\(\int \frac {(20+x^2-x^3) \log (6)+(20-x^2+2 x^3) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} (-20 \log (6)+(20 e^x x-20 \log (6)) \log (x))}{5 x^2 \log (6) \log ^2(x)} \, dx\) [7910]

Optimal result

Integrand size = 76, antiderivative size = 34 \[ \int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx=\frac {\frac {4 \left (-1+e^{\frac {e^x}{\log (6)}}\right )}{x}+\frac {1}{5} \left (-x+x^2\right )}{\log (x)} \]

[Out]

(1/5*x^2-1/5*x+4*(exp(exp(x)/ln(6))-1)/x)/ln(x)

Rubi [F]

\[ \int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx=\int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx \]

[In]

Int[((20 + x^2 - x^3)*Log[6] + (20 - x^2 + 2*x^3)*Log[6]*Log[x] + E^(E^x/Log[6])*(-20*Log[6] + (20*E^x*x - 20*
Log[6])*Log[x]))/(5*x^2*Log[6]*Log[x]^2),x]

[Out]

(4*E^(E^x/Log[6]))/(x*Log[x]) + Defer[Int][(20 + x^2 - x^3)/(x^2*Log[x]^2), x]/5 + Defer[Int][(20 - x^2 + 2*x^
3)/(x^2*Log[x]), x]/5

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{x^2 \log ^2(x)} \, dx}{5 \log (6)} \\ & = \frac {\int \left (\frac {\log (6) \left (20+x^2-x^3+20 \log (x)-x^2 \log (x)+2 x^3 \log (x)\right )}{x^2 \log ^2(x)}+\frac {20 e^{\frac {e^x}{\log (6)}} \left (-\log (6)+e^x x \log (x)-\log (6) \log (x)\right )}{x^2 \log ^2(x)}\right ) \, dx}{5 \log (6)} \\ & = \frac {1}{5} \int \frac {20+x^2-x^3+20 \log (x)-x^2 \log (x)+2 x^3 \log (x)}{x^2 \log ^2(x)} \, dx+\frac {4 \int \frac {e^{\frac {e^x}{\log (6)}} \left (-\log (6)+e^x x \log (x)-\log (6) \log (x)\right )}{x^2 \log ^2(x)} \, dx}{\log (6)} \\ & = \frac {4 e^{\frac {e^x}{\log (6)}}}{x \log (x)}+\frac {1}{5} \int \left (\frac {20+x^2-x^3}{x^2 \log ^2(x)}+\frac {20-x^2+2 x^3}{x^2 \log (x)}\right ) \, dx \\ & = \frac {4 e^{\frac {e^x}{\log (6)}}}{x \log (x)}+\frac {1}{5} \int \frac {20+x^2-x^3}{x^2 \log ^2(x)} \, dx+\frac {1}{5} \int \frac {20-x^2+2 x^3}{x^2 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx=\frac {-20+20 e^{\frac {e^x}{\log (6)}}-x^2+x^3}{5 x \log (x)} \]

[In]

Integrate[((20 + x^2 - x^3)*Log[6] + (20 - x^2 + 2*x^3)*Log[6]*Log[x] + E^(E^x/Log[6])*(-20*Log[6] + (20*E^x*x
- 20*Log[6])*Log[x]))/(5*x^2*Log[6]*Log[x]^2),x]

[Out]

(-20 + 20*E^(E^x/Log[6]) - x^2 + x^3)/(5*x*Log[x])

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29

method	result	size
parallelrisch	\(\frac {\ln \left (6\right ) x^{3}-x^{2} \ln \left (6\right )+20 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{\ln \left (6\right )}} \ln \left (6\right )-20 \ln \left (6\right )}{5 \ln \left (6\right ) x \ln \left (x \right )}\)	\(44\)
risch	\(\frac {x^{3} \ln \left (3\right )+x^{3} \ln \left (2\right )-x^{2} \ln \left (3\right )-x^{2} \ln \left (2\right )-20 \ln \left (3\right )-20 \ln \left (2\right )}{5 \left (\ln \left (2\right )+\ln \left (3\right )\right ) x \ln \left (x \right )}+\frac {4 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{\ln \left (2\right )+\ln \left (3\right )}}}{\ln \left (x \right ) x}\)	\(73\)

[In]

int(1/5*(((20*exp(x)*x-20*ln(6))*ln(x)-20*ln(6))*exp(exp(x)/ln(6))+(2*x^3-x^2+20)*ln(6)*ln(x)+(-x^3+x^2+20)*ln
(6))/x^2/ln(6)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/5/ln(6)/x*(ln(6)*x^3-x^2*ln(6)+20*exp(exp(x)/ln(6))*ln(6)-20*ln(6))/ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx=\frac {x^{3} - x^{2} + 20 \, e^{\left (\frac {e^{x}}{\log \left (6\right )}\right )} - 20}{5 \, x \log \left (x\right )} \]

[In]

integrate(1/5*(((20*exp(x)*x-20*log(6))*log(x)-20*log(6))*exp(exp(x)/log(6))+(2*x^3-x^2+20)*log(6)*log(x)+(-x^
3+x^2+20)*log(6))/x^2/log(6)/log(x)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx=\frac {x^{3} - x^{2} - 20}{5 x \log {\left (x \right )}} + \frac {4 e^{\frac {e^{x}}{\log {\left (6 \right )}}}}{x \log {\left (x \right )}} \]

[In]

integrate(1/5*(((20*exp(x)*x-20*ln(6))*ln(x)-20*ln(6))*exp(exp(x)/ln(6))+(2*x**3-x**2+20)*ln(6)*ln(x)+(-x**3+x
**2+20)*ln(6))/x**2/ln(6)/ln(x)**2,x)

[Out]

(x**3 - x**2 - 20)/(5*x*log(x)) + 4*exp(exp(x)/log(6))/(x*log(x))

Maxima [F]

\[ \int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx=\int { \frac {{\left (2 \, x^{3} - x^{2} + 20\right )} \log \left (6\right ) \log \left (x\right ) + 20 \, {\left ({\left (x e^{x} - \log \left (6\right )\right )} \log \left (x\right ) - \log \left (6\right )\right )} e^{\left (\frac {e^{x}}{\log \left (6\right )}\right )} - {\left (x^{3} - x^{2} - 20\right )} \log \left (6\right )}{5 \, x^{2} \log \left (6\right ) \log \left (x\right )^{2}} \,d x } \]

[In]

integrate(1/5*(((20*exp(x)*x-20*log(6))*log(x)-20*log(6))*exp(exp(x)/log(6))+(2*x^3-x^2+20)*log(6)*log(x)+(-x^
3+x^2+20)*log(6))/x^2/log(6)/log(x)^2,x, algorithm="maxima")

[Out]

1/5*(gamma(-1, -log(x))*log(6) - 2*gamma(-1, -2*log(x))*log(6) - 20*gamma(-1, log(x))*log(6) + 20*(log(3) + lo
g(2))*e^(e^x/(log(3) + log(2)))/(x*log(x)) + integrate((2*x^3*(log(3) + log(2)) - x^2*(log(3) + log(2)) + 20*l
og(3) + 20*log(2))/(x^2*log(x)), x))/log(6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (28) = 56\).

Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71 \[ \int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx=\frac {{\left (x^{3} e^{x} \log \left (6\right ) - x^{2} e^{x} \log \left (6\right ) - 20 \, e^{x} \log \left (6\right ) + 20 \, e^{\left (\frac {x \log \left (6\right ) + e^{x}}{\log \left (6\right )}\right )} \log \left (6\right )\right )} e^{\left (-x\right )}}{5 \, x \log \left (6\right ) \log \left (x\right )} \]

[In]

integrate(1/5*(((20*exp(x)*x-20*log(6))*log(x)-20*log(6))*exp(exp(x)/log(6))+(2*x^3-x^2+20)*log(6)*log(x)+(-x^
3+x^2+20)*log(6))/x^2/log(6)/log(x)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (20+x^2-x^3\right ) \log (6)+\left (20-x^2+2 x^3\right ) \log (6) \log (x)+e^{\frac {e^x}{\log (6)}} \left (-20 \log (6)+\left (20 e^x x-20 \log (6)\right ) \log (x)\right )}{5 x^2 \log (6) \log ^2(x)} \, dx=\int \frac {\frac {\ln \left (6\right )\,\left (-x^3+x^2+20\right )}{5}-\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{\ln \left (6\right )}}\,\left (20\,\ln \left (6\right )+\ln \left (x\right )\,\left (20\,\ln \left (6\right )-20\,x\,{\mathrm {e}}^x\right )\right )}{5}+\frac {\ln \left (6\right )\,\ln \left (x\right )\,\left (2\,x^3-x^2+20\right )}{5}}{x^2\,\ln \left (6\right )\,{\ln \left (x\right )}^2} \,d x \]

[In]

int(((log(6)*(x^2 - x^3 + 20))/5 - (exp(exp(x)/log(6))*(20*log(6) + log(x)*(20*log(6) - 20*x*exp(x))))/5 + (lo
g(6)*log(x)*(2*x^3 - x^2 + 20))/5)/(x^2*log(6)*log(x)^2),x)

[Out]

int(((log(6)*(x^2 - x^3 + 20))/5 - (exp(exp(x)/log(6))*(20*log(6) + log(x)*(20*log(6) - 20*x*exp(x))))/5 + (lo
g(6)*log(x)*(2*x^3 - x^2 + 20))/5)/(x^2*log(6)*log(x)^2), x)