3.438 \(\int \frac {-x+2 x^3}{1-x^2+x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac {1}{2} \log \left (x^4-x^2+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1587} \[ \frac {1}{2} \log \left (x^4-x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-x + 2*x^3)/(1 - x^2 + x^4),x]

[Out]

Log[1 - x^2 + x^4]/2

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {1}{2} \log \left (x^4-x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 - x^2 + x^4]/2

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fricas [A]  time = 0.47, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^4 - x^2 + 1)

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giac [A]  time = 0.28, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^4 - x^2 + 1)

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \frac {\ln \left (x^{4}-x^{2}+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.65, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^4 - x^2 + 1)

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mupad [B]  time = 0.04, size = 13, normalized size = 0.87 \[ \frac {\ln \left (x^4-x^2+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 2*x^3)/(x^4 - x^2 + 1),x)

[Out]

log(x^4 - x^2 + 1)/2

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sympy [A]  time = 0.09, size = 10, normalized size = 0.67 \[ \frac {\log {\left (x^{4} - x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**4 - x**2 + 1)/2

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