3.3.23 \(\int \frac {-1+x^2}{(1+x^2) \sqrt [3]{x^2+x^4}} \, dx\)

Optimal. Leaf size=23 \[ -\frac {3 \left (x^4+x^2\right )^{2/3}}{x \left (x^2+1\right )} \]

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Rubi [A]  time = 0.08, antiderivative size = 14, normalized size of antiderivative = 0.61, 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 = {2056, 449} \begin {gather*} -\frac {3 x}{\sqrt [3]{x^4+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)/((1 + x^2)*(x^2 + x^4)^(1/3)),x]

[Out]

(-3*x)/(x^2 + x^4)^(1/3)

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 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]

Rubi steps

\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {-1+x^2}{x^{2/3} \left (1+x^2\right )^{4/3}} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {3 x}{\sqrt [3]{x^2+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 14, normalized size = 0.61 \begin {gather*} -\frac {3 x}{\sqrt [3]{x^4+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^2)/((1 + x^2)*(x^2 + x^4)^(1/3)),x]

[Out]

(-3*x)/(x^2 + x^4)^(1/3)

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IntegrateAlgebraic [A]  time = 0.10, size = 23, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^2+x^4\right )^{2/3}}{x \left (1+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(x^2 + x^4)^(1/3)),x]

[Out]

(-3*(x^2 + x^4)^(2/3))/(x*(1 + x^2))

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fricas [A]  time = 0.43, size = 18, normalized size = 0.78 \begin {gather*} -\frac {3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^4+x^2)^(1/3),x, algorithm="fricas")

[Out]

-3*(x^4 + x^2)^(2/3)/(x^3 + x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^4+x^2)^(1/3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.07, size = 13, normalized size = 0.57

method	result	size
gosper	\(-\frac {3 x}{\left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\)	\(13\)
risch	\(-\frac {3 x}{\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}\)	\(15\)
trager	\(-\frac {3 \left (x^{4}+x^{2}\right )^{\frac {2}{3}}}{x \left (x^{2}+1\right )}\)	\(22\)
meijerg	\(-3 \hypergeom \left (\left [\frac {1}{6}, \frac {4}{3}\right ], \left [\frac {7}{6}\right ], -x^{2}\right ) x^{\frac {1}{3}}+\frac {3 \hypergeom \left (\left [\frac {7}{6}, \frac {4}{3}\right ], \left [\frac {13}{6}\right ], -x^{2}\right ) x^{\frac {7}{3}}}{7}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)/(x^2+1)/(x^4+x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^4+x^2)^(1/3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.15, size = 21, normalized size = 0.91 \begin {gather*} -\frac {3\,{\left (x^4+x^2\right )}^{2/3}}{x\,\left (x^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*(x^2 + x^4)^(2/3))/(x*(x^2 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)/(x**2+1)/(x**4+x**2)**(1/3),x)

[Out]

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

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