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

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Rubi [A]  time = 0.14, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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)

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fricas [A]  time = 0.42, size = 30, normalized size = 0.83 \[ -\frac {2 \, x \sqrt {\frac {x^{4} - 2 \, x^{2} + 1}{x^{3} + x}}}{x^{2} - 1} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {{\left (x^{2} - 1\right )}^{2}}{{\left (x^{2} + 1\right )} x}}}{x^{2} + 1}\,{d x} \]

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)

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maple [A]  time = 0.01, size = 34, normalized size = 0.94 \[ -\frac {2 \sqrt {\frac {\left (x^{2}-1\right )^{2}}{\left (x^{2}+1\right ) x}}\, x}{\left (x -1\right ) \left (x +1\right )} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {{\left (x^{2} - 1\right )}^{2}}{{\left (x^{2} + 1\right )} x}}}{x^{2} + 1}\,{d x} \]

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)

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mupad [B]  time = 3.49, size = 48, normalized size = 1.33 \[ -\frac {\left (2\,x^3+2\,x\right )\,\sqrt {\frac {1}{x^2+1}}\,\sqrt {{\left (x^2-1\right )}^2}\,\sqrt {\frac {1}{x}}}{\left (x^2-1\right )\,\left (x^2+1\right )} \]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {\left (x - 1\right )^{2} \left (x + 1\right )^{2}}{x^{3} + x}}}{x^{2} + 1}\, dx \]

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)

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