\(\int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx\) [325]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x^2 (a+b \text {arccosh}(c x))}+\frac {2 \sqrt {1-c x} \text {Int}\left (\frac {1}{x^3 (a+b \text {arccosh}(c x))},x\right )}{b c \sqrt {-1+c x}} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.96 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.97 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 7.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 427, normalized size of antiderivative = 15.25 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.86 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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