3.17.64 $\int \frac{-6+16x-12x^2}{-21+3x-4x^2+2x^3}dx$

Optimal. Leaf size=24 $$\log \left(\frac{1}{{(-1-x+2(4+\frac{1}{3}(2-x)x^2))}^2}\right)$$

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.75, number of steps used = 1, number of rules used = 1, integrand size = 28, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.036, Rules used = {1587} $$\begin{array}{c}-2\log (-2x^3+4x^2-3x+21)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-6 + 16*x - 12*x^2)/(-21 + 3*x - 4*x^2 + 2*x^3),x]

[Out]

-2*Log[21 - 3*x + 4*x^2 - 2*x^3]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =-2\log (21-3x+4x^2-2x^3)\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.75 $$\begin{array}{c}-2\log (21-3x+4x^2-2x^3)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 16*x - 12*x^2)/(-21 + 3*x - 4*x^2 + 2*x^3),x]

[Out]

-2*Log[21 - 3*x + 4*x^2 - 2*x^3]

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fricas [A]  time = 0.89, size = 18, normalized size = 0.75 $$\begin{array}{c}-2\log (2x^3-4x^2+3x-21)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^2+16*x-6)/(2*x^3-4*x^2+3*x-21),x, algorithm="fricas")

[Out]

-2*log(2*x^3 - 4*x^2 + 3*x - 21)

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giac [A]  time = 0.25, size = 19, normalized size = 0.79 $$\begin{array}{c}-2\log \left(|2x^3-4x^2+3x-21|\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^2+16*x-6)/(2*x^3-4*x^2+3*x-21),x, algorithm="giac")

[Out]

-2*log(abs(2*x^3 - 4*x^2 + 3*x - 21))

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maple [A]  time = 0.02, size = 19, normalized size = 0.79

method	result	size
default	$-2\ln (2x^3-4x^2+3x-21)$	$19$
norman	$-2\ln (2x^3-4x^2+3x-21)$	$19$
risch	$-2\ln (2x^3-4x^2+3x-21)$	$19$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*x^2+16*x-6)/(2*x^3-4*x^2+3*x-21),x,method=_RETURNVERBOSE)

[Out]

-2*ln(2*x^3-4*x^2+3*x-21)

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maxima [A]  time = 0.42, size = 18, normalized size = 0.75 $$\begin{array}{c}-2\log (2x^3-4x^2+3x-21)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^2+16*x-6)/(2*x^3-4*x^2+3*x-21),x, algorithm="maxima")

[Out]

-2*log(2*x^3 - 4*x^2 + 3*x - 21)

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mupad [B]  time = 0.07, size = 16, normalized size = 0.67 $$\begin{array}{c}-2\ln (x^3-2x^2+\frac{3x}{2}-\frac{21}{2})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x^2 - 16*x + 6)/(3*x - 4*x^2 + 2*x^3 - 21),x)

[Out]

-2*log((3*x)/2 - 2*x^2 + x^3 - 21/2)

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sympy [A]  time = 0.09, size = 19, normalized size = 0.79 $$\begin{array}{c}-2\log (2x^3-4x^2+3x-21)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x**2+16*x-6)/(2*x**3-4*x**2+3*x-21),x)

[Out]

-2*log(2*x**3 - 4*x**2 + 3*x - 21)

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