\(\int \frac {x^m (1-c^2 x^2)^{3/2}}{a+b \text {arccosh}(c x)} \, dx\) [310]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate(x^m*(-c^2*x^2+1)^(3/2)/(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 = 2.74 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^m\,{\left (1-c^2\,x^2\right )}^{3/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]

[In]

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

[Out]

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