\(\int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx\) [416]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 378, normalized size of antiderivative = 14.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.78 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {c^2\,x^2+1}}{x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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