3.888 \(\int \frac{-1+x^2}{(1+x^2) \sqrt{x+x^3}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{2 x}{\sqrt{x^3+x}} \]

[Out]

(-2*x)/Sqrt[x + x^3]

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Rubi [A]  time = 0.0701814, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2056, 449} \[ -\frac{2 x}{\sqrt{x^3+x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[x + x^3]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, 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]

Rubi steps

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

Mathematica [A]  time = 0.0103464, size = 12, normalized size = 1. \[ -\frac{2 x}{\sqrt{x^3+x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[x + x^3]

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Maple [A]  time = 0.004, size = 11, normalized size = 0.9 \begin{align*} -2\,{\frac{x}{\sqrt{{x}^{3}+x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)/(x^2+1)/(x^3+x)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^3+x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.50459, size = 38, normalized size = 3.17 \begin{align*} -\frac{2 \, \sqrt{x^{3} + x}}{x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^3+x)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(x^3 + x)/(x^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)/(x**2+1)/(x**3+x)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^3+x)^(1/2),x, algorithm="giac")

[Out]

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