\(\int \frac {1}{(c-a^2 c x^2) \text {arccosh}(a x)^2} \, dx\) [318]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.18 (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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.63 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx=- \frac {\int \frac {1}{a^{2} x^{2} \operatorname {acosh}^{2}{\left (a x \right )} - \operatorname {acosh}^{2}{\left (a x \right )}}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 259, normalized size of antiderivative = 12.95 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.70 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^2\,\left (c-a^2\,c\,x^2\right )} \,d x \]

[In]

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

[Out]

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