\(\int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx\) [327]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx=\text {Int}\left (\frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

Int[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])^2),x]

[Out]

Defer[Int][Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.62 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 1.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\sqrt {-c^{2} x^{2}+1}}{x^{4} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]

[In]

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

[Out]

int((-c^2*x^2+1)^(1/2)/x^4/(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 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 48.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 456, normalized size of antiderivative = 16.29 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(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^6 + sqrt(c*
x + 1)*sqrt(c*x - 1)*a*b*c^2*x^5 - a*b*c*x^4 + (b^2*c^3*x^6 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x^5 - b^2*c*
x^4)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate(((2*c^3*x^3 - 5*c*x)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(2
*c^4*x^4 - 5*c^2*x^2 + 2)*(c*x + 1)*sqrt(c*x - 1) + (2*c^5*x^5 - 5*c^3*x^3 + 3*c*x)*sqrt(c*x + 1))*sqrt(-c*x +
1)/(a*b*c^5*x^9 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^7 - 2*a*b*c^3*x^7 + a*b*c*x^5 + 2*(a*b*c^4*x^8 - a*b*c^2*x^6)
*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^9 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^7 - 2*b^2*c^3*x^7 + b^2*c*x^5 + 2*
(b^2*c^4*x^8 - b^2*c^2*x^6)*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 = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate(sqrt(-c^2*x^2 + 1)/((b*arccosh(c*x) + a)^2*x^4), x)

Mupad [N/A]

Not integrable

Time = 3.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{x^4\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int((1 - c^2*x^2)^(1/2)/(x^4*(a + b*acosh(c*x))^2),x)

[Out]

int((1 - c^2*x^2)^(1/2)/(x^4*(a + b*acosh(c*x))^2), x)