3.54.15 \(\int \frac {8 x-12 x^2+4 x^3}{e+4 x^2-4 x^3+x^4} \, dx\)

Optimal. Leaf size=12 \[ \log \left (e+\left (-2 x+x^2\right )^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.33, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1587} \begin {gather*} \log \left (x^4-4 x^3+4 x^2+e\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*x - 12*x^2 + 4*x^3)/(E + 4*x^2 - 4*x^3 + x^4),x]

[Out]

Log[E + 4*x^2 - 4*x^3 + x^4]

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 {gather*} \begin {aligned} \text {integral} &=\log \left (e+4 x^2-4 x^3+x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} \log \left (e+(-2+x)^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x - 12*x^2 + 4*x^3)/(E + 4*x^2 - 4*x^3 + x^4),x]

[Out]

Log[E + (-2 + x)^2*x^2]

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fricas [A]  time = 0.46, size = 17, normalized size = 1.42 \begin {gather*} \log \left (x^{4} - 4 \, x^{3} + 4 \, x^{2} + e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-12*x^2+8*x)/(exp(1)+x^4-4*x^3+4*x^2),x, algorithm="fricas")

[Out]

log(x^4 - 4*x^3 + 4*x^2 + e)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-12*x^2+8*x)/(exp(1)+x^4-4*x^3+4*x^2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.06, size = 18, normalized size = 1.50

method	result	size
derivativedivides	\(\ln \left ({\mathrm e}+x^{4}-4 x^{3}+4 x^{2}\right )\)	\(18\)
default	\(\ln \left ({\mathrm e}+x^{4}-4 x^{3}+4 x^{2}\right )\)	\(18\)
norman	\(\ln \left ({\mathrm e}+x^{4}-4 x^{3}+4 x^{2}\right )\)	\(18\)
risch	\(\ln \left ({\mathrm e}+x^{4}-4 x^{3}+4 x^{2}\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3-12*x^2+8*x)/(exp(1)+x^4-4*x^3+4*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(exp(1)+x^4-4*x^3+4*x^2)

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maxima [A]  time = 0.37, size = 17, normalized size = 1.42 \begin {gather*} \log \left (x^{4} - 4 \, x^{3} + 4 \, x^{2} + e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-12*x^2+8*x)/(exp(1)+x^4-4*x^3+4*x^2),x, algorithm="maxima")

[Out]

log(x^4 - 4*x^3 + 4*x^2 + e)

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mupad [B]  time = 0.06, size = 17, normalized size = 1.42 \begin {gather*} \ln \left (x^4-4\,x^3+4\,x^2+\mathrm {e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - 12*x^2 + 4*x^3)/(exp(1) + 4*x^2 - 4*x^3 + x^4),x)

[Out]

log(exp(1) + 4*x^2 - 4*x^3 + x^4)

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sympy [A]  time = 0.15, size = 17, normalized size = 1.42 \begin {gather*} \log {\left (x^{4} - 4 x^{3} + 4 x^{2} + e \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3-12*x**2+8*x)/(exp(1)+x**4-4*x**3+4*x**2),x)

[Out]

log(x**4 - 4*x**3 + 4*x**2 + E)

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