\(\int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} (-6 e^5 x^4-18 e^5 x^2 \log (5))} \, dx\) [6304]

Optimal result

Integrand size = 99, antiderivative size = 28 \[ \int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} \left (-6 e^5 x^4-18 e^5 x^2 \log (5)\right )} \, dx=5-\frac {3}{-x^2+3 \left (e^{\frac {1}{e^5 x}}-\log (5)\right )} \]

[Out]

5-3/(3*exp(1/x/exp(5))-3*ln(5)-x^2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6820, 12, 6818} \[ \int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} \left (-6 e^5 x^4-18 e^5 x^2 \log (5)\right )} \, dx=-\frac {3}{-x^2+3 e^{\frac {1}{e^5 x}}-\log (125)} \]

[In]

Int[(-9*E^(1/(E^5*x)) - 6*E^5*x^3)/(9*E^(5 + 2/(E^5*x))*x^2 + E^5*x^6 + 6*E^5*x^4*Log[5] + 9*E^5*x^2*Log[5]^2
+ E^(1/(E^5*x))*(-6*E^5*x^4 - 18*E^5*x^2*Log[5])),x]

[Out]

-3/(3*E^(1/(E^5*x)) - x^2 - Log[125])

Rule 12

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

Rule 6818

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 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left (-3 e^{\frac {1}{e^5 x}}-2 e^5 x^3\right )}{e^5 x^2 \left (3 e^{\frac {1}{e^5 x}}-x^2-\log (125)\right )^2} \, dx \\ & = \frac {3 \int \frac {-3 e^{\frac {1}{e^5 x}}-2 e^5 x^3}{x^2 \left (3 e^{\frac {1}{e^5 x}}-x^2-\log (125)\right )^2} \, dx}{e^5} \\ & = -\frac {3}{3 e^{\frac {1}{e^5 x}}-x^2-\log (125)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} \left (-6 e^5 x^4-18 e^5 x^2 \log (5)\right )} \, dx=\frac {3}{-3 e^{\frac {1}{e^5 x}}+x^2+\log (125)} \]

[In]

Integrate[(-9*E^(1/(E^5*x)) - 6*E^5*x^3)/(9*E^(5 + 2/(E^5*x))*x^2 + E^5*x^6 + 6*E^5*x^4*Log[5] + 9*E^5*x^2*Log
[5]^2 + E^(1/(E^5*x))*(-6*E^5*x^4 - 18*E^5*x^2*Log[5])),x]

[Out]

3/(-3*E^(1/(E^5*x)) + x^2 + Log[125])

Maple [A] (verified)

Time = 3.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {3}{x^{2}+3 \ln \left (5\right )-3 \,{\mathrm e}^{\frac {{\mathrm e}^{-5}}{x}}}\)	\(22\)
norman	\(\frac {3}{x^{2}+3 \ln \left (5\right )-3 \,{\mathrm e}^{\frac {{\mathrm e}^{-5}}{x}}}\)	\(24\)
parallelrisch	\(\frac {3}{x^{2}+3 \ln \left (5\right )-3 \,{\mathrm e}^{\frac {{\mathrm e}^{-5}}{x}}}\)	\(24\)

[In]

int((-9*exp(1/x/exp(5))-6*x^3*exp(5))/(9*x^2*exp(5)*exp(1/x/exp(5))^2+(-18*x^2*exp(5)*ln(5)-6*x^4*exp(5))*exp(
1/x/exp(5))+9*x^2*exp(5)*ln(5)^2+6*x^4*exp(5)*ln(5)+x^6*exp(5)),x,method=_RETURNVERBOSE)

[Out]

3/(x^2+3*ln(5)-3*exp(exp(-5)/x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} \left (-6 e^5 x^4-18 e^5 x^2 \log (5)\right )} \, dx=\frac {3}{x^{2} - 3 \, e^{\left (\frac {e^{\left (-5\right )}}{x}\right )} + 3 \, \log \left (5\right )} \]

[In]

integrate((-9*exp(1/x/exp(5))-6*x^3*exp(5))/(9*x^2*exp(5)*exp(1/x/exp(5))^2+(-18*x^2*exp(5)*log(5)-6*x^4*exp(5
))*exp(1/x/exp(5))+9*x^2*exp(5)*log(5)^2+6*x^4*exp(5)*log(5)+x^6*exp(5)),x, algorithm="fricas")

[Out]

3/(x^2 - 3*e^(e^(-5)/x) + 3*log(5))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} \left (-6 e^5 x^4-18 e^5 x^2 \log (5)\right )} \, dx=- \frac {3}{- x^{2} + 3 e^{\frac {1}{x e^{5}}} - 3 \log {\left (5 \right )}} \]

[In]

integrate((-9*exp(1/x/exp(5))-6*x**3*exp(5))/(9*x**2*exp(5)*exp(1/x/exp(5))**2+(-18*x**2*exp(5)*ln(5)-6*x**4*e
xp(5))*exp(1/x/exp(5))+9*x**2*exp(5)*ln(5)**2+6*x**4*exp(5)*ln(5)+x**6*exp(5)),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} \left (-6 e^5 x^4-18 e^5 x^2 \log (5)\right )} \, dx=\frac {3}{x^{2} - 3 \, e^{\left (\frac {e^{\left (-5\right )}}{x}\right )} + 3 \, \log \left (5\right )} \]

[In]

integrate((-9*exp(1/x/exp(5))-6*x^3*exp(5))/(9*x^2*exp(5)*exp(1/x/exp(5))^2+(-18*x^2*exp(5)*log(5)-6*x^4*exp(5
))*exp(1/x/exp(5))+9*x^2*exp(5)*log(5)^2+6*x^4*exp(5)*log(5)+x^6*exp(5)),x, algorithm="maxima")

[Out]

3/(x^2 - 3*e^(e^(-5)/x) + 3*log(5))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} \left (-6 e^5 x^4-18 e^5 x^2 \log (5)\right )} \, dx=-\frac {3}{x^{2} {\left (\frac {3 \, e^{\left (\frac {e^{\left (-5\right )}}{x}\right )}}{x^{2}} - \frac {3 \, \log \left (5\right )}{x^{2}} - 1\right )}} \]

[In]

integrate((-9*exp(1/x/exp(5))-6*x^3*exp(5))/(9*x^2*exp(5)*exp(1/x/exp(5))^2+(-18*x^2*exp(5)*log(5)-6*x^4*exp(5
))*exp(1/x/exp(5))+9*x^2*exp(5)*log(5)^2+6*x^4*exp(5)*log(5)+x^6*exp(5)),x, algorithm="giac")

[Out]

-3/(x^2*(3*e^(e^(-5)/x)/x^2 - 3*log(5)/x^2 - 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {-9 e^{\frac {1}{e^5 x}}-6 e^5 x^3}{9 e^{5+\frac {2}{e^5 x}} x^2+e^5 x^6+6 e^5 x^4 \log (5)+9 e^5 x^2 \log ^2(5)+e^{\frac {1}{e^5 x}} \left (-6 e^5 x^4-18 e^5 x^2 \log (5)\right )} \, dx=\int -\frac {9\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-5}}{x}}+6\,x^3\,{\mathrm {e}}^5}{x^6\,{\mathrm {e}}^5-{\mathrm {e}}^{\frac {{\mathrm {e}}^{-5}}{x}}\,\left (6\,{\mathrm {e}}^5\,x^4+18\,{\mathrm {e}}^5\,\ln \left (5\right )\,x^2\right )+6\,x^4\,{\mathrm {e}}^5\,\ln \left (5\right )+9\,x^2\,{\mathrm {e}}^5\,{\ln \left (5\right )}^2+9\,x^2\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{-5}}{x}}\,{\mathrm {e}}^5} \,d x \]

[In]

int(-(9*exp(exp(-5)/x) + 6*x^3*exp(5))/(x^6*exp(5) - exp(exp(-5)/x)*(6*x^4*exp(5) + 18*x^2*exp(5)*log(5)) + 6*
x^4*exp(5)*log(5) + 9*x^2*exp(5)*log(5)^2 + 9*x^2*exp((2*exp(-5))/x)*exp(5)),x)

[Out]

int(-(9*exp(exp(-5)/x) + 6*x^3*exp(5))/(x^6*exp(5) - exp(exp(-5)/x)*(6*x^4*exp(5) + 18*x^2*exp(5)*log(5)) + 6*
x^4*exp(5)*log(5) + 9*x^2*exp(5)*log(5)^2 + 9*x^2*exp((2*exp(-5))/x)*exp(5)), x)