3.42.52 \(\int \frac {4 x^2+e^{625 x^6} (2-7500 x^6)}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} (-12 x^2-6 x \log (4))} \, dx\)

Optimal. Leaf size=25 \[ \frac {2}{3 \left (\frac {e^{625 x^6}}{x}-2 x-\log (4)\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 1.47, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 e^{1250 x^6}+12 x^4+12 x^3 \log (4)+3 x^2 \log ^2(4)+e^{625 x^6} \left (-12 x^2-6 x \log (4)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x^2 + E^(625*x^6)*(2 - 7500*x^6))/(3*E^(1250*x^6) + 12*x^4 + 12*x^3*Log[4] + 3*x^2*Log[4]^2 + E^(625*x^
6)*(-12*x^2 - 6*x*Log[4])),x]

[Out]

(2*Defer[Int][(E^(625*x^6) - 2*x^2 - x*Log[4])^(-1), x])/3 + (2*Log[4]*Defer[Int][x/(-E^(625*x^6) + 2*x^2 + x*
Log[4])^2, x])/3 + (8*Defer[Int][x^2/(-E^(625*x^6) + 2*x^2 + x*Log[4])^2, x])/3 - 2500*Log[4]*Defer[Int][x^7/(
-E^(625*x^6) + 2*x^2 + x*Log[4])^2, x] - 5000*Defer[Int][x^8/(-E^(625*x^6) + 2*x^2 + x*Log[4])^2, x] + 2500*De
fer[Int][x^6/(-E^(625*x^6) + 2*x^2 + x*Log[4]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{3 \left (e^{625 x^6}-x (2 x+\log (4))\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {4 x^2+e^{625 x^6} \left (2-7500 x^6\right )}{\left (e^{625 x^6}-x (2 x+\log (4))\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {2 \left (-1+3750 x^6\right )}{-e^{625 x^6}+2 x^2+x \log (4)}-\frac {2 x \left (-4 x+7500 x^7-\log (4)+3750 x^6 \log (4)\right )}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2}\right ) \, dx\\ &=\frac {2}{3} \int \frac {-1+3750 x^6}{-e^{625 x^6}+2 x^2+x \log (4)} \, dx-\frac {2}{3} \int \frac {x \left (-4 x+7500 x^7-\log (4)+3750 x^6 \log (4)\right )}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2} \, dx\\ &=-\left (\frac {2}{3} \int \left (-\frac {4 x^2}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2}+\frac {7500 x^8}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2}-\frac {x \log (4)}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2}+\frac {3750 x^7 \log (4)}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2}\right ) \, dx\right )+\frac {2}{3} \int \left (\frac {1}{e^{625 x^6}-2 x^2-x \log (4)}+\frac {3750 x^6}{-e^{625 x^6}+2 x^2+x \log (4)}\right ) \, dx\\ &=\frac {2}{3} \int \frac {1}{e^{625 x^6}-2 x^2-x \log (4)} \, dx+\frac {8}{3} \int \frac {x^2}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2} \, dx+2500 \int \frac {x^6}{-e^{625 x^6}+2 x^2+x \log (4)} \, dx-5000 \int \frac {x^8}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2} \, dx+\frac {1}{3} (2 \log (4)) \int \frac {x}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2} \, dx-(2500 \log (4)) \int \frac {x^7}{\left (-e^{625 x^6}+2 x^2+x \log (4)\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.41, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 x}{3 \left (e^{625 x^6}-2 x^2-x \log (4)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^2 + E^(625*x^6)*(2 - 7500*x^6))/(3*E^(1250*x^6) + 12*x^4 + 12*x^3*Log[4] + 3*x^2*Log[4]^2 + E^(
625*x^6)*(-12*x^2 - 6*x*Log[4])),x]

[Out]

(2*x)/(3*(E^(625*x^6) - 2*x^2 - x*Log[4]))

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 24, normalized size = 0.96 \begin {gather*} -\frac {2 \, x}{3 \, {\left (2 \, x^{2} + 2 \, x \log \relax (2) - e^{\left (625 \, x^{6}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*log(2)-12*x^2)*exp(625*x^6)+12*x^2*log(2
)^2+24*x^3*log(2)+12*x^4),x, algorithm="fricas")

[Out]

-2/3*x/(2*x^2 + 2*x*log(2) - e^(625*x^6))

________________________________________________________________________________________

giac [A]  time = 0.33, size = 24, normalized size = 0.96 \begin {gather*} -\frac {2 \, x}{3 \, {\left (2 \, x^{2} + 2 \, x \log \relax (2) - e^{\left (625 \, x^{6}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*log(2)-12*x^2)*exp(625*x^6)+12*x^2*log(2
)^2+24*x^3*log(2)+12*x^4),x, algorithm="giac")

[Out]

-2/3*x/(2*x^2 + 2*x*log(2) - e^(625*x^6))

________________________________________________________________________________________

maple [A]  time = 0.14, size = 25, normalized size = 1.00

method	result	size
norman	\(-\frac {2 x}{3 \left (2 x \ln \relax (2)+2 x^{2}-{\mathrm e}^{625 x^{6}}\right )}\)	\(25\)
risch	\(-\frac {2 x}{3 \left (2 x \ln \relax (2)+2 x^{2}-{\mathrm e}^{625 x^{6}}\right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*ln(2)-12*x^2)*exp(625*x^6)+12*x^2*ln(2)^2+24*x
^3*ln(2)+12*x^4),x,method=_RETURNVERBOSE)

[Out]

-2/3*x/(2*x*ln(2)+2*x^2-exp(625*x^6))

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 24, normalized size = 0.96 \begin {gather*} -\frac {2 \, x}{3 \, {\left (2 \, x^{2} + 2 \, x \log \relax (2) - e^{\left (625 \, x^{6}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7500*x^6+2)*exp(625*x^6)+4*x^2)/(3*exp(625*x^6)^2+(-12*x*log(2)-12*x^2)*exp(625*x^6)+12*x^2*log(2
)^2+24*x^3*log(2)+12*x^4),x, algorithm="maxima")

[Out]

-2/3*x/(2*x^2 + 2*x*log(2) - e^(625*x^6))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {{\mathrm {e}}^{625\,x^6}\,\left (7500\,x^6-2\right )-4\,x^2}{3\,{\mathrm {e}}^{1250\,x^6}+12\,x^2\,{\ln \relax (2)}^2+24\,x^3\,\ln \relax (2)-{\mathrm {e}}^{625\,x^6}\,\left (12\,x^2+12\,\ln \relax (2)\,x\right )+12\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(625*x^6)*(7500*x^6 - 2) - 4*x^2)/(3*exp(1250*x^6) + 12*x^2*log(2)^2 + 24*x^3*log(2) - exp(625*x^6)*(
12*x*log(2) + 12*x^2) + 12*x^4),x)

[Out]

-int((exp(625*x^6)*(7500*x^6 - 2) - 4*x^2)/(3*exp(1250*x^6) + 12*x^2*log(2)^2 + 24*x^3*log(2) - exp(625*x^6)*(
12*x*log(2) + 12*x^2) + 12*x^4), x)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 22, normalized size = 0.88 \begin {gather*} \frac {2 x}{- 6 x^{2} - 6 x \log {\relax (2 )} + 3 e^{625 x^{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7500*x**6+2)*exp(625*x**6)+4*x**2)/(3*exp(625*x**6)**2+(-12*x*ln(2)-12*x**2)*exp(625*x**6)+12*x**
2*ln(2)**2+24*x**3*ln(2)+12*x**4),x)

[Out]

2*x/(-6*x**2 - 6*x*log(2) + 3*exp(625*x**6))

________________________________________________________________________________________