acosh(a*x)**3), x)

Maxima [N/A]

Not integrable

Time = 1.10 (sec) , antiderivative size = 883, normalized size of antiderivative = 88.30 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx=\int { \frac {1}{x^{2} \operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx=\int { \frac {1}{x^{2} \operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

integrate(1/(x^2*arccosh(a*x)^3), x)

Mupad [N/A]

Not integrable

Time = 2.66 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^3} \, dx=\int \frac {1}{x^2\,{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]

[In]

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

[Out]

int(1/(x^2*acosh(a*x)^3), x)