3.889 \(\int \frac{\sqrt{\frac{(-1+x^2)^2}{x (1+x^2)}}}{1+x^2} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x \left (x^2+1\right )}}}{1-x^2} \]

[Out]

(2*x*Sqrt[(1 - x^2)^2/(x*(1 + x^2))])/(1 - x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.139978, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {6718, 449} \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x \left (x^2+1\right )}}}{1-x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(-1 + x^2)^2/(x*(1 + x^2))]/(1 + x^2),x]

[Out]

(2*x*Sqrt[(1 - x^2)^2/(x*(1 + x^2))])/(1 - x^2)

Rule 6718

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n*z^q)^FracP
art[p])/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p])), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !FreeQ[w, x] &&  !FreeQ[z, x]

Rule 449

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{\frac{\left (-1+x^2\right )^2}{x \left (1+x^2\right )}}}{1+x^2} \, dx &=\frac{\left (\sqrt{x} \sqrt{\frac{\left (-1+x^2\right )^2}{x \left (1+x^2\right )}} \sqrt{1+x^2}\right ) \int \frac{-1+x^2}{\sqrt{x} \left (1+x^2\right )^{3/2}} \, dx}{-1+x^2}\\ &=\frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x \left (1+x^2\right )}}}{1-x^2}\\ \end{align*}

Mathematica [A]  time = 0.0196528, size = 29, normalized size = 0.81 \[ -\frac{2 x \sqrt{\frac{\left (x^2-1\right )^2}{x^3+x}}}{x^2-1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-1 + x^2)^2/(x*(1 + x^2))]/(1 + x^2),x]

[Out]

(-2*x*Sqrt[(-1 + x^2)^2/(x + x^3)])/(-1 + x^2)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 34, normalized size = 0.9 \begin{align*} -2\,{\frac{x}{ \left ( x-1 \right ) \left ( 1+x \right ) }\sqrt{{\frac{ \left ({x}^{2}-1 \right ) ^{2}}{x \left ({x}^{2}+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-1)^2/x/(x^2+1))^(1/2)/(x^2+1),x)

[Out]

-2*x/(x-1)/(1+x)*((x^2-1)^2/x/(x^2+1))^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{{\left (x^{2} + 1\right )} x}}}{x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-1)^2/x/(x^2+1))^(1/2)/(x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt((x^2 - 1)^2/((x^2 + 1)*x))/(x^2 + 1), x)

________________________________________________________________________________________

Fricas [A]  time = 1.45234, size = 68, normalized size = 1.89 \begin{align*} -\frac{2 \, x \sqrt{\frac{x^{4} - 2 \, x^{2} + 1}{x^{3} + x}}}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-1)^2/x/(x^2+1))^(1/2)/(x^2+1),x, algorithm="fricas")

[Out]

-2*x*sqrt((x^4 - 2*x^2 + 1)/(x^3 + x))/(x^2 - 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}{x^{3} + x}}}{x^{2} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-1)**2/x/(x**2+1))**(1/2)/(x**2+1),x)

[Out]

Integral(sqrt((x - 1)**2*(x + 1)**2/(x**3 + x))/(x**2 + 1), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{{\left (x^{2} + 1\right )} x}}}{x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-1)^2/x/(x^2+1))^(1/2)/(x^2+1),x, algorithm="giac")

[Out]

integrate(sqrt((x^2 - 1)^2/((x^2 + 1)*x))/(x^2 + 1), x)