\(\int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx\) [415]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.16 (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 = {} \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 10.99 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {c^{2} x^{2} + 1}}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 392, normalized size of antiderivative = 14.52 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

-((c^2*x^2 + 1)^2 + (c^3*x^3 + c*x)*sqrt(c^2*x^2 + 1))/(a*b*c^3*x^3 + sqrt(c^2*x^2 + 1)*a*b*c^2*x^2 + a*b*c*x
+ (b^2*c^3*x^3 + sqrt(c^2*x^2 + 1)*b^2*c^2*x^2 + b^2*c*x)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((c^3*x^3
- 2*c*x)*(c^2*x^2 + 1)^(3/2) + (2*c^4*x^4 - c^2*x^2 - 1)*(c^2*x^2 + 1) + (c^5*x^5 + c^3*x^3)*sqrt(c^2*x^2 + 1)
)/(a*b*c^5*x^6 + (c^2*x^2 + 1)*a*b*c^3*x^4 + 2*a*b*c^3*x^4 + a*b*c*x^2 + (b^2*c^5*x^6 + (c^2*x^2 + 1)*b^2*c^3*
x^4 + 2*b^2*c^3*x^4 + b^2*c*x^2 + 2*(b^2*c^4*x^5 + b^2*c^2*x^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)
) + 2*(a*b*c^4*x^5 + a*b*c^2*x^3)*sqrt(c^2*x^2 + 1)), x)

Giac [F(-2)]

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

[In]

integrate((c^2*x^2+1)^(1/2)/x/(a+b*arcsinh(c*x))^2,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 = 2.82 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {c^2\,x^2+1}}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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