3.52.91 \(\int \frac {e^{18} (96+32 e^3)+e^{36} (9 x+6 e^3 x+e^6 x)+e^{18} (6+2 e^3) \log (x)}{65536 x+e^{18} (1536 x^2+512 e^3 x^2)+e^{36} (9 x^3+6 e^3 x^3+e^6 x^3)+(16384 x+e^{18} (192 x^2+64 e^3 x^2)) \log (x)+(1536 x+e^{18} (6 x^2+2 e^3 x^2)) \log ^2(x)+64 x \log ^3(x)+x \log ^4(x)} \, dx\)

Optimal. Leaf size=25 \[ 5-\frac {1}{x+\frac {(16+\log (x))^2}{e^{18} \left (3+e^3\right )}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.24, antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 3, integrand size = 159, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6688, 12, 6686} \begin {gather*} -\frac {e^{18} \left (3+e^3\right )}{e^{18} \left (3+e^3\right ) x+\log ^2(x)+32 \log (x)+256} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^18*(96 + 32*E^3) + E^36*(9*x + 6*E^3*x + E^6*x) + E^18*(6 + 2*E^3)*Log[x])/(65536*x + E^18*(1536*x^2 +
512*E^3*x^2) + E^36*(9*x^3 + 6*E^3*x^3 + E^6*x^3) + (16384*x + E^18*(192*x^2 + 64*E^3*x^2))*Log[x] + (1536*x +
E^18*(6*x^2 + 2*E^3*x^2))*Log[x]^2 + 64*x*Log[x]^3 + x*Log[x]^4),x]

[Out]

-((E^18*(3 + E^3))/(256 + E^18*(3 + E^3)*x + 32*Log[x] + Log[x]^2))

Rule 12

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, 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 {e^{18} \left (3+e^3\right ) \left (32+3 e^{18} \left (1+\frac {e^3}{3}\right ) x+2 \log (x)\right )}{x \left (256+3 e^{18} \left (1+\frac {e^3}{3}\right ) x+32 \log (x)+\log ^2(x)\right )^2} \, dx\\ &=\left (e^{18} \left (3+e^3\right )\right ) \int \frac {32+3 e^{18} \left (1+\frac {e^3}{3}\right ) x+2 \log (x)}{x \left (256+3 e^{18} \left (1+\frac {e^3}{3}\right ) x+32 \log (x)+\log ^2(x)\right )^2} \, dx\\ &=-\frac {e^{18} \left (3+e^3\right )}{256+e^{18} \left (3+e^3\right ) x+32 \log (x)+\log ^2(x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 33, normalized size = 1.32 \begin {gather*} -\frac {e^{18} \left (3+e^3\right )}{256+\left (3 e^{18}+e^{21}\right ) x+32 \log (x)+\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^18*(96 + 32*E^3) + E^36*(9*x + 6*E^3*x + E^6*x) + E^18*(6 + 2*E^3)*Log[x])/(65536*x + E^18*(1536*
x^2 + 512*E^3*x^2) + E^36*(9*x^3 + 6*E^3*x^3 + E^6*x^3) + (16384*x + E^18*(192*x^2 + 64*E^3*x^2))*Log[x] + (15
36*x + E^18*(6*x^2 + 2*E^3*x^2))*Log[x]^2 + 64*x*Log[x]^3 + x*Log[x]^4),x]

[Out]

-((E^18*(3 + E^3))/(256 + (3*E^18 + E^21)*x + 32*Log[x] + Log[x]^2))

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 30, normalized size = 1.20 \begin {gather*} -\frac {e^{21} + 3 \, e^{18}}{x e^{21} + 3 \, x e^{18} + \log \relax (x)^{2} + 32 \, \log \relax (x) + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(3)+6)*exp(18)*log(x)+(x*exp(3)^2+6*x*exp(3)+9*x)*exp(18)^2+(32*exp(3)+96)*exp(18))/(x*log(x)
^4+64*x*log(x)^3+((2*x^2*exp(3)+6*x^2)*exp(18)+1536*x)*log(x)^2+((64*x^2*exp(3)+192*x^2)*exp(18)+16384*x)*log(
x)+(x^3*exp(3)^2+6*x^3*exp(3)+9*x^3)*exp(18)^2+(512*x^2*exp(3)+1536*x^2)*exp(18)+65536*x),x, algorithm="fricas
")

[Out]

-(e^21 + 3*e^18)/(x*e^21 + 3*x*e^18 + log(x)^2 + 32*log(x) + 256)

________________________________________________________________________________________

giac [A]  time = 6.75, size = 30, normalized size = 1.20 \begin {gather*} -\frac {2 \, {\left (e^{21} + 3 \, e^{18}\right )}}{x e^{21} + 3 \, x e^{18} + \log \relax (x)^{2} + 32 \, \log \relax (x) + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(3)+6)*exp(18)*log(x)+(x*exp(3)^2+6*x*exp(3)+9*x)*exp(18)^2+(32*exp(3)+96)*exp(18))/(x*log(x)
^4+64*x*log(x)^3+((2*x^2*exp(3)+6*x^2)*exp(18)+1536*x)*log(x)^2+((64*x^2*exp(3)+192*x^2)*exp(18)+16384*x)*log(
x)+(x^3*exp(3)^2+6*x^3*exp(3)+9*x^3)*exp(18)^2+(512*x^2*exp(3)+1536*x^2)*exp(18)+65536*x),x, algorithm="giac")

[Out]

-2*(e^21 + 3*e^18)/(x*e^21 + 3*x*e^18 + log(x)^2 + 32*log(x) + 256)

________________________________________________________________________________________

maple [A]  time = 0.23, size = 36, normalized size = 1.44

method	result	size
norman	\(\frac {-{\mathrm e}^{3} {\mathrm e}^{18}-3 \,{\mathrm e}^{18}}{{\mathrm e}^{3} {\mathrm e}^{18} x +\ln \relax (x )^{2}+3 \,{\mathrm e}^{18} x +32 \ln \relax (x )+256}\)	\(36\)
risch	\(-\frac {{\mathrm e}^{18} {\mathrm e}^{3}}{x \,{\mathrm e}^{21}+3 \,{\mathrm e}^{18} x +\ln \relax (x )^{2}+32 \ln \relax (x )+256}-\frac {3 \,{\mathrm e}^{18}}{x \,{\mathrm e}^{21}+3 \,{\mathrm e}^{18} x +\ln \relax (x )^{2}+32 \ln \relax (x )+256}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*exp(3)+6)*exp(18)*ln(x)+(x*exp(3)^2+6*x*exp(3)+9*x)*exp(18)^2+(32*exp(3)+96)*exp(18))/(x*ln(x)^4+64*x*
ln(x)^3+((2*x^2*exp(3)+6*x^2)*exp(18)+1536*x)*ln(x)^2+((64*x^2*exp(3)+192*x^2)*exp(18)+16384*x)*ln(x)+(x^3*exp
(3)^2+6*x^3*exp(3)+9*x^3)*exp(18)^2+(512*x^2*exp(3)+1536*x^2)*exp(18)+65536*x),x,method=_RETURNVERBOSE)

[Out]

(-exp(3)*exp(18)-3*exp(18))/(exp(3)*exp(18)*x+ln(x)^2+3*exp(18)*x+32*ln(x)+256)

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 30, normalized size = 1.20 \begin {gather*} -\frac {e^{21} + 3 \, e^{18}}{x {\left (e^{21} + 3 \, e^{18}\right )} + \log \relax (x)^{2} + 32 \, \log \relax (x) + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(3)+6)*exp(18)*log(x)+(x*exp(3)^2+6*x*exp(3)+9*x)*exp(18)^2+(32*exp(3)+96)*exp(18))/(x*log(x)
^4+64*x*log(x)^3+((2*x^2*exp(3)+6*x^2)*exp(18)+1536*x)*log(x)^2+((64*x^2*exp(3)+192*x^2)*exp(18)+16384*x)*log(
x)+(x^3*exp(3)^2+6*x^3*exp(3)+9*x^3)*exp(18)^2+(512*x^2*exp(3)+1536*x^2)*exp(18)+65536*x),x, algorithm="maxima
")

[Out]

-(e^21 + 3*e^18)/(x*(e^21 + 3*e^18) + log(x)^2 + 32*log(x) + 256)

________________________________________________________________________________________

mupad [B]  time = 10.87, size = 84, normalized size = 3.36 \begin {gather*} \frac {\frac {{\left (3\,{\mathrm {e}}^{18}+{\mathrm {e}}^{21}\right )}^2\,x^3}{256}+\left (\frac {3\,{\mathrm {e}}^{18}}{256}+\frac {{\mathrm {e}}^{21}}{256}\right )\,x^2\,{\ln \relax (x)}^2+\left (\frac {3\,{\mathrm {e}}^{18}}{8}+\frac {{\mathrm {e}}^{21}}{8}\right )\,x^2\,\ln \relax (x)}{32\,x^2\,\ln \relax (x)+x^2\,{\ln \relax (x)}^2+3\,x^3\,{\mathrm {e}}^{18}+x^3\,{\mathrm {e}}^{21}+256\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(36)*(9*x + 6*x*exp(3) + x*exp(6)) + exp(18)*(32*exp(3) + 96) + exp(18)*log(x)*(2*exp(3) + 6))/(65536*
x + 64*x*log(x)^3 + x*log(x)^4 + log(x)*(16384*x + exp(18)*(64*x^2*exp(3) + 192*x^2)) + exp(36)*(6*x^3*exp(3)
+ x^3*exp(6) + 9*x^3) + log(x)^2*(1536*x + exp(18)*(2*x^2*exp(3) + 6*x^2)) + exp(18)*(512*x^2*exp(3) + 1536*x^
2)),x)

[Out]

((x^3*(3*exp(18) + exp(21))^2)/256 + x^2*log(x)*((3*exp(18))/8 + exp(21)/8) + x^2*log(x)^2*((3*exp(18))/256 +
exp(21)/256))/(32*x^2*log(x) + x^2*log(x)^2 + 3*x^3*exp(18) + x^3*exp(21) + 256*x^2)

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 32, normalized size = 1.28 \begin {gather*} \frac {- e^{21} - 3 e^{18}}{3 x e^{18} + x e^{21} + \log {\relax (x )}^{2} + 32 \log {\relax (x )} + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(3)+6)*exp(18)*ln(x)+(x*exp(3)**2+6*x*exp(3)+9*x)*exp(18)**2+(32*exp(3)+96)*exp(18))/(x*ln(x)
**4+64*x*ln(x)**3+((2*x**2*exp(3)+6*x**2)*exp(18)+1536*x)*ln(x)**2+((64*x**2*exp(3)+192*x**2)*exp(18)+16384*x)
*ln(x)+(x**3*exp(3)**2+6*x**3*exp(3)+9*x**3)*exp(18)**2+(512*x**2*exp(3)+1536*x**2)*exp(18)+65536*x),x)

[Out]

(-exp(21) - 3*exp(18))/(3*x*exp(18) + x*exp(21) + log(x)**2 + 32*log(x) + 256)

________________________________________________________________________________________