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

Optimal. Leaf size=23 \[ \log \left (-x+\frac {1}{4} e^4 x^3 (4+x)+3 \log (\log (4))\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {1587} \begin {gather*} \log \left (-e^4 \left (x^4+4 x^3\right )+4 x-12 \log (\log (4))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[4*x - E^4*(4*x^3 + x^4) - 12*Log[Log[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 (4 x-e^4 \left (4 x^3+x^4\right )-12 \log (\log (4))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.09 \begin {gather*} \log \left (-4 x+4 e^4 x^3+e^4 x^4+12 \log (\log (4))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-4*x + 4*E^4*x^3 + E^4*x^4 + 12*Log[Log[4]]]

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fricas [A]  time = 0.62, size = 24, normalized size = 1.04 \begin {gather*} \log \left ({\left (x^{4} + 4 \, x^{3}\right )} e^{4} - 4 \, x + 12 \, \log \left (2 \, \log \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((x^4 + 4*x^3)*e^4 - 4*x + 12*log(2*log(2)))

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giac [A]  time = 0.21, size = 25, normalized size = 1.09 \begin {gather*} \log \left ({\left | {\left (x^{4} + 4 \, x^{3}\right )} e^{4} - 4 \, x + 12 \, \log \left (2 \, \log \relax (2)\right ) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs((x^4 + 4*x^3)*e^4 - 4*x + 12*log(2*log(2))))

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maple [A]  time = 0.06, size = 27, normalized size = 1.17

method	result	size
derivativedivides	\(\ln \left (12 \ln \left (2 \ln \relax (2)\right )+\left (x^{4}+4 x^{3}\right ) {\mathrm e}^{4}-4 x \right )\)	\(27\)
risch	\(\ln \left (x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}+12 \ln \relax (2)+12 \ln \left (\ln \relax (2)\right )-4 x \right )\)	\(28\)
default	\(\ln \left (x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}+12 \ln \left (2 \ln \relax (2)\right )-4 x \right )\)	\(30\)
norman	\(\ln \left (x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}+12 \ln \left (2 \ln \relax (2)\right )-4 x \right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(12*ln(2*ln(2))+(x^4+4*x^3)*exp(2)^2-4*x)

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maxima [A]  time = 0.35, size = 25, normalized size = 1.09 \begin {gather*} \log \left (x^{4} e^{4} + 4 \, x^{3} e^{4} - 4 \, x + 12 \, \log \left (2 \, \log \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^4*e^4 + 4*x^3*e^4 - 4*x + 12*log(2*log(2)))

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mupad [B]  time = 0.22, size = 23, normalized size = 1.00 \begin {gather*} \ln \left ({\mathrm {e}}^4\,x^4+4\,{\mathrm {e}}^4\,x^3-4\,x+\ln \left ({\ln \relax (4)}^{12}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(log(4)^12) - 4*x + 4*x^3*exp(4) + x^4*exp(4))

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sympy [A]  time = 1.64, size = 31, normalized size = 1.35 \begin {gather*} \log {\left (x^{4} e^{4} + 4 x^{3} e^{4} - 4 x + 12 \log {\left (\log {\relax (2 )} \right )} + 12 \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**4*exp(4) + 4*x**3*exp(4) - 4*x + 12*log(log(2)) + 12*log(2))

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