\(\int \frac {e^{-x} (4+(-4+4 e^x x^6) \log (x)+(-20-4 x) \log (x) \log (\frac {\log (x)}{x}))}{x^6 \log (x)} \, dx\) [7481]

Optimal result

Integrand size = 43, antiderivative size = 22 \[ \int \frac {e^{-x} \left (4+\left (-4+4 e^x x^6\right ) \log (x)+(-20-4 x) \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{x^6 \log (x)} \, dx=5+4 \left (x+\frac {e^{-x} \log \left (\frac {\log (x)}{x}\right )}{x^5}\right ) \]

[Out]

4/x^5/exp(x)*ln(ln(x)/x)+4*x+5

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6874, 2326} \[ \int \frac {e^{-x} \left (4+\left (-4+4 e^x x^6\right ) \log (x)+(-20-4 x) \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{x^6 \log (x)} \, dx=\frac {4 e^{-x} \log \left (\frac {\log (x)}{x}\right )}{x^5}+4 x \]

[In]

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

[Out]

4*x + (4*Log[Log[x]/x])/(E^x*x^5)

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6874

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-x} \left (4+\left (-4+4 e^x x^6\right ) \log (x)+(-20-4 x) \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{x^6 \log (x)} \, dx=4 x+\frac {4 e^{-x} \log \left (\frac {\log (x)}{x}\right )}{x^5} \]

[In]

Integrate[(4 + (-4 + 4*E^x*x^6)*Log[x] + (-20 - 4*x)*Log[x]*Log[Log[x]/x])/(E^x*x^6*Log[x]),x]

[Out]

4*x + (4*Log[Log[x]/x])/(E^x*x^5)

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18

method	result	size
parallelrisch	\(\frac {\left (4 x^{6} {\mathrm e}^{x}+4 \ln \left (\frac {\ln \left (x \right )}{x}\right )\right ) {\mathrm e}^{-x}}{x^{5}}\)	\(26\)
risch	\(\frac {4 \,{\mathrm e}^{-x} \ln \left (\ln \left (x \right )\right )}{x^{5}}-\frac {2 \left (-2 x^{6} {\mathrm e}^{x}-i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+i \pi \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+2 \ln \left (x \right )\right ) {\mathrm e}^{-x}}{x^{5}}\)	\(119\)

[In]

int(((-4*x-20)*ln(x)*ln(ln(x)/x)+(4*x^6*exp(x)-4)*ln(x)+4)/x^6/exp(x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-x} \left (4+\left (-4+4 e^x x^6\right ) \log (x)+(-20-4 x) \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{x^6 \log (x)} \, dx=\frac {4 \, {\left (x^{6} e^{x} + \log \left (\frac {\log \left (x\right )}{x}\right )\right )} e^{\left (-x\right )}}{x^{5}} \]

[In]

integrate(((-4*x-20)*log(x)*log(log(x)/x)+(4*x^6*exp(x)-4)*log(x)+4)/x^6/exp(x)/log(x),x, algorithm="fricas")

[Out]

4*(x^6*e^x + log(log(x)/x))*e^(-x)/x^5

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-x} \left (4+\left (-4+4 e^x x^6\right ) \log (x)+(-20-4 x) \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{x^6 \log (x)} \, dx=4 x + \frac {4 e^{- x} \log {\left (\frac {\log {\left (x \right )}}{x} \right )}}{x^{5}} \]

[In]

integrate(((-4*x-20)*ln(x)*ln(ln(x)/x)+(4*x**6*exp(x)-4)*ln(x)+4)/x**6/exp(x)/ln(x),x)

[Out]

4*x + 4*exp(-x)*log(log(x)/x)/x**5

Maxima [F]

\[ \int \frac {e^{-x} \left (4+\left (-4+4 e^x x^6\right ) \log (x)+(-20-4 x) \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{x^6 \log (x)} \, dx=\int { -\frac {4 \, {\left ({\left (x + 5\right )} \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right ) - {\left (x^{6} e^{x} - 1\right )} \log \left (x\right ) - 1\right )} e^{\left (-x\right )}}{x^{6} \log \left (x\right )} \,d x } \]

[In]

integrate(((-4*x-20)*log(x)*log(log(x)/x)+(4*x^6*exp(x)-4)*log(x)+4)/x^6/exp(x)/log(x),x, algorithm="maxima")

[Out]

4*x - 4*(e^(-x)*log(x) - e^(-x)*log(log(x)))/x^5 + 4*gamma(-5, x) + 4*integrate(e^(-x)/x^6, x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-x} \left (4+\left (-4+4 e^x x^6\right ) \log (x)+(-20-4 x) \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{x^6 \log (x)} \, dx=\frac {4 \, {\left (x^{6} - e^{\left (-x\right )} \log \left (x\right ) + e^{\left (-x\right )} \log \left (\log \left (x\right )\right )\right )}}{x^{5}} \]

[In]

integrate(((-4*x-20)*log(x)*log(log(x)/x)+(4*x^6*exp(x)-4)*log(x)+4)/x^6/exp(x)/log(x),x, algorithm="giac")

[Out]

4*(x^6 - e^(-x)*log(x) + e^(-x)*log(log(x)))/x^5

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-x} \left (4+\left (-4+4 e^x x^6\right ) \log (x)+(-20-4 x) \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{x^6 \log (x)} \, dx=\int \frac {{\mathrm {e}}^{-x}\,\left (\ln \left (x\right )\,\left (4\,x^6\,{\mathrm {e}}^x-4\right )-\ln \left (\frac {\ln \left (x\right )}{x}\right )\,\ln \left (x\right )\,\left (4\,x+20\right )+4\right )}{x^6\,\ln \left (x\right )} \,d x \]

[In]

int((exp(-x)*(log(x)*(4*x^6*exp(x) - 4) - log(log(x)/x)*log(x)*(4*x + 20) + 4))/(x^6*log(x)),x)

[Out]

int((exp(-x)*(log(x)*(4*x^6*exp(x) - 4) - log(log(x)/x)*log(x)*(4*x + 20) + 4))/(x^6*log(x)), x)