3.53.59 \(\int \frac {-x-625 x^5-\log (2)+(2500 x^5-\log (2)) \log (5 x)}{x^2 \log ^2(5 x)} \, dx\)

Optimal. Leaf size=19 \[ \frac {x+625 x^5+\log (2)}{x \log (5 x)} \]

________________________________________________________________________________________

Rubi [F]  time = 0.28, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-x-625 x^5-\log (2)+\left (2500 x^5-\log (2)\right ) \log (5 x)}{x^2 \log ^2(5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x - 625*x^5 - Log[2] + (2500*x^5 - Log[2])*Log[5*x])/(x^2*Log[5*x]^2),x]

[Out]

4*ExpIntegralEi[4*Log[5*x]] - 5*ExpIntegralEi[-Log[5*x]]*Log[2] + Defer[Int][(-x - 625*x^5 - Log[2])/(x^2*Log[
5*x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-x-625 x^5-\log (2)}{x^2 \log ^2(5 x)}+\frac {2500 x^5-\log (2)}{x^2 \log (5 x)}\right ) \, dx\\ &=\int \frac {-x-625 x^5-\log (2)}{x^2 \log ^2(5 x)} \, dx+\int \frac {2500 x^5-\log (2)}{x^2 \log (5 x)} \, dx\\ &=\int \left (\frac {2500 x^3}{\log (5 x)}-\frac {\log (2)}{x^2 \log (5 x)}\right ) \, dx+\int \frac {-x-625 x^5-\log (2)}{x^2 \log ^2(5 x)} \, dx\\ &=2500 \int \frac {x^3}{\log (5 x)} \, dx-\log (2) \int \frac {1}{x^2 \log (5 x)} \, dx+\int \frac {-x-625 x^5-\log (2)}{x^2 \log ^2(5 x)} \, dx\\ &=4 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (5 x)\right )-(5 \log (2)) \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (5 x)\right )+\int \frac {-x-625 x^5-\log (2)}{x^2 \log ^2(5 x)} \, dx\\ &=4 \text {Ei}(4 \log (5 x))-5 \text {Ei}(-\log (5 x)) \log (2)+\int \frac {-x-625 x^5-\log (2)}{x^2 \log ^2(5 x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 19, normalized size = 1.00 \begin {gather*} \frac {x+625 x^5+\log (2)}{x \log (5 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x - 625*x^5 - Log[2] + (2500*x^5 - Log[2])*Log[5*x])/(x^2*Log[5*x]^2),x]

[Out]

(x + 625*x^5 + Log[2])/(x*Log[5*x])

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 19, normalized size = 1.00 \begin {gather*} \frac {625 \, x^{5} + x + \log \relax (2)}{x \log \left (5 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(2)+2500*x^5)*log(5*x)-log(2)-625*x^5-x)/x^2/log(5*x)^2,x, algorithm="fricas")

[Out]

(625*x^5 + x + log(2))/(x*log(5*x))

________________________________________________________________________________________

giac [A]  time = 0.21, size = 19, normalized size = 1.00 \begin {gather*} \frac {625 \, x^{5} + x + \log \relax (2)}{x \log \left (5 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(2)+2500*x^5)*log(5*x)-log(2)-625*x^5-x)/x^2/log(5*x)^2,x, algorithm="giac")

[Out]

(625*x^5 + x + log(2))/(x*log(5*x))

________________________________________________________________________________________

maple [A]  time = 0.09, size = 20, normalized size = 1.05

method	result	size
norman	\(\frac {x +625 x^{5}+\ln \relax (2)}{x \ln \left (5 x \right )}\)	\(20\)
risch	\(\frac {x +625 x^{5}+\ln \relax (2)}{x \ln \left (5 x \right )}\)	\(20\)
derivativedivides	\(\frac {625 x^{4}}{\ln \left (5 x \right )}+5 \ln \relax (2) \expIntegralEi \left (1, \ln \left (5 x \right )\right )-5 \ln \relax (2) \left (-\frac {1}{5 x \ln \left (5 x \right )}+\expIntegralEi \left (1, \ln \left (5 x \right )\right )\right )+\frac {1}{\ln \left (5 x \right )}\)	\(51\)
default	\(\frac {625 x^{4}}{\ln \left (5 x \right )}+5 \ln \relax (2) \expIntegralEi \left (1, \ln \left (5 x \right )\right )-5 \ln \relax (2) \left (-\frac {1}{5 x \ln \left (5 x \right )}+\expIntegralEi \left (1, \ln \left (5 x \right )\right )\right )+\frac {1}{\ln \left (5 x \right )}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-ln(2)+2500*x^5)*ln(5*x)-ln(2)-625*x^5-x)/x^2/ln(5*x)^2,x,method=_RETURNVERBOSE)

[Out]

(x+625*x^5+ln(2))/x/ln(5*x)

________________________________________________________________________________________

maxima [C]  time = 0.38, size = 47, normalized size = 2.47 \begin {gather*} -5 \, {\rm Ei}\left (-\log \left (5 \, x\right )\right ) \log \relax (2) + 5 \, \Gamma \left (-1, \log \left (5 \, x\right )\right ) \log \relax (2) + \frac {1}{\log \left (5 \, x\right )} + 4 \, {\rm Ei}\left (4 \, \log \left (5 \, x\right )\right ) - 4 \, \Gamma \left (-1, -4 \, \log \left (5 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(2)+2500*x^5)*log(5*x)-log(2)-625*x^5-x)/x^2/log(5*x)^2,x, algorithm="maxima")

[Out]

-5*Ei(-log(5*x))*log(2) + 5*gamma(-1, log(5*x))*log(2) + 1/log(5*x) + 4*Ei(4*log(5*x)) - 4*gamma(-1, -4*log(5*
x))

________________________________________________________________________________________

mupad [B]  time = 3.55, size = 19, normalized size = 1.00 \begin {gather*} \frac {625\,x^5+x+\ln \relax (2)}{x\,\ln \left (5\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + log(2) + log(5*x)*(log(2) - 2500*x^5) + 625*x^5)/(x^2*log(5*x)^2),x)

[Out]

(x + log(2) + 625*x^5)/(x*log(5*x))

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 15, normalized size = 0.79 \begin {gather*} \frac {625 x^{5} + x + \log {\relax (2 )}}{x \log {\left (5 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(2)+2500*x**5)*ln(5*x)-ln(2)-625*x**5-x)/x**2/ln(5*x)**2,x)

[Out]

(625*x**5 + x + log(2))/(x*log(5*x))

________________________________________________________________________________________