3.74.21 \(\int \frac {e^2 (8-8 x+3750 x^2-400 x^3+10 x^4)-4 e^2 \log (\frac {3}{x})}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+(-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6) \log (\frac {3}{x})+4 x^2 \log ^2(\frac {3}{x})} \, dx\)

Optimal. Leaf size=30 \[ \frac {e^2}{x \left (-1+x-\frac {1}{2} (-25+x)^2 x^2+\log \left (\frac {3}{x}\right )\right )} \]

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Rubi [A]  time = 0.49, antiderivative size = 36, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6688, 12, 6687} \begin {gather*} -\frac {2 e^2}{x \left (x^4-50 x^3+625 x^2-2 x-2 \log \left (\frac {3}{x}\right )+2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^2*(8 - 8*x + 3750*x^2 - 400*x^3 + 10*x^4) - 4*E^2*Log[3/x])/(4*x^2 - 8*x^3 + 2504*x^4 - 2700*x^5 + 3908
29*x^6 - 62504*x^7 + 3750*x^8 - 100*x^9 + x^10 + (-8*x^2 + 8*x^3 - 2500*x^4 + 200*x^5 - 4*x^6)*Log[3/x] + 4*x^
2*Log[3/x]^2),x]

[Out]

(-2*E^2)/(x*(2 - 2*x + 625*x^2 - 50*x^3 + x^4 - 2*Log[3/x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^2 \left (4-4 x+1875 x^2-200 x^3+5 x^4-2 \log \left (\frac {3}{x}\right )\right )}{x^2 \left (2-2 x+625 x^2-50 x^3+x^4-2 \log \left (\frac {3}{x}\right )\right )^2} \, dx\\ &=\left (2 e^2\right ) \int \frac {4-4 x+1875 x^2-200 x^3+5 x^4-2 \log \left (\frac {3}{x}\right )}{x^2 \left (2-2 x+625 x^2-50 x^3+x^4-2 \log \left (\frac {3}{x}\right )\right )^2} \, dx\\ &=-\frac {2 e^2}{x \left (2-2 x+625 x^2-50 x^3+x^4-2 \log \left (\frac {3}{x}\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.33, size = 38, normalized size = 1.27 \begin {gather*} \frac {2 e^2}{x \left (-2+2 x-625 x^2+50 x^3-x^4+2 \log \left (\frac {3}{x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^2*(8 - 8*x + 3750*x^2 - 400*x^3 + 10*x^4) - 4*E^2*Log[3/x])/(4*x^2 - 8*x^3 + 2504*x^4 - 2700*x^5
+ 390829*x^6 - 62504*x^7 + 3750*x^8 - 100*x^9 + x^10 + (-8*x^2 + 8*x^3 - 2500*x^4 + 200*x^5 - 4*x^6)*Log[3/x]
+ 4*x^2*Log[3/x]^2),x]

[Out]

(2*E^2)/(x*(-2 + 2*x - 625*x^2 + 50*x^3 - x^4 + 2*Log[3/x]))

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fricas [A]  time = 1.03, size = 37, normalized size = 1.23 \begin {gather*} -\frac {2 \, e^{2}}{x^{5} - 50 \, x^{4} + 625 \, x^{3} - 2 \, x^{2} - 2 \, x \log \left (\frac {3}{x}\right ) + 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(2)*log(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x^2*log(3/x)^2+(-4*x^6+200*x^5-2500*x
^4+8*x^3-8*x^2)*log(3/x)+x^10-100*x^9+3750*x^8-62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x, algorith
m="fricas")

[Out]

-2*e^2/(x^5 - 50*x^4 + 625*x^3 - 2*x^2 - 2*x*log(3/x) + 2*x)

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giac [A]  time = 0.22, size = 42, normalized size = 1.40 \begin {gather*} \frac {2 \, e^{2}}{x^{5} {\left (\frac {50}{x} - \frac {625}{x^{2}} + \frac {2}{x^{3}} + \frac {2 \, \log \left (\frac {3}{x}\right )}{x^{4}} - \frac {2}{x^{4}} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(2)*log(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x^2*log(3/x)^2+(-4*x^6+200*x^5-2500*x
^4+8*x^3-8*x^2)*log(3/x)+x^10-100*x^9+3750*x^8-62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x, algorith
m="giac")

[Out]

2*e^2/(x^5*(50/x - 625/x^2 + 2/x^3 + 2*log(3/x)/x^4 - 2/x^4 - 1))

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maple [A]  time = 0.07, size = 36, normalized size = 1.20

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{2}}{x \left (x^{4}-50 x^{3}+625 x^{2}-2 x -2 \ln \left (\frac {3}{x}\right )+2\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*exp(2)*ln(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x^2*ln(3/x)^2+(-4*x^6+200*x^5-2500*x^4+8*x^3
-8*x^2)*ln(3/x)+x^10-100*x^9+3750*x^8-62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x,method=_RETURNVERB
OSE)

[Out]

-2*exp(2)/x/(x^4-50*x^3+625*x^2-2*x-2*ln(3/x)+2)

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maxima [A]  time = 0.49, size = 37, normalized size = 1.23 \begin {gather*} -\frac {2 \, e^{2}}{x^{5} - 50 \, x^{4} + 625 \, x^{3} - 2 \, x^{2} - 2 \, x {\left (\log \relax (3) - 1\right )} + 2 \, x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(2)*log(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x^2*log(3/x)^2+(-4*x^6+200*x^5-2500*x
^4+8*x^3-8*x^2)*log(3/x)+x^10-100*x^9+3750*x^8-62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x, algorith
m="maxima")

[Out]

-2*e^2/(x^5 - 50*x^4 + 625*x^3 - 2*x^2 - 2*x*(log(3) - 1) + 2*x*log(x))

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mupad [B]  time = 4.66, size = 37, normalized size = 1.23 \begin {gather*} \frac {2\,{\mathrm {e}}^2}{x\,\left (2\,x+2\,\ln \left (\frac {3}{x}\right )-625\,x^2+50\,x^3-x^4-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*exp(2)*log(3/x) - exp(2)*(3750*x^2 - 8*x - 400*x^3 + 10*x^4 + 8))/(4*x^2*log(3/x)^2 - log(3/x)*(8*x^2
- 8*x^3 + 2500*x^4 - 200*x^5 + 4*x^6) + 4*x^2 - 8*x^3 + 2504*x^4 - 2700*x^5 + 390829*x^6 - 62504*x^7 + 3750*x^
8 - 100*x^9 + x^10),x)

[Out]

(2*exp(2))/(x*(2*x + 2*log(3/x) - 625*x^2 + 50*x^3 - x^4 - 2))

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sympy [A]  time = 0.20, size = 34, normalized size = 1.13 \begin {gather*} \frac {2 e^{2}}{- x^{5} + 50 x^{4} - 625 x^{3} + 2 x^{2} + 2 x \log {\left (\frac {3}{x} \right )} - 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(2)*ln(3/x)+(10*x**4-400*x**3+3750*x**2-8*x+8)*exp(2))/(4*x**2*ln(3/x)**2+(-4*x**6+200*x**5-2
500*x**4+8*x**3-8*x**2)*ln(3/x)+x**10-100*x**9+3750*x**8-62504*x**7+390829*x**6-2700*x**5+2504*x**4-8*x**3+4*x
**2),x)

[Out]

2*exp(2)/(-x**5 + 50*x**4 - 625*x**3 + 2*x**2 + 2*x*log(3/x) - 2*x)

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