3.48.35 \(\int \frac {-36 x-x^4+e^x (36+x^3)+(-e^x x^3+x^4) \log (x)+(2 x^5-2 e^x x^5+(-x^5+e^x x^5) \log (3 e^{\frac {9}{x^4}})+(x^4-e^x x^4) \log (x)) \log (\frac {-2 x+x \log (3 e^{\frac {9}{x^4}})-\log (x)}{x})}{-2 e^{2 x} x^5+4 e^x x^6-2 x^7+(e^{2 x} x^5-2 e^x x^6+x^7) \log (3 e^{\frac {9}{x^4}})+(-e^{2 x} x^4+2 e^x x^5-x^6) \log (x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {\log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{-e^x+x} \]

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Rubi [F]  time = 6.07, 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 {-36 x-x^4+e^x \left (36+x^3\right )+\left (-e^x x^3+x^4\right ) \log (x)+\left (2 x^5-2 e^x x^5+\left (-x^5+e^x x^5\right ) \log \left (3 e^{\frac {9}{x^4}}\right )+\left (x^4-e^x x^4\right ) \log (x)\right ) \log \left (\frac {-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)}{x}\right )}{-2 e^{2 x} x^5+4 e^x x^6-2 x^7+\left (e^{2 x} x^5-2 e^x x^6+x^7\right ) \log \left (3 e^{\frac {9}{x^4}}\right )+\left (-e^{2 x} x^4+2 e^x x^5-x^6\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-36*x - x^4 + E^x*(36 + x^3) + (-(E^x*x^3) + x^4)*Log[x] + (2*x^5 - 2*E^x*x^5 + (-x^5 + E^x*x^5)*Log[3*E^
(9/x^4)] + (x^4 - E^x*x^4)*Log[x])*Log[(-2*x + x*Log[3*E^(9/x^4)] - Log[x])/x])/(-2*E^(2*x)*x^5 + 4*E^x*x^6 -
2*x^7 + (E^(2*x)*x^5 - 2*E^x*x^6 + x^7)*Log[3*E^(9/x^4)] + (-(E^(2*x)*x^4) + 2*E^x*x^5 - x^6)*Log[x]),x]

[Out]

36*Defer[Int][1/((E^x - x)*x^4*(-2*x + x*Log[3*E^(9/x^4)] - Log[x])), x] + Defer[Int][1/((E^x - x)*x*(-2*x + x
*Log[3*E^(9/x^4)] - Log[x])), x] - Defer[Int][Log[x]/((E^x - x)*x*(-2*x + x*Log[3*E^(9/x^4)] - Log[x])), x] -
Defer[Int][Log[-2 + Log[3*E^(9/x^4)] - Log[x]/x]/(E^x - x)^2, x] + Defer[Int][(x*Log[-2 + Log[3*E^(9/x^4)] - L
og[x]/x])/(E^x - x)^2, x] - 2*Defer[Int][(x*Log[-2 + Log[3*E^(9/x^4)] - Log[x]/x])/((E^x - x)*(-2*x + x*Log[3*
E^(9/x^4)] - Log[x])), x] + Defer[Int][(x*Log[3*E^(9/x^4)]*Log[-2 + Log[3*E^(9/x^4)] - Log[x]/x])/((E^x - x)*(
-2*x + x*Log[3*E^(9/x^4)] - Log[x])), x] - Defer[Int][(Log[x]*Log[-2 + Log[3*E^(9/x^4)] - Log[x]/x])/((E^x - x
)*(-2*x + x*Log[3*E^(9/x^4)] - Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\left (\left (e^x-x\right ) \left (36+x^3\right )\right )-\left (-1+e^x\right ) x^5 \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )\right ) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )+x^3 \log (x) \left (e^x-x+\left (-1+e^x\right ) x \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )\right )}{\left (e^x-x\right )^2 x^4 \left (2 x-x \log \left (3 e^{\frac {9}{x^4}}\right )+\log (x)\right )} \, dx\\ &=\int \left (\frac {(-1+x) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right )^2}+\frac {36+x^3-x^3 \log (x)-2 x^5 \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )+x^5 \log \left (3 e^{\frac {9}{x^4}}\right ) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )-x^4 \log (x) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right ) x^4 \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )}\right ) \, dx\\ &=\int \frac {(-1+x) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right )^2} \, dx+\int \frac {36+x^3-x^3 \log (x)-2 x^5 \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )+x^5 \log \left (3 e^{\frac {9}{x^4}}\right ) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )-x^4 \log (x) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right ) x^4 \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )} \, dx\\ &=\int \left (-\frac {\log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right )^2}+\frac {x \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right )^2}\right ) \, dx+\int \left (\frac {36}{\left (e^x-x\right ) x^4 \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )}+\frac {1}{\left (e^x-x\right ) x \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )}-\frac {\log (x)}{\left (e^x-x\right ) x \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )}-\frac {2 x \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right ) \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )}+\frac {x \log \left (3 e^{\frac {9}{x^4}}\right ) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right ) \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )}-\frac {\log (x) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right ) \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right ) \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )} \, dx\right )+36 \int \frac {1}{\left (e^x-x\right ) x^4 \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )} \, dx+\int \frac {1}{\left (e^x-x\right ) x \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )} \, dx-\int \frac {\log (x)}{\left (e^x-x\right ) x \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )} \, dx-\int \frac {\log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right )^2} \, dx+\int \frac {x \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right )^2} \, dx+\int \frac {x \log \left (3 e^{\frac {9}{x^4}}\right ) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right ) \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )} \, dx-\int \frac {\log (x) \log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{\left (e^x-x\right ) \left (-2 x+x \log \left (3 e^{\frac {9}{x^4}}\right )-\log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 31, normalized size = 1.03 \begin {gather*} -\frac {\log \left (-2+\log \left (3 e^{\frac {9}{x^4}}\right )-\frac {\log (x)}{x}\right )}{e^x-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36*x - x^4 + E^x*(36 + x^3) + (-(E^x*x^3) + x^4)*Log[x] + (2*x^5 - 2*E^x*x^5 + (-x^5 + E^x*x^5)*Lo
g[3*E^(9/x^4)] + (x^4 - E^x*x^4)*Log[x])*Log[(-2*x + x*Log[3*E^(9/x^4)] - Log[x])/x])/(-2*E^(2*x)*x^5 + 4*E^x*
x^6 - 2*x^7 + (E^(2*x)*x^5 - 2*E^x*x^6 + x^7)*Log[3*E^(9/x^4)] + (-(E^(2*x)*x^4) + 2*E^x*x^5 - x^6)*Log[x]),x]

[Out]

-(Log[-2 + Log[3*E^(9/x^4)] - Log[x]/x]/(E^x - x))

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fricas [A]  time = 0.59, size = 34, normalized size = 1.13 \begin {gather*} \frac {\log \left (\frac {x^{4} \log \relax (3) - 2 \, x^{4} - x^{3} \log \relax (x) + 9}{x^{4}}\right )}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^5*exp(x)-x^5)*log(3*exp(9/x^4))+(-exp(x)*x^4+x^4)*log(x)-2*x^5*exp(x)+2*x^5)*log((x*log(3*exp(9
/x^4))-2*x-log(x))/x)+(-exp(x)*x^3+x^4)*log(x)+(x^3+36)*exp(x)-x^4-36*x)/((x^5*exp(x)^2-2*x^6*exp(x)+x^7)*log(
3*exp(9/x^4))+(-exp(x)^2*x^4+2*x^5*exp(x)-x^6)*log(x)-2*x^5*exp(x)^2+4*x^6*exp(x)-2*x^7),x, algorithm="fricas"
)

[Out]

log((x^4*log(3) - 2*x^4 - x^3*log(x) + 9)/x^4)/(x - e^x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^5*exp(x)-x^5)*log(3*exp(9/x^4))+(-exp(x)*x^4+x^4)*log(x)-2*x^5*exp(x)+2*x^5)*log((x*log(3*exp(9
/x^4))-2*x-log(x))/x)+(-exp(x)*x^3+x^4)*log(x)+(x^3+36)*exp(x)-x^4-36*x)/((x^5*exp(x)^2-2*x^6*exp(x)+x^7)*log(
3*exp(9/x^4))+(-exp(x)^2*x^4+2*x^5*exp(x)-x^6)*log(x)-2*x^5*exp(x)^2+4*x^6*exp(x)-2*x^7),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (x^{5} {\mathrm e}^{x}-x^{5}\right ) \ln \left (3 \,{\mathrm e}^{\frac {9}{x^{4}}}\right )+\left (-{\mathrm e}^{x} x^{4}+x^{4}\right ) \ln \relax (x )-2 x^{5} {\mathrm e}^{x}+2 x^{5}\right ) \ln \left (\frac {x \ln \left (3 \,{\mathrm e}^{\frac {9}{x^{4}}}\right )-2 x -\ln \relax (x )}{x}\right )+\left (-{\mathrm e}^{x} x^{3}+x^{4}\right ) \ln \relax (x )+\left (x^{3}+36\right ) {\mathrm e}^{x}-x^{4}-36 x}{\left (x^{5} {\mathrm e}^{2 x}-2 x^{6} {\mathrm e}^{x}+x^{7}\right ) \ln \left (3 \,{\mathrm e}^{\frac {9}{x^{4}}}\right )+\left (-{\mathrm e}^{2 x} x^{4}+2 x^{5} {\mathrm e}^{x}-x^{6}\right ) \ln \relax (x )-2 x^{5} {\mathrm e}^{2 x}+4 x^{6} {\mathrm e}^{x}-2 x^{7}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^5*exp(x)-x^5)*ln(3*exp(9/x^4))+(-exp(x)*x^4+x^4)*ln(x)-2*x^5*exp(x)+2*x^5)*ln((x*ln(3*exp(9/x^4))-2*x
-ln(x))/x)+(-exp(x)*x^3+x^4)*ln(x)+(x^3+36)*exp(x)-x^4-36*x)/((x^5*exp(x)^2-2*x^6*exp(x)+x^7)*ln(3*exp(9/x^4))
+(-exp(x)^2*x^4+2*x^5*exp(x)-x^6)*ln(x)-2*x^5*exp(x)^2+4*x^6*exp(x)-2*x^7),x)

[Out]

int((((x^5*exp(x)-x^5)*ln(3*exp(9/x^4))+(-exp(x)*x^4+x^4)*ln(x)-2*x^5*exp(x)+2*x^5)*ln((x*ln(3*exp(9/x^4))-2*x
-ln(x))/x)+(-exp(x)*x^3+x^4)*ln(x)+(x^3+36)*exp(x)-x^4-36*x)/((x^5*exp(x)^2-2*x^6*exp(x)+x^7)*ln(3*exp(9/x^4))
+(-exp(x)^2*x^4+2*x^5*exp(x)-x^6)*ln(x)-2*x^5*exp(x)^2+4*x^6*exp(x)-2*x^7),x)

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maxima [A]  time = 0.54, size = 32, normalized size = 1.07 \begin {gather*} \frac {\log \left (x^{4} {\left (\log \relax (3) - 2\right )} - x^{3} \log \relax (x) + 9\right ) - 4 \, \log \relax (x)}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^5*exp(x)-x^5)*log(3*exp(9/x^4))+(-exp(x)*x^4+x^4)*log(x)-2*x^5*exp(x)+2*x^5)*log((x*log(3*exp(9
/x^4))-2*x-log(x))/x)+(-exp(x)*x^3+x^4)*log(x)+(x^3+36)*exp(x)-x^4-36*x)/((x^5*exp(x)^2-2*x^6*exp(x)+x^7)*log(
3*exp(9/x^4))+(-exp(x)^2*x^4+2*x^5*exp(x)-x^6)*log(x)-2*x^5*exp(x)^2+4*x^6*exp(x)-2*x^7),x, algorithm="maxima"
)

[Out]

(log(x^4*(log(3) - 2) - x^3*log(x) + 9) - 4*log(x))/(x - e^x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int -\frac {36\,x+\ln \left (-\frac {2\,x+\ln \relax (x)-x\,\ln \left (3\,{\mathrm {e}}^{\frac {9}{x^4}}\right )}{x}\right )\,\left (2\,x^5\,{\mathrm {e}}^x-\ln \left (3\,{\mathrm {e}}^{\frac {9}{x^4}}\right )\,\left (x^5\,{\mathrm {e}}^x-x^5\right )-2\,x^5+\ln \relax (x)\,\left (x^4\,{\mathrm {e}}^x-x^4\right )\right )-{\mathrm {e}}^x\,\left (x^3+36\right )+x^4+\ln \relax (x)\,\left (x^3\,{\mathrm {e}}^x-x^4\right )}{\ln \relax (x)\,\left (x^4\,{\mathrm {e}}^{2\,x}-2\,x^5\,{\mathrm {e}}^x+x^6\right )-4\,x^6\,{\mathrm {e}}^x+2\,x^5\,{\mathrm {e}}^{2\,x}-\ln \left (3\,{\mathrm {e}}^{\frac {9}{x^4}}\right )\,\left (x^5\,{\mathrm {e}}^{2\,x}-2\,x^6\,{\mathrm {e}}^x+x^7\right )+2\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((36*x + log(-(2*x + log(x) - x*log(3*exp(9/x^4)))/x)*(2*x^5*exp(x) - log(3*exp(9/x^4))*(x^5*exp(x) - x^5)
- 2*x^5 + log(x)*(x^4*exp(x) - x^4)) - exp(x)*(x^3 + 36) + x^4 + log(x)*(x^3*exp(x) - x^4))/(log(x)*(x^4*exp(2
*x) - 2*x^5*exp(x) + x^6) - 4*x^6*exp(x) + 2*x^5*exp(2*x) - log(3*exp(9/x^4))*(x^5*exp(2*x) - 2*x^6*exp(x) + x
^7) + 2*x^7),x)

[Out]

-int(-(36*x + log(-(2*x + log(x) - x*log(3*exp(9/x^4)))/x)*(2*x^5*exp(x) - log(3*exp(9/x^4))*(x^5*exp(x) - x^5
) - 2*x^5 + log(x)*(x^4*exp(x) - x^4)) - exp(x)*(x^3 + 36) + x^4 + log(x)*(x^3*exp(x) - x^4))/(log(x)*(x^4*exp
(2*x) - 2*x^5*exp(x) + x^6) - 4*x^6*exp(x) + 2*x^5*exp(2*x) - log(3*exp(9/x^4))*(x^5*exp(2*x) - 2*x^6*exp(x) +
x^7) + 2*x^7), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**5*exp(x)-x**5)*ln(3*exp(9/x**4))+(-exp(x)*x**4+x**4)*ln(x)-2*x**5*exp(x)+2*x**5)*ln((x*ln(3*ex
p(9/x**4))-2*x-ln(x))/x)+(-exp(x)*x**3+x**4)*ln(x)+(x**3+36)*exp(x)-x**4-36*x)/((x**5*exp(x)**2-2*x**6*exp(x)+
x**7)*ln(3*exp(9/x**4))+(-exp(x)**2*x**4+2*x**5*exp(x)-x**6)*ln(x)-2*x**5*exp(x)**2+4*x**6*exp(x)-2*x**7),x)

[Out]

Timed out

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