\(\int \frac {(1-c^2 x^2)^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx\) [282]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.10 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

Defer[Int][(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])), 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))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 8.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 27.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((-c^2*x^2+1)^(3/2)/x^3/(a+b*arccosh(c*x)),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 = 3.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 (a+b \text {arccosh}(c x))} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]

[In]

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

[Out]

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