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

Optimal. Leaf size=18 \[ -\frac{1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]

[Out]

-x + ArcTan[x] + Log[x] - Log[1 + x^2]/2

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Rubi [A]  time = 0.0251952, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1802, 635, 203, 260} \[ -\frac{1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^3)/(x*(1 + x^2)),x]

[Out]

-x + ArcTan[x] + Log[x] - Log[1 + x^2]/2

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]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.0047565, size = 18, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^3)/(x*(1 + x^2)),x]

[Out]

-x + ArcTan[x] + Log[x] - Log[1 + x^2]/2

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Maple [A]  time = 0.006, size = 17, normalized size = 0.9 \begin{align*} -x+\arctan \left ( x \right ) +\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42209, size = 22, normalized size = 1.22 \begin{align*} -x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.05672, size = 59, normalized size = 3.28 \begin{align*} -x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.110263, size = 15, normalized size = 0.83 \begin{align*} - x + \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11354, size = 23, normalized size = 1.28 \begin{align*} -x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\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/(x^2+1),x, algorithm="giac")

[Out]

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