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

Optimal. Leaf size=8 \[ -\log (1-x) \]

[Out]

-Log[1 - x]

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Rubi [A]  time = 0.0074175, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1586, 31} \[ -\log (1-x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x + x^2 + x^3)/(1 - x^4),x]

[Out]

-Log[1 - x]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0008638, size = 8, normalized size = 1. \[ -\log (1-x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - x]

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Maple [A]  time = 0., size = 7, normalized size = 0.9 \begin{align*} -\ln \left ( -1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(-1+x)

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Maxima [A]  time = 0.943333, size = 8, normalized size = 1. \begin{align*} -\log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 1)

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Fricas [A]  time = 1.70181, size = 18, normalized size = 2.25 \begin{align*} -\log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 1)

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Sympy [A]  time = 0.056644, size = 5, normalized size = 0.62 \begin{align*} - \log{\left (x - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 1)

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Giac [A]  time = 1.05278, size = 9, normalized size = 1.12 \begin{align*} -\log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(x - 1))