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

Optimal. Leaf size=21 \[ \frac{x^2}{2}+\frac{1}{2} \log \left (x^2+1\right )-3 x \]

[Out]

-3*x + x^2/2 + Log[1 + x^2]/2

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Rubi [A]  time = 0.0145165, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {1810, 260} \[ \frac{x^2}{2}+\frac{1}{2} \log \left (x^2+1\right )-3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3*x + x^2/2 + Log[1 + x^2]/2

Rule 1810

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

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

Mathematica [A]  time = 0.0044459, size = 21, normalized size = 1. \[ \frac{x^2}{2}+\frac{1}{2} \log \left (x^2+1\right )-3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3*x + x^2/2 + Log[1 + x^2]/2

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Maple [A]  time = 0.001, size = 18, normalized size = 0.9 \begin{align*} -3\,x+{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47398, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} - 3 \, x + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 3*x + 1/2*log(x^2 + 1)

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Fricas [A]  time = 1.43708, size = 46, normalized size = 2.19 \begin{align*} \frac{1}{2} \, x^{2} - 3 \, x + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 3*x + 1/2*log(x^2 + 1)

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Sympy [A]  time = 0.076529, size = 15, normalized size = 0.71 \begin{align*} \frac{x^{2}}{2} - 3 x + \frac{\log{\left (x^{2} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 - 3*x + log(x**2 + 1)/2

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Giac [A]  time = 1.09907, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} - 3 \, x + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 3*x + 1/2*log(x^2 + 1)