\(\int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx\) [273]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.24 (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 = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.80 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.70 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]

[In]

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

[Out]

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