\(\int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx\) [360]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx=-\frac {\text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b}+\text {Int}\left (\frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\sqrt {c^{2} x^{2} + 1}}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((c^2*x^2+1)^(1/2)/x/(a+b*arcsinh(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 = 2.55 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\sqrt {c^2\,x^2+1}}{x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \]

[In]

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

[Out]

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