\(\int \frac {(1+c^2 x^2)^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx\) [424]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.18 (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 = {} \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.68 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

integral((c^2*x^2 + 1)^(3/2)/(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 = 3.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 443, normalized size of antiderivative = 16.41 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

-((c^4*x^4 + 2*c^2*x^2 + 1)*(c^2*x^2 + 1) + (c^5*x^5 + 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 + sq
rt(c^2*x^2 + 1))) + integrate(((2*c^5*x^5 - c^3*x^3 - 3*c*x)*(c^2*x^2 + 1)^(3/2) + 2*(2*c^6*x^6 + c^4*x^4 - 2*
c^2*x^2 - 1)*(c^2*x^2 + 1) + (2*c^7*x^7 + 3*c^5*x^5 - 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 = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.74 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {{\left (c^2\,x^2+1\right )}^{3/2}}{x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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