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

Optimal. Leaf size=30 \[ \frac{x^2}{2}+x+\frac{2}{1-x}+\log (1-x)-\log (x+1) \]

[Out]

2/(1 - x) + x + x^2/2 + Log[1 - x] - Log[1 + x]

________________________________________________________________________________________

Rubi [A]  time = 0.0298122, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2074} \[ \frac{x^2}{2}+x+\frac{2}{1-x}+\log (1-x)-\log (x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(1 - x) + x + x^2/2 + Log[1 - x] - Log[1 + x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

Mathematica [A]  time = 0.0158656, size = 29, normalized size = 0.97 \[ \frac{1}{2} (x+1)^2-\frac{2}{x-1}+\log (1-x)-\log (x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(-1 + x) + (1 + x)^2/2 + Log[1 - x] - Log[1 + x]

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 25, normalized size = 0.8 \begin{align*} x+{\frac{{x}^{2}}{2}}-\ln \left ( 1+x \right ) +\ln \left ( -1+x \right ) -2\, \left ( -1+x \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.925685, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{2} \, x^{2} + x - \frac{2}{x - 1} - \log \left (x + 1\right ) + \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.07323, size = 109, normalized size = 3.63 \begin{align*} \frac{x^{3} + x^{2} - 2 \,{\left (x - 1\right )} \log \left (x + 1\right ) + 2 \,{\left (x - 1\right )} \log \left (x - 1\right ) - 2 \, x - 4}{2 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.090176, size = 20, normalized size = 0.67 \begin{align*} \frac{x^{2}}{2} + x + \log{\left (x - 1 \right )} - \log{\left (x + 1 \right )} - \frac{2}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.05145, size = 35, normalized size = 1.17 \begin{align*} \frac{1}{2} \, x^{2} + x - \frac{2}{x - 1} - \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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