3.37.46 \(\int \frac {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+(e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))) \log (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5})}{e^5 (3 x^3+x^4)+e^5 x^3 \log (x)+(3 x^3+x^4+x^3 \log (x)) \log (-3-x-\log (x))} \, dx\) [3646]

Optimal. Leaf size=22 \[ \frac {\log \left (x+\frac {x \log (-3-x-\log (x))}{e^5}\right )}{x^2} \]

[Out]

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

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Rubi [F]
time = 2.57, 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 {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+\left (e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))\right ) \log \left (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5}\right )}{e^5 \left (3 x^3+x^4\right )+e^5 x^3 \log (x)+\left (3 x^3+x^4+x^3 \log (x)\right ) \log (-3-x-\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Log[(x*(E^5 + Log[-3 - x - Log[x]]))/E^5]/x^2 - E^5*Defer[Int][1/(x^3*(E^5 + Log[-3 - x - Log[x]])), x] + 3*E^
5*Defer[Int][1/(x^3*(3 + x + Log[x])*(E^5 + Log[-3 - x - Log[x]])), x] + E^5*Defer[Int][1/(x^2*(3 + x + Log[x]
)*(E^5 + Log[-3 - x - Log[x]])), x] + E^5*Defer[Int][Log[x]/(x^3*(3 + x + Log[x])*(E^5 + Log[-3 - x - Log[x]])
), x]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log \left (x+\frac {x \log (-3-x-\log (x))}{e^5}\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x + (x*Log[-3 - x - Log[x]])/E^5]/x^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 6.71, size = 169, normalized size = 7.68

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )}{x^{2}}+\frac {-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right )^{2}-10-i \pi \mathrm {csgn}\left (i x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right )^{3}+2 \ln \left (x \right )}{2 x^{2}}\)	\(169\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x^2*ln(exp(5)+ln(-ln(x)-3-x))+1/2*(-I*Pi*csgn(I*x)*csgn(I*(exp(5)+ln(-ln(x)-3-x)))*csgn(I*x*(exp(5)+ln(-ln(x
)-3-x)))+I*Pi*csgn(I*x)*csgn(I*x*(exp(5)+ln(-ln(x)-3-x)))^2+I*Pi*csgn(I*(exp(5)+ln(-ln(x)-3-x)))*csgn(I*x*(exp
(5)+ln(-ln(x)-3-x)))^2-10-I*Pi*csgn(I*x*(exp(5)+ln(-ln(x)-3-x)))^3+2*ln(x))/x^2

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Maxima [A]
time = 0.32, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log \left (x\right ) + \log \left (e^{5} + \log \left (-x - \log \left (x\right ) - 3\right )\right ) - 5}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(x) + log(e^5 + log(-x - log(x) - 3)) - 5)/x^2

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Fricas [A]
time = 0.37, size = 25, normalized size = 1.14 \begin {gather*} \frac {\log \left ({\left (x e^{5} + x \log \left (-x - \log \left (x\right ) - 3\right )\right )} e^{\left (-5\right )}\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 3.97, size = 24, normalized size = 1.09 \begin {gather*} \frac {\log {\left (\frac {x \log {\left (- x - \log {\left (x \right )} - 3 \right )} + x e^{5}}{e^{5}} \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.49, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log \left (x\right ) + \log \left (e^{5} + \log \left (-x - \log \left (x\right ) - 3\right )\right ) - 5}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(x) + log(e^5 + log(-x - log(x) - 3)) - 5)/x^2

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Mupad [B]
time = 2.70, size = 22, normalized size = 1.00 \begin {gather*} \frac {\ln \left (x\,\left ({\mathrm {e}}^5+\ln \left (-x-\ln \left (x\right )-3\right )\right )\right )-5}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + log(- x - log(x) - 3)*(x + log(x) + 3) + exp(5)*(x + 3) + exp(5)*log(x) - log(exp(-5)*(x*exp(5) + x*l
og(- x - log(x) - 3)))*(2*exp(5)*log(x) + log(- x - log(x) - 3)*(2*x + 2*log(x) + 6) + exp(5)*(2*x + 6)) + 1)/
(exp(5)*(3*x^3 + x^4) + log(- x - log(x) - 3)*(x^3*log(x) + 3*x^3 + x^4) + x^3*exp(5)*log(x)),x)

[Out]

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

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