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

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

[Out]

-ln(1-x)

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 31

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

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*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ -\log (1-x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - x]

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fricas [A]  time = 0.74, size = 6, normalized size = 0.75 \[ -\log \left (x - 1\right ) \]

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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giac [A]  time = 0.15, size = 7, normalized size = 0.88 \[ -\log \left ({\left | x - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.04, size = 7, normalized size = 0.88 \[ -\ln \left (x -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x-1)

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maxima [A]  time = 1.35, size = 6, normalized size = 0.75 \[ -\log \left (x - 1\right ) \]

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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mupad [B]  time = 0.02, size = 6, normalized size = 0.75 \[ -\ln \left (x-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + x^2 + x^3 + 1)/(x^4 - 1),x)

[Out]

-log(x - 1)

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sympy [A]  time = 0.07, size = 5, normalized size = 0.62 \[ - \log {\left (x - 1 \right )} \]

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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