3.17.85 $\int \frac{3-8x+x^2-2x^3}{3+x^2}dx$

Optimal. Leaf size=19 $$2+x-\log \left(e^{4+x^2}(3+x^2)\right)$$

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.79, 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} $$\begin{array}{c}-x^2-\log (x^2+3)+x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(3 - 8*x + x^2 - 2*x^3)/(3 + x^2),x]

[Out]

x - x^2 - Log[3 + x^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]

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]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (1-2x-\frac{2x}{3+x^2})dx\\ & =x-x^2-2\int \frac{x}{3+x^2}dx\\ & =x-x^2-\log (3+x^2)\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.79 $$\begin{array}{c}x-x^2-\log (3+x^2)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(3 - 8*x + x^2 - 2*x^3)/(3 + x^2),x]

[Out]

x - x^2 - Log[3 + x^2]

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fricas [A]  time = 0.81, size = 15, normalized size = 0.79 $$\begin{array}{c}-x^2+x-\log (x^2+3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+x^2-8*x+3)/(x^2+3),x, algorithm="fricas")

[Out]

-x^2 + x - log(x^2 + 3)

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giac [A]  time = 0.23, size = 15, normalized size = 0.79 $$\begin{array}{c}-x^2+x-\log (x^2+3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+x^2-8*x+3)/(x^2+3),x, algorithm="giac")

[Out]

-x^2 + x - log(x^2 + 3)

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maple [A]  time = 0.30, size = 16, normalized size = 0.84

method	result	size
default	$x-x^2-\ln (x^2+3)$	$16$
norman	$x-x^2-\ln (x^2+3)$	$16$
risch	$x-x^2-\ln (x^2+3)$	$16$
meijerg	$\sqrt{3}\arctan \left(\frac{x\sqrt{3}}{3}\right)-x^2-\ln (1+\frac{x^2}{3})+\frac{\sqrt{3}(\frac{2x\sqrt{3}}{3}-2\arctan \left(\frac{x\sqrt{3}}{3}\right))}{2}$	$49$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^3+x^2-8*x+3)/(x^2+3),x,method=_RETURNVERBOSE)

[Out]

x-x^2-ln(x^2+3)

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maxima [A]  time = 0.37, size = 15, normalized size = 0.79 $$\begin{array}{c}-x^2+x-\log (x^2+3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+x^2-8*x+3)/(x^2+3),x, algorithm="maxima")

[Out]

-x^2 + x - log(x^2 + 3)

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mupad [B]  time = 1.07, size = 15, normalized size = 0.79 $$\begin{array}{c}x-\ln (x^2+3)-x^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x - x^2 + 2*x^3 - 3)/(x^2 + 3),x)

[Out]

x - log(x^2 + 3) - x^2

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sympy [A]  time = 0.11, size = 10, normalized size = 0.53 $$\begin{array}{c}-x^2+x-\log (x^2+3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3+x**2-8*x+3)/(x**2+3),x)

[Out]

-x**2 + x - log(x**2 + 3)

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