\(\int \frac {x^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\) [312]

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 = 1.02 (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 = 1.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[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 = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {x^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x^{m}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

[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/(a*c^2*x^2 + (b*c^2*x^2 - b)*arccosh(c*x) - a), x)

Sympy [N/A]

Not integrable

Time = 2.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {x^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x^{m}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x^{m}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x^m}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]

[In]

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

[Out]

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