\(\int \frac {1}{(c-a^2 c x^2)^2 \text {arccosh}(a x)} \, dx\) [266]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \operatorname {arcosh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {acosh}{\left (a x \right )} - 2 a^{2} x^{2} \operatorname {acosh}{\left (a x \right )} + \operatorname {acosh}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \operatorname {arcosh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \operatorname {arcosh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)} \, dx=\int \frac {1}{\mathrm {acosh}\left (a\,x\right )\,{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]

[In]

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

[Out]

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