\(\int \frac {(1-c^2 x^2)^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx\) [333]

Optimal result

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

[Out]

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

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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [F(-1)]

Timed out. \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx=\text {\$Aborted} \]

[In]

Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])^2),x]

[Out]

$Aborted

Maple [N/A] (verified)

Not integrable

Time = 1.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]

[In]

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

[Out]

int((-c^2*x^2+1)^(3/2)/x^3/(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 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{3}} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(3/2)/x^3/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 176.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

integrate((-c**2*x**2+1)**(3/2)/x**3/(a+b*acosh(c*x))**2,x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(3/2)/(x**3*(a + b*acosh(c*x))**2), x)

Maxima [N/A]

Not integrable

Time = 0.92 (sec) , antiderivative size = 483, normalized size of antiderivative = 17.25 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{3}} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(3/2)/x^3/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*x^2+1)^(3/2)/x^3/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 3.39 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int((1 - c^2*x^2)^(3/2)/(x^3*(a + b*acosh(c*x))^2),x)

[Out]

int((1 - c^2*x^2)^(3/2)/(x^3*(a + b*acosh(c*x))^2), x)