### 3.196 $$\int \frac{-x+2 x^3}{1-x^2+x^4} \, dx$$

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

[Out]

Log[1 - x^2 + x^4]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0092598, antiderivative size = 15, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0061354, size = 15, normalized size = 1. $\frac{1}{2} \log \left (x^4-x^2+1\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[1 - x^2 + x^4]/2

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{2}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 0.924786, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.81189, size = 32, normalized size = 2.13 \begin{align*} \frac{1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.083604, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{2} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.06469, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \end{align*}

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)