\(\int \frac {1}{(c-a^2 c x^2)^2 \text {arccosh}(a x)^2} \, dx\) [319]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx=-\frac {1}{a c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}-\frac {3 a \text {Int}\left (\frac {x}{(-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)},x\right )}{c^2} \]

[Out]

-1/a/c^2/(a*x-1)^(3/2)/(a*x+1)^(3/2)/arccosh(a*x)-3*a*Unintegrable(x/(a*x-1)^(5/2)/(a*x+1)^(5/2)/arccosh(a*x),
x)/c^2

Rubi [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx \]

[In]

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

[Out]

-(1/(a*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])) - (3*a*Defer[Int][x/((-1 + a*x)^(5/2)*(1 + a*x)^(5/
2)*ArcCosh[a*x]), x])/c^2

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}-\frac {(3 a) \int \frac {x}{(-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)} \, dx}{c^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 8.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{2} \operatorname {arccosh}\left (a x \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 12.41 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {acosh}^{2}{\left (a x \right )} - 2 a^{2} x^{2} \operatorname {acosh}^{2}{\left (a x \right )} + \operatorname {acosh}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 351, normalized size of antiderivative = 17.55 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

-(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))/((a^5*c^2*x^4 - 2*a^3*c^2*x^2 + a*c^2 + (a^4*c^2*x^3 - a^2*c^2*x)*sqrt(a*
x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) - integrate((3*a^4*x^4 - 2*a^2*x^2 + (3*a^2*x^2
- 1)*(a*x + 1)*(a*x - 1) + 3*(2*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - 1)/((a^8*c^2*x^8 - 4*a^6*c^2*x^6
+ 6*a^4*c^2*x^4 - 4*a^2*c^2*x^2 + (a^6*c^2*x^6 - 2*a^4*c^2*x^4 + a^2*c^2*x^2)*(a*x + 1)*(a*x - 1) + 2*(a^7*c^2
*x^7 - 3*a^5*c^2*x^5 + 3*a^3*c^2*x^3 - a*c^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + c^2)*log(a*x + sqrt(a*x + 1)*sqr
t(a*x - 1))), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^2\,{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]

[In]

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

[Out]

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