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

Optimal. Leaf size=22 \[ 5+4 \left (x+\frac {e^{-x} \log \left (\frac {\log (x)}{x}\right )}{x^5}\right ) \]

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Rubi [A]  time = 0.52, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6742, 2288} \begin {gather*} \frac {4 e^{-x} \log \left (\frac {\log (x)}{x}\right )}{x^5}+4 x \end {gather*}

Antiderivative was successfully verified.

[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 2288

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 21, normalized size = 0.95 \begin {gather*} 4 x+\frac {4 e^{-x} \log \left (\frac {\log (x)}{x}\right )}{x^5} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.51, size = 23, normalized size = 1.05 \begin {gather*} \frac {4 \, {\left (x^{6} e^{x} + \log \left (\frac {\log \relax (x)}{x}\right )\right )} e^{\left (-x\right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.13, size = 25, normalized size = 1.14 \begin {gather*} \frac {4 \, {\left (x^{6} - e^{\left (-x\right )} \log \relax (x) + e^{\left (-x\right )} \log \left (\log \relax (x)\right )\right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [C]  time = 0.08, size = 119, normalized size = 5.41

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

Verification of antiderivative is not currently implemented for this CAS.

[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]

4/x^5*exp(-x)*ln(ln(x))-2*(-2*x^6*exp(x)+I*Pi*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))-I*Pi*csgn(I/x)*csgn(I/x*
ln(x))^2-I*Pi*csgn(I*ln(x))*csgn(I/x*ln(x))^2+I*Pi*csgn(I/x*ln(x))^3+2*ln(x))/x^5*exp(-x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 4 \, x - \frac {4 \, {\left (e^{\left (-x\right )} \log \relax (x) - e^{\left (-x\right )} \log \left (\log \relax (x)\right )\right )}}{x^{5}} + 4 \, \Gamma \left (-5, x\right ) + 4 \, \int \frac {e^{\left (-x\right )}}{x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{-x}\,\left (\ln \relax (x)\,\left (4\,x^6\,{\mathrm {e}}^x-4\right )-\ln \left (\frac {\ln \relax (x)}{x}\right )\,\ln \relax (x)\,\left (4\,x+20\right )+4\right )}{x^6\,\ln \relax (x)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.45, size = 17, normalized size = 0.77 \begin {gather*} 4 x + \frac {4 e^{- x} \log {\left (\frac {\log {\relax (x )}}{x} \right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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