3.260 \(\int \frac {1+2 x^2+x^5}{-x+x^3} \, dx\)

Optimal. Leaf size=25 \[ \frac {x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]

[Out]

x+1/3*x^3+2*ln(1-x)-ln(x)+ln(1+x)

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1593, 1802} \[ \frac {x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2 + x^5)/(-x + x^3),x]

[Out]

x + x^3/3 + 2*Log[1 - x] - Log[x] + Log[1 + 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+2 x^2+x^5}{-x+x^3} \, dx &=\int \frac {1+2 x^2+x^5}{x \left (-1+x^2\right )} \, dx\\ &=\int \left (1+\frac {2}{-1+x}-\frac {1}{x}+x^2+\frac {1}{1+x}\right ) \, dx\\ &=x+\frac {x^3}{3}+2 \log (1-x)-\log (x)+\log (1+x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ \frac {x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^2 + x^5)/(-x + x^3),x]

[Out]

x + x^3/3 + 2*Log[1 - x] - Log[x] + Log[1 + x]

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fricas [A]  time = 0.76, size = 21, normalized size = 0.84 \[ \frac {1}{3} \, x^{3} + x + \log \left (x + 1\right ) + 2 \, \log \left (x - 1\right ) - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+2*x^2+1)/(x^3-x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.25, size = 24, normalized size = 0.96 \[ \frac {1}{3} \, x^{3} + x + \log \left ({\left | x + 1 \right |}\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+2*x^2+1)/(x^3-x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 22, normalized size = 0.88 \[ \frac {x^{3}}{3}+x -\ln \relax (x )+2 \ln \left (x -1\right )+\ln \left (x +1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+2*x^2+1)/(x^3-x),x)

[Out]

1/3*x^3+x+2*ln(x-1)+ln(x+1)-ln(x)

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maxima [A]  time = 1.15, size = 21, normalized size = 0.84 \[ \frac {1}{3} \, x^{3} + x + \log \left (x + 1\right ) + 2 \, \log \left (x - 1\right ) - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+2*x^2+1)/(x^3-x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 30, normalized size = 1.20 \[ x+2\,\ln \left (x-1\right )+\frac {x^3}{3}+\mathrm {atan}\left (\frac {48{}\mathrm {i}}{11\,\left (22\,x-2\right )}+\frac {13}{11}{}\mathrm {i}\right )\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^2 + x^5 + 1)/(x - x^3),x)

[Out]

x + 2*log(x - 1) + atan(48i/(11*(22*x - 2)) + 13i/11)*2i + x^3/3

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sympy [A]  time = 0.13, size = 20, normalized size = 0.80 \[ \frac {x^{3}}{3} + x - \log {\relax (x )} + 2 \log {\left (x - 1 \right )} + \log {\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+2*x**2+1)/(x**3-x),x)

[Out]

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

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