3.8.61 \(\int \frac {\log ^{\frac {5}{x}}(5) (-2 x-5 \log (\log (5)))}{30 x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac {\log ^{\frac {5}{x}}(5)}{30 x^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 2288} \begin {gather*} \frac {\log ^{\frac {5}{x}}(5)}{30 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Log[5]^(5/x)*(-2*x - 5*Log[Log[5]]))/(30*x^4),x]

[Out]

Log[5]^(5/x)/(30*x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{30} \int \frac {\log ^{\frac {5}{x}}(5) (-2 x-5 \log (\log (5)))}{x^4} \, dx\\ &=\frac {\log ^{\frac {5}{x}}(5)}{30 x^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log ^{\frac {5}{x}}(5)}{30 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Log[5]^(5/x)*(-2*x - 5*Log[Log[5]]))/(30*x^4),x]

[Out]

Log[5]^(5/x)/(30*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \relax (5)^{\frac {5}{x}}}{30 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/30*(-5*log(log(5))-2*x)*exp(5*log(log(5))/x)/x^4,x, algorithm="fricas")

[Out]

1/30*log(5)^(5/x)/x^2

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, x + 5 \, \log \left (\log \relax (5)\right )\right )} \log \relax (5)^{\frac {5}{x}}}{30 \, x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/30*(-5*log(log(5))-2*x)*exp(5*log(log(5))/x)/x^4,x, algorithm="giac")

[Out]

integrate(-1/30*(2*x + 5*log(log(5)))*log(5)^(5/x)/x^4, x)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 14, normalized size = 0.93

method	result	size
risch	\(\frac {\ln \relax (5)^{\frac {5}{x}}}{30 x^{2}}\)	\(14\)
gosper	\(\frac {{\mathrm e}^{\frac {5 \ln \left (\ln \relax (5)\right )}{x}}}{30 x^{2}}\)	\(15\)
derivativedivides	\(\frac {{\mathrm e}^{\frac {5 \ln \left (\ln \relax (5)\right )}{x}}}{30 x^{2}}\)	\(15\)
default	\(\frac {{\mathrm e}^{\frac {5 \ln \left (\ln \relax (5)\right )}{x}}}{30 x^{2}}\)	\(15\)
norman	\(\frac {{\mathrm e}^{\frac {5 \ln \left (\ln \relax (5)\right )}{x}}}{30 x^{2}}\)	\(15\)
meijerg	\(-\frac {2-\frac {\left (\frac {75 \ln \left (\ln \relax (5)\right )^{2}}{x^{2}}-\frac {30 \ln \left (\ln \relax (5)\right )}{x}+6\right ) {\mathrm e}^{\frac {5 \ln \left (\ln \relax (5)\right )}{x}}}{3}}{750 \ln \left (\ln \relax (5)\right )^{2}}+\frac {1-\frac {\left (2-\frac {10 \ln \left (\ln \relax (5)\right )}{x}\right ) {\mathrm e}^{\frac {5 \ln \left (\ln \relax (5)\right )}{x}}}{2}}{375 \ln \left (\ln \relax (5)\right )^{2}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/30*(-5*ln(ln(5))-2*x)*exp(5*ln(ln(5))/x)/x^4,x,method=_RETURNVERBOSE)

[Out]

1/30/x^2*ln(5)^(5/x)

________________________________________________________________________________________

maxima [C]  time = 0.67, size = 35, normalized size = 2.33 \begin {gather*} \frac {\Gamma \left (3, -\frac {5 \, \log \left (\log \relax (5)\right )}{x}\right )}{750 \, \log \left (\log \relax (5)\right )^{2}} - \frac {\Gamma \left (2, -\frac {5 \, \log \left (\log \relax (5)\right )}{x}\right )}{375 \, \log \left (\log \relax (5)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/30*(-5*log(log(5))-2*x)*exp(5*log(log(5))/x)/x^4,x, algorithm="maxima")

[Out]

1/750*gamma(3, -5*log(log(5))/x)/log(log(5))^2 - 1/375*gamma(2, -5*log(log(5))/x)/log(log(5))^2

________________________________________________________________________________________

mupad [B]  time = 0.57, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\ln \relax (5)}^{5/x}}{30\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((5*log(log(5)))/x)*(2*x + 5*log(log(5))))/(30*x^4),x)

[Out]

log(5)^(5/x)/(30*x^2)

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 14, normalized size = 0.93 \begin {gather*} \frac {e^{\frac {5 \log {\left (\log {\relax (5 )} \right )}}{x}}}{30 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/30*(-5*ln(ln(5))-2*x)*exp(5*ln(ln(5))/x)/x**4,x)

[Out]

exp(5*log(log(5))/x)/(30*x**2)

________________________________________________________________________________________