\(\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))} \, dx\) [272]

Optimal result

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

[Out]

-Chi((a+b*arccosh(c*x))/b)*cosh(a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)+Shi((a+b*arccosh(c*x))/b)*sinh(a/b)*(c*x-1
)^(1/2)/b/(-c*x+1)^(1/2)+Unintegrable(1/x/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.31 (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 = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.97 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((-c^2*x^2+1)^(1/2)/x/(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 = 3.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]

[In]

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

[Out]

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