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

Optimal. Leaf size=14 \[ -\frac{2 x}{\sqrt{x \left (x^2+1\right )}} \]

[Out]

(-2*x)/Sqrt[x*(1 + x^2)]

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Rubi [A]  time = 0.149441, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6719, 449} \[ -\frac{2 x}{\sqrt{x \left (x^2+1\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[x*(1 + x^2)]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, 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{-1+x^2}{\left (1+x^2\right ) \sqrt{x \left (1+x^2\right )}} \, 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 \left (1+x^2\right )}}\\ &=-\frac{2 x}{\sqrt{x \left (1+x^2\right )}}\\ \end{align*}

Mathematica [A]  time = 0.0257639, size = 12, normalized size = 0.86 \[ -\frac{2 x}{\sqrt{x^3+x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[x + x^3]

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Maple [A]  time = 0.006, size = 13, normalized size = 0.9 \begin{align*} -2\,{\frac{x}{\sqrt{x \left ({x}^{2}+1 \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]  time = 1.45797, size = 38, normalized size = 2.71 \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*(x^2+1))^(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*(x**2+1))**(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{{\left (x^{2} + 1\right )} 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*(x^2+1))^(1/2),x, algorithm="giac")

[Out]

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