\(\int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx\) [417]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\text {Int}\left (\frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2},x\right ) \]

[Out]

Unintegrable((c^2*x^2+1)^(1/2)/x^3/(a+b*arcsinh(c*x))^2,x)

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Int[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])^2),x]

[Out]

Defer[Int][Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 16.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Integrate[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

\[\int \frac {\sqrt {c^{2} x^{2}+1}}{x^{3} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}d x\]

[In]

int((c^2*x^2+1)^(1/2)/x^3/(a+b*arcsinh(c*x))^2,x)

[Out]

int((c^2*x^2+1)^(1/2)/x^3/(a+b*arcsinh(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}} \,d x } \]

[In]

integrate((c^2*x^2+1)^(1/2)/x^3/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)/(b^2*x^3*arcsinh(c*x)^2 + 2*a*b*x^3*arcsinh(c*x) + a^2*x^3), x)

Sympy [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {c^{2} x^{2} + 1}}{x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

integrate((c**2*x**2+1)**(1/2)/x**3/(a+b*asinh(c*x))**2,x)

[Out]

Integral(sqrt(c**2*x**2 + 1)/(x**3*(a + b*asinh(c*x))**2), x)

Maxima [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 403, normalized size of antiderivative = 14.93 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}} \,d x } \]

[In]

integrate((c^2*x^2+1)^(1/2)/x^3/(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^5 + sqrt(c^2*x^2 + 1)*a*b*c^2*x^4 + a*b*c*x^
3 + (b^2*c^3*x^5 + sqrt(c^2*x^2 + 1)*b^2*c^2*x^4 + b^2*c*x^3)*log(c*x + sqrt(c^2*x^2 + 1))) - integrate(((c^3*
x^3 + 4*c*x)*(c^2*x^2 + 1)^(3/2) + (2*c^4*x^4 + 7*c^2*x^2 + 3)*(c^2*x^2 + 1) + (c^5*x^5 + 3*c^3*x^3 + 2*c*x)*s
qrt(c^2*x^2 + 1))/(a*b*c^5*x^8 + (c^2*x^2 + 1)*a*b*c^3*x^6 + 2*a*b*c^3*x^6 + a*b*c*x^4 + (b^2*c^5*x^8 + (c^2*x
^2 + 1)*b^2*c^3*x^6 + 2*b^2*c^3*x^6 + b^2*c*x^4 + 2*(b^2*c^4*x^7 + b^2*c^2*x^5)*sqrt(c^2*x^2 + 1))*log(c*x + s
qrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^7 + a*b*c^2*x^5)*sqrt(c^2*x^2 + 1)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*x^2+1)^(1/2)/x^3/(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 = 2.80 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {c^2\,x^2+1}}{x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int((c^2*x^2 + 1)^(1/2)/(x^3*(a + b*asinh(c*x))^2),x)

[Out]

int((c^2*x^2 + 1)^(1/2)/(x^3*(a + b*asinh(c*x))^2), x)