3.11.35 \(\int \frac {-4 x \log (5)+x \log (5) \log (x)-4 \log ^2(x)+\log ^3(x)+(x \log (5)-8 \log (x)+7 \log ^2(x)-\log ^3(x)) \log (4 x+e^5 x)}{(-4 x^2 \log (5)+x^2 \log (5) \log (x)-4 x \log ^2(x)+x \log ^3(x)) \log (4 x+e^5 x)} \, dx\)

Optimal. Leaf size=25 \[ \log \left ((-4+\log (x)) \left (\log (5)+\frac {\log ^2(x)}{x}\right ) \log \left (\left (4+e^5\right ) x\right )\right ) \]

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Rubi [F]  time = 4.57, 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 {-4 x \log (5)+x \log (5) \log (x)-4 \log ^2(x)+\log ^3(x)+\left (x \log (5)-8 \log (x)+7 \log ^2(x)-\log ^3(x)\right ) \log \left (4 x+e^5 x\right )}{\left (-4 x^2 \log (5)+x^2 \log (5) \log (x)-4 x \log ^2(x)+x \log ^3(x)\right ) \log \left (4 x+e^5 x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x*Log[5] + x*Log[5]*Log[x] - 4*Log[x]^2 + Log[x]^3 + (x*Log[5] - 8*Log[x] + 7*Log[x]^2 - Log[x]^3)*Log
[4*x + E^5*x])/((-4*x^2*Log[5] + x^2*Log[5]*Log[x] - 4*x*Log[x]^2 + x*Log[x]^3)*Log[4*x + E^5*x]),x]

[Out]

-Log[x] + Log[4 - Log[x]] + Log[x*Log[5] + Log[x]^2] - 4*Log[5]*Defer[Int][1/((-4 + Log[x])*(x*Log[5] + Log[x]
^2)*Log[(4 + E^5)*x]), x] + Log[5]*Defer[Int][Log[x]/((-4 + Log[x])*(x*Log[5] + Log[x]^2)*Log[(4 + E^5)*x]), x
] - 4*Defer[Int][Log[x]^2/(x*(-4 + Log[x])*(x*Log[5] + Log[x]^2)*Log[(4 + E^5)*x]), x] + Defer[Int][Log[x]^3/(
x*(-4 + Log[x])*(x*Log[5] + Log[x]^2)*Log[(4 + E^5)*x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x \log (5)+x \log (5) \log (x)-4 \log ^2(x)+\log ^3(x)+\left (x \log (5)-8 \log (x)+7 \log ^2(x)-\log ^3(x)\right ) \log \left (4 x+e^5 x\right )}{\left (-4 x^2 \log (5)+x^2 \log (5) \log (x)-4 x \log ^2(x)+x \log ^3(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx\\ &=\int \frac {4 x \log (5)-x \log (5) \log (x)+4 \log ^2(x)-\log ^3(x)-\left (x \log (5)-8 \log (x)+7 \log ^2(x)-\log ^3(x)\right ) \log \left (4 x+e^5 x\right )}{x (4-\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx\\ &=\int \left (\frac {x \log (5)-8 \log (x)+7 \log ^2(x)-\log ^3(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right )}-\frac {4 \log (5)}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )}+\frac {\log (5) \log (x)}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )}-\frac {4 \log ^2(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )}+\frac {\log ^3(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )}\right ) \, dx\\ &=-\left (4 \int \frac {\log ^2(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx\right )+\log (5) \int \frac {\log (x)}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx-(4 \log (5)) \int \frac {1}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\int \frac {x \log (5)-8 \log (x)+7 \log ^2(x)-\log ^3(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right )} \, dx+\int \frac {\log ^3(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx\\ &=-\left (4 \int \frac {\log ^2(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx\right )+\log (5) \int \frac {\log (x)}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx-(4 \log (5)) \int \frac {1}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\int \left (-\frac {1}{x}+\frac {1}{x (-4+\log (x))}+\frac {x \log (5)+2 \log (x)}{x \left (x \log (5)+\log ^2(x)\right )}\right ) \, dx+\int \frac {\log ^3(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx\\ &=-\log (x)-4 \int \frac {\log ^2(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\log (5) \int \frac {\log (x)}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx-(4 \log (5)) \int \frac {1}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\int \frac {1}{x (-4+\log (x))} \, dx+\int \frac {x \log (5)+2 \log (x)}{x \left (x \log (5)+\log ^2(x)\right )} \, dx+\int \frac {\log ^3(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx\\ &=-\log (x)+\log \left (x \log (5)+\log ^2(x)\right )-4 \int \frac {\log ^2(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\log (5) \int \frac {\log (x)}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx-(4 \log (5)) \int \frac {1}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\int \frac {\log ^3(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-4+\log (x)\right )\\ &=-\log (x)+\log (4-\log (x))+\log \left (x \log (5)+\log ^2(x)\right )-4 \int \frac {\log ^2(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\log (5) \int \frac {\log (x)}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx-(4 \log (5)) \int \frac {1}{(-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx+\int \frac {\log ^3(x)}{x (-4+\log (x)) \left (x \log (5)+\log ^2(x)\right ) \log \left (\left (4+e^5\right ) x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 31, normalized size = 1.24 \begin {gather*} -\log (x)+\log (4-\log (x))+\log \left (x \log (5)+\log ^2(x)\right )+\log \left (\log \left (\left (4+e^5\right ) x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x*Log[5] + x*Log[5]*Log[x] - 4*Log[x]^2 + Log[x]^3 + (x*Log[5] - 8*Log[x] + 7*Log[x]^2 - Log[x]^
3)*Log[4*x + E^5*x])/((-4*x^2*Log[5] + x^2*Log[5]*Log[x] - 4*x*Log[x]^2 + x*Log[x]^3)*Log[4*x + E^5*x]),x]

[Out]

-Log[x] + Log[4 - Log[x]] + Log[x*Log[5] + Log[x]^2] + Log[Log[(4 + E^5)*x]]

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fricas [A]  time = 0.80, size = 29, normalized size = 1.16 \begin {gather*} \log \left (x \log \relax (5) + \log \relax (x)^{2}\right ) - \log \relax (x) + \log \left (\log \relax (x) + \log \left (e^{5} + 4\right )\right ) + \log \left (\log \relax (x) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(x)^3+7*log(x)^2-8*log(x)+x*log(5))*log(x*exp(5)+4*x)+log(x)^3-4*log(x)^2+x*log(5)*log(x)-4*x*
log(5))/(x*log(x)^3-4*x*log(x)^2+x^2*log(5)*log(x)-4*x^2*log(5))/log(x*exp(5)+4*x),x, algorithm="fricas")

[Out]

log(x*log(5) + log(x)^2) - log(x) + log(log(x) + log(e^5 + 4)) + log(log(x) - 4)

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giac [B]  time = 1.04, size = 74, normalized size = 2.96 \begin {gather*} \log \left (x \log \relax (5) \log \relax (x)^{2} + \log \relax (x)^{4} + x \log \relax (5) \log \relax (x) \log \left (e^{5} + 4\right ) + \log \relax (x)^{3} \log \left (e^{5} + 4\right ) - 4 \, x \log \relax (5) \log \relax (x) - 4 \, \log \relax (x)^{3} - 4 \, x \log \relax (5) \log \left (e^{5} + 4\right ) - 4 \, \log \relax (x)^{2} \log \left (e^{5} + 4\right )\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(x)^3+7*log(x)^2-8*log(x)+x*log(5))*log(x*exp(5)+4*x)+log(x)^3-4*log(x)^2+x*log(5)*log(x)-4*x*
log(5))/(x*log(x)^3-4*x*log(x)^2+x^2*log(5)*log(x)-4*x^2*log(5))/log(x*exp(5)+4*x),x, algorithm="giac")

[Out]

log(x*log(5)*log(x)^2 + log(x)^4 + x*log(5)*log(x)*log(e^5 + 4) + log(x)^3*log(e^5 + 4) - 4*x*log(5)*log(x) -
4*log(x)^3 - 4*x*log(5)*log(e^5 + 4) - 4*log(x)^2*log(e^5 + 4)) - log(x)

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maple [A]  time = 0.08, size = 29, normalized size = 1.16

method	result	size
default	\(\ln \left (\ln \left (\left (4+{\mathrm e}^{5}\right ) x \right )\right )-\ln \relax (x )+\ln \left (\ln \relax (x )-4\right )+\ln \left (x \ln \relax (5)+\ln \relax (x )^{2}\right )\)	\(29\)
risch	\(-\ln \relax (x )+\ln \left (\ln \relax (x )^{2} \ln \relax (5) x +\ln \relax (x )^{4}-4 x \ln \relax (5) \ln \relax (x )-4 \ln \relax (x )^{3}\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-ln(x)^3+7*ln(x)^2-8*ln(x)+x*ln(5))*ln(x*exp(5)+4*x)+ln(x)^3-4*ln(x)^2+x*ln(5)*ln(x)-4*x*ln(5))/(x*ln(x)
^3-4*x*ln(x)^2+x^2*ln(5)*ln(x)-4*x^2*ln(5))/ln(x*exp(5)+4*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.09, size = 29, normalized size = 1.16 \begin {gather*} \log \left (x \log \relax (5) + \log \relax (x)^{2}\right ) - \log \relax (x) + \log \left (\log \relax (x) + \log \left (e^{5} + 4\right )\right ) + \log \left (\log \relax (x) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(x)^3+7*log(x)^2-8*log(x)+x*log(5))*log(x*exp(5)+4*x)+log(x)^3-4*log(x)^2+x*log(5)*log(x)-4*x*
log(5))/(x*log(x)^3-4*x*log(x)^2+x^2*log(5)*log(x)-4*x^2*log(5))/log(x*exp(5)+4*x),x, algorithm="maxima")

[Out]

log(x*log(5) + log(x)^2) - log(x) + log(log(x) + log(e^5 + 4)) + log(log(x) - 4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \left (4\,x+x\,{\mathrm {e}}^5\right )\,\left ({\ln \relax (x)}^3-7\,{\ln \relax (x)}^2+8\,\ln \relax (x)-x\,\ln \relax (5)\right )+4\,x\,\ln \relax (5)+4\,{\ln \relax (x)}^2-{\ln \relax (x)}^3-x\,\ln \relax (5)\,\ln \relax (x)}{\ln \left (4\,x+x\,{\mathrm {e}}^5\right )\,\left (-\ln \relax (5)\,x^2\,\ln \relax (x)+4\,\ln \relax (5)\,x^2-x\,{\ln \relax (x)}^3+4\,x\,{\ln \relax (x)}^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4*x + x*exp(5))*(8*log(x) - x*log(5) - 7*log(x)^2 + log(x)^3) + 4*x*log(5) + 4*log(x)^2 - log(x)^3 -
x*log(5)*log(x))/(log(4*x + x*exp(5))*(4*x*log(x)^2 - x*log(x)^3 + 4*x^2*log(5) - x^2*log(5)*log(x))),x)

[Out]

int((log(4*x + x*exp(5))*(8*log(x) - x*log(5) - 7*log(x)^2 + log(x)^3) + 4*x*log(5) + 4*log(x)^2 - log(x)^3 -
x*log(5)*log(x))/(log(4*x + x*exp(5))*(4*x*log(x)^2 - x*log(x)^3 + 4*x^2*log(5) - x^2*log(5)*log(x))), x)

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sympy [B]  time = 0.90, size = 76, normalized size = 3.04 \begin {gather*} - \log {\relax (x )} + \log {\left (- 4 x \log {\relax (5 )} \log {\left (4 + e^{5} \right )} + \left (- 4 x \log {\relax (5 )} + x \log {\relax (5 )} \log {\left (4 + e^{5} \right )}\right ) \log {\relax (x )} + \left (x \log {\relax (5 )} - 4 \log {\left (4 + e^{5} \right )}\right ) \log {\relax (x )}^{2} + \log {\relax (x )}^{4} + \left (-4 + \log {\left (4 + e^{5} \right )}\right ) \log {\relax (x )}^{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(x)**3+7*ln(x)**2-8*ln(x)+x*ln(5))*ln(x*exp(5)+4*x)+ln(x)**3-4*ln(x)**2+x*ln(5)*ln(x)-4*x*ln(5)
)/(x*ln(x)**3-4*x*ln(x)**2+x**2*ln(5)*ln(x)-4*x**2*ln(5))/ln(x*exp(5)+4*x),x)

[Out]

-log(x) + log(-4*x*log(5)*log(4 + exp(5)) + (-4*x*log(5) + x*log(5)*log(4 + exp(5)))*log(x) + (x*log(5) - 4*lo
g(4 + exp(5)))*log(x)**2 + log(x)**4 + (-4 + log(4 + exp(5)))*log(x)**3)

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