3.32.43 \(\int \frac {(e^e (4-4 x)+80 x) \log ^3(\frac {1}{10} (e^e (-15-x)+20 x+e^e \log (x)))}{20 x^2+e^e (-15 x-x^2)+e^e x \log (x)} \, dx\)

Optimal. Leaf size=26 \[ \log ^4\left (2 x+\frac {1}{2} e^e \left (-3+\frac {1}{5} (-x+\log (x))\right )\right ) \]

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Rubi [A]  time = 0.27, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6686} \begin {gather*} \log ^4\left (\frac {1}{10} \left (20 x-e^e (x+15)+e^e \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((E^E*(4 - 4*x) + 80*x)*Log[(E^E*(-15 - x) + 20*x + E^E*Log[x])/10]^3)/(20*x^2 + E^E*(-15*x - x^2) + E^E*x
*Log[x]),x]

[Out]

Log[(20*x - E^E*(15 + x) + E^E*Log[x])/10]^4

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log ^4\left (\frac {1}{10} \left (20 x-e^e (15+x)+e^e \log (x)\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 25, normalized size = 0.96 \begin {gather*} \log ^4\left (\frac {1}{10} \left (20 x-e^e (15+x)+e^e \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((E^E*(4 - 4*x) + 80*x)*Log[(E^E*(-15 - x) + 20*x + E^E*Log[x])/10]^3)/(20*x^2 + E^E*(-15*x - x^2) +
E^E*x*Log[x]),x]

[Out]

Log[(20*x - E^E*(15 + x) + E^E*Log[x])/10]^4

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fricas [A]  time = 0.55, size = 22, normalized size = 0.85 \begin {gather*} \log \left (-\frac {1}{10} \, {\left (x + 15\right )} e^{e} + \frac {1}{10} \, e^{e} \log \relax (x) + 2 \, x\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+4)*exp(exp(1))+80*x)*log(1/10*exp(exp(1))*log(x)+1/10*(-x-15)*exp(exp(1))+2*x)^3/(x*exp(exp(1
))*log(x)+(-x^2-15*x)*exp(exp(1))+20*x^2),x, algorithm="fricas")

[Out]

log(-1/10*(x + 15)*e^e + 1/10*e^e*log(x) + 2*x)^4

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giac [B]  time = 0.78, size = 111, normalized size = 4.27 \begin {gather*} -4 \, \log \left (10\right )^{3} \log \left (-x e^{e} + e^{e} \log \relax (x) + 20 \, x - 15 \, e^{e}\right ) + 6 \, \log \left (10\right )^{2} \log \left (-x e^{e} + e^{e} \log \relax (x) + 20 \, x - 15 \, e^{e}\right )^{2} - 4 \, \log \left (10\right ) \log \left (-x e^{e} + e^{e} \log \relax (x) + 20 \, x - 15 \, e^{e}\right )^{3} + \log \left (-x e^{e} + e^{e} \log \relax (x) + 20 \, x - 15 \, e^{e}\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+4)*exp(exp(1))+80*x)*log(1/10*exp(exp(1))*log(x)+1/10*(-x-15)*exp(exp(1))+2*x)^3/(x*exp(exp(1
))*log(x)+(-x^2-15*x)*exp(exp(1))+20*x^2),x, algorithm="giac")

[Out]

-4*log(10)^3*log(-x*e^e + e^e*log(x) + 20*x - 15*e^e) + 6*log(10)^2*log(-x*e^e + e^e*log(x) + 20*x - 15*e^e)^2
- 4*log(10)*log(-x*e^e + e^e*log(x) + 20*x - 15*e^e)^3 + log(-x*e^e + e^e*log(x) + 20*x - 15*e^e)^4

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maple [A]  time = 0.13, size = 25, normalized size = 0.96

method	result	size
risch	\(\ln \left (\frac {{\mathrm e}^{{\mathrm e}} \ln \relax (x )}{10}+\frac {\left (-x -15\right ) {\mathrm e}^{{\mathrm e}}}{10}+2 x \right )^{4}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x+4)*exp(exp(1))+80*x)*ln(1/10*exp(exp(1))*ln(x)+1/10*(-x-15)*exp(exp(1))+2*x)^3/(x*exp(exp(1))*ln(x)
+(-x^2-15*x)*exp(exp(1))+20*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(1/10*exp(exp(1))*ln(x)+1/10*(-x-15)*exp(exp(1))+2*x)^4

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maxima [B]  time = 0.61, size = 381, normalized size = 14.65 \begin {gather*} -12 \, {\left (\log \relax (5) + \log \relax (2)\right )} e \log \left (-x {\left (e^{e} - 20\right )} + e^{e} \log \relax (x) - 15 \, e^{e}\right )^{2} + 4 \, e \log \left (-x {\left (e^{e} - 20\right )} + e^{e} \log \relax (x) - 15 \, e^{e}\right )^{3} - \log \left (-x {\left (e^{e} - 20\right )} + e^{e} \log \relax (x) - 15 \, e^{e}\right )^{4} + 4 \, \log \left (-{\left (x {\left (e^{e} - 20\right )} - e^{e} \log \relax (x) + 15 \, e^{e}\right )} e^{\left (-e\right )}\right ) \log \left (-\frac {1}{10} \, {\left (x + 15\right )} e^{e} + \frac {1}{10} \, e^{e} \log \relax (x) + 2 \, x\right )^{3} - 6 \, {\left (\log \left (-x {\left (e^{e} - 20\right )} + e^{e} \log \relax (x) - 15 \, e^{e}\right )^{2} - 2 \, \log \left (-x {\left (e^{e} - 20\right )} + e^{e} \log \relax (x) - 15 \, e^{e}\right ) \log \left (-\frac {1}{10} \, {\left (x + 15\right )} e^{e} + \frac {1}{10} \, e^{e} \log \relax (x) + 2 \, x\right ) + 2 \, \log \left (-{\left (x {\left (e^{e} - 20\right )} - e^{e} \log \relax (x) + 15 \, e^{e}\right )} e^{\left (-e\right )}\right ) \log \left (-\frac {1}{10} \, {\left (x + 15\right )} e^{e} + \frac {1}{10} \, e^{e} \log \relax (x) + 2 \, x\right )\right )} \log \left (-\frac {1}{10} \, {\left (x + 15\right )} e^{e} + \frac {1}{10} \, e^{e} \log \relax (x) + 2 \, x\right )^{2} + 4 \, {\left (6 \, {\left (\log \relax (5) + \log \relax (2)\right )} e \log \left (-x {\left (e^{e} - 20\right )} + e^{e} \log \relax (x) - 15 \, e^{e}\right ) - 3 \, e \log \left (-x {\left (e^{e} - 20\right )} + e^{e} \log \relax (x) - 15 \, e^{e}\right )^{2} + \log \left (-x {\left (e^{e} - 20\right )} + e^{e} \log \relax (x) - 15 \, e^{e}\right )^{3}\right )} \log \left (-\frac {1}{10} \, {\left (x + 15\right )} e^{e} + \frac {1}{10} \, e^{e} \log \relax (x) + 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+4)*exp(exp(1))+80*x)*log(1/10*exp(exp(1))*log(x)+1/10*(-x-15)*exp(exp(1))+2*x)^3/(x*exp(exp(1
))*log(x)+(-x^2-15*x)*exp(exp(1))+20*x^2),x, algorithm="maxima")

[Out]

-12*(log(5) + log(2))*e*log(-x*(e^e - 20) + e^e*log(x) - 15*e^e)^2 + 4*e*log(-x*(e^e - 20) + e^e*log(x) - 15*e
^e)^3 - log(-x*(e^e - 20) + e^e*log(x) - 15*e^e)^4 + 4*log(-(x*(e^e - 20) - e^e*log(x) + 15*e^e)*e^(-e))*log(-
1/10*(x + 15)*e^e + 1/10*e^e*log(x) + 2*x)^3 - 6*(log(-x*(e^e - 20) + e^e*log(x) - 15*e^e)^2 - 2*log(-x*(e^e -
20) + e^e*log(x) - 15*e^e)*log(-1/10*(x + 15)*e^e + 1/10*e^e*log(x) + 2*x) + 2*log(-(x*(e^e - 20) - e^e*log(x
) + 15*e^e)*e^(-e))*log(-1/10*(x + 15)*e^e + 1/10*e^e*log(x) + 2*x))*log(-1/10*(x + 15)*e^e + 1/10*e^e*log(x)
+ 2*x)^2 + 4*(6*(log(5) + log(2))*e*log(-x*(e^e - 20) + e^e*log(x) - 15*e^e) - 3*e*log(-x*(e^e - 20) + e^e*log
(x) - 15*e^e)^2 + log(-x*(e^e - 20) + e^e*log(x) - 15*e^e)^3)*log(-1/10*(x + 15)*e^e + 1/10*e^e*log(x) + 2*x)

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mupad [B]  time = 3.27, size = 22, normalized size = 0.85 \begin {gather*} {\ln \left (2\,x-\frac {{\mathrm {e}}^{\mathrm {e}}\,\left (x+15\right )}{10}+\frac {{\mathrm {e}}^{\mathrm {e}}\,\ln \relax (x)}{10}\right )}^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2*x - (exp(exp(1))*(x + 15))/10 + (exp(exp(1))*log(x))/10)^3*(80*x - exp(exp(1))*(4*x - 4)))/(20*x^2
- exp(exp(1))*(15*x + x^2) + x*exp(exp(1))*log(x)),x)

[Out]

log(2*x - (exp(exp(1))*(x + 15))/10 + (exp(exp(1))*log(x))/10)^4

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sympy [A]  time = 0.44, size = 29, normalized size = 1.12 \begin {gather*} \log {\left (2 x + \left (- \frac {x}{10} - \frac {3}{2}\right ) e^{e} + \frac {e^{e} \log {\relax (x )}}{10} \right )}^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+4)*exp(exp(1))+80*x)*ln(1/10*exp(exp(1))*ln(x)+1/10*(-x-15)*exp(exp(1))+2*x)**3/(x*exp(exp(1)
)*ln(x)+(-x**2-15*x)*exp(exp(1))+20*x**2),x)

[Out]

log(2*x + (-x/10 - 3/2)*exp(E) + exp(E)*log(x)/10)**4

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