3.370 \(\int \frac{1+x^3}{-x+x^3} \, dx\)

Optimal. Leaf size=12 \[ x+\log (1-x)-\log (x) \]

[Out]

x + Log[1 - x] - Log[x]

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Rubi [A]  time = 0.0293969, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1593, 1802} \[ x+\log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^3)/(-x + x^3),x]

[Out]

x + Log[1 - x] - Log[x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.0044697, size = 12, normalized size = 1. \[ x+\log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^3)/(-x + x^3),x]

[Out]

x + Log[1 - x] - Log[x]

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Maple [A]  time = 0.005, size = 11, normalized size = 0.9 \begin{align*} x+\ln \left ( x-1 \right ) -\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+ln(x-1)-ln(x)

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Maxima [A]  time = 1.05532, size = 14, normalized size = 1.17 \begin{align*} x + \log \left (x - 1\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x - 1) - log(x)

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Fricas [A]  time = 1.71472, size = 34, normalized size = 2.83 \begin{align*} x + \log \left (x - 1\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x - 1) - log(x)

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Sympy [A]  time = 0.089337, size = 8, normalized size = 0.67 \begin{align*} x - \log{\left (x \right )} + \log{\left (x - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - log(x) + log(x - 1)

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Giac [A]  time = 1.12206, size = 16, normalized size = 1.33 \begin{align*} x + \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(abs(x - 1)) - log(abs(x))