\(\int \frac {x^m \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx\) [311]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {x^m \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^m \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^m \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^m \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{m}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.67 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {x^m \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^{m} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate(x^m*(-c^2*x^2+1)^(1/2)/(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 = 2.71 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^m\,\sqrt {1-c^2\,x^2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]

[In]

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

[Out]

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