3.60.100 \(\int \frac {-5+(5+4 x \log (\frac {5}{4+e})) \log (\frac {5+4 x \log (\frac {5}{4+e})}{x \log (\frac {5}{4+e})})}{10+8 x \log (\frac {5}{4+e})} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{2} x \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6742, 2448, 263, 31} \begin {gather*} \frac {1}{2} x \log \left (\frac {5}{x \log \left (\frac {5}{4+e}\right )}+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + (5 + 4*x*Log[5/(4 + E)])*Log[(5 + 4*x*Log[5/(4 + E)])/(x*Log[5/(4 + E)])])/(10 + 8*x*Log[5/(4 + E)])
,x]

[Out]

(x*Log[4 + 5/(x*Log[5/(4 + E)])])/2

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, 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 (-\frac {5}{2 \left (5+4 x \log \left (\frac {5}{4+e}\right )\right )}+\frac {1}{2} \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right )\right ) \, dx\\ &=-\frac {5 \log \left (5+4 x \log \left (\frac {5}{4+e}\right )\right )}{8 \log \left (\frac {5}{4+e}\right )}+\frac {1}{2} \int \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right ) \, dx\\ &=\frac {1}{2} x \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right )-\frac {5 \log \left (5+4 x \log \left (\frac {5}{4+e}\right )\right )}{8 \log \left (\frac {5}{4+e}\right )}+\frac {5 \int \frac {1}{x \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right )} \, dx}{2 \log \left (\frac {5}{4+e}\right )}\\ &=\frac {1}{2} x \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right )-\frac {5 \log \left (5+4 x \log \left (\frac {5}{4+e}\right )\right )}{8 \log \left (\frac {5}{4+e}\right )}+\frac {5 \int \frac {1}{4 x+\frac {5}{\log \left (\frac {5}{4+e}\right )}} \, dx}{2 \log \left (\frac {5}{4+e}\right )}\\ &=\frac {1}{2} x \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 67, normalized size = 2.91 \begin {gather*} \frac {5 \log (x)+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right )-5 \log \left (5+4 x \log \left (\frac {5}{4+e}\right )\right )}{8 \log \left (\frac {5}{4+e}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + (5 + 4*x*Log[5/(4 + E)])*Log[(5 + 4*x*Log[5/(4 + E)])/(x*Log[5/(4 + E)])])/(10 + 8*x*Log[5/(4
+ E)]),x]

[Out]

(5*Log[x] + (5 + 4*x*Log[5/(4 + E)])*Log[4 + 5/(x*Log[5/(4 + E)])] - 5*Log[5 + 4*x*Log[5/(4 + E)]])/(8*Log[5/(
4 + E)])

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fricas [A]  time = 0.61, size = 33, normalized size = 1.43 \begin {gather*} \frac {1}{2} \, x \log \left (\frac {4 \, x \log \left (\frac {5}{e + 4}\right ) + 5}{x \log \left (\frac {5}{e + 4}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(5/(exp(1)+4))+5)*log((4*x*log(5/(exp(1)+4))+5)/x/log(5/(exp(1)+4)))-5)/(8*x*log(5/(exp(1)+
4))+10),x, algorithm="fricas")

[Out]

1/2*x*log((4*x*log(5/(e + 4)) + 5)/(x*log(5/(e + 4))))

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giac [B]  time = 0.20, size = 176, normalized size = 7.65 \begin {gather*} -\frac {5 \, \log \left (\frac {5}{e + 4}\right ) \log \left (\frac {4 \, x \log \left (\frac {5}{e + 4}\right ) + 5}{x \log \left (\frac {5}{e + 4}\right )}\right )}{2 \, {\left (4 \, \log \relax (5)^{2} - 8 \, \log \relax (5) \log \left (e + 4\right ) + 4 \, \log \left (e + 4\right )^{2} - \frac {{\left (4 \, x \log \left (\frac {5}{e + 4}\right ) + 5\right )} \log \relax (5)^{2}}{x \log \left (\frac {5}{e + 4}\right )} + \frac {2 \, {\left (4 \, x \log \left (\frac {5}{e + 4}\right ) + 5\right )} \log \relax (5) \log \left (e + 4\right )}{x \log \left (\frac {5}{e + 4}\right )} - \frac {{\left (4 \, x \log \left (\frac {5}{e + 4}\right ) + 5\right )} \log \left (e + 4\right )^{2}}{x \log \left (\frac {5}{e + 4}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(5/(exp(1)+4))+5)*log((4*x*log(5/(exp(1)+4))+5)/x/log(5/(exp(1)+4)))-5)/(8*x*log(5/(exp(1)+
4))+10),x, algorithm="giac")

[Out]

-5/2*log(5/(e + 4))*log((4*x*log(5/(e + 4)) + 5)/(x*log(5/(e + 4))))/(4*log(5)^2 - 8*log(5)*log(e + 4) + 4*log
(e + 4)^2 - (4*x*log(5/(e + 4)) + 5)*log(5)^2/(x*log(5/(e + 4))) + 2*(4*x*log(5/(e + 4)) + 5)*log(5)*log(e + 4
)/(x*log(5/(e + 4))) - (4*x*log(5/(e + 4)) + 5)*log(e + 4)^2/(x*log(5/(e + 4))))

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maple [A]  time = 0.32, size = 34, normalized size = 1.48

method	result	size
norman	\(\frac {x \ln \left (\frac {4 x \ln \left (\frac {5}{{\mathrm e}+4}\right )+5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}\right )}{2}\)	\(34\)
risch	\(\frac {x \ln \left (\frac {4 x \left (\ln \relax (5)-\ln \left ({\mathrm e}+4\right )\right )+5}{x \left (\ln \relax (5)-\ln \left ({\mathrm e}+4\right )\right )}\right )}{2}\)	\(36\)
derivativedivides	\(-\frac {-\frac {\ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \ln \left (\frac {5}{{\mathrm e}+4}\right ) x}{4}+\frac {5 \ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right )}{4}}{2 \ln \left (\frac {5}{{\mathrm e}+4}\right )}\)	\(85\)
default	\(-\frac {-\frac {\ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \ln \left (\frac {5}{{\mathrm e}+4}\right ) x}{4}+\frac {5 \ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right )}{4}}{2 \ln \left (\frac {5}{{\mathrm e}+4}\right )}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x*ln(5/(exp(1)+4))+5)*ln((4*x*ln(5/(exp(1)+4))+5)/x/ln(5/(exp(1)+4)))-5)/(8*x*ln(5/(exp(1)+4))+10),x,m
ethod=_RETURNVERBOSE)

[Out]

1/2*x*ln((4*x*ln(5/(exp(1)+4))+5)/x/ln(5/(exp(1)+4)))

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maxima [B]  time = 0.49, size = 128, normalized size = 5.57 \begin {gather*} -\frac {4 \, x {\left (\log \relax (5) - \log \left (e + 4\right )\right )} \log \relax (x) + 4 \, {\left (\log \relax (5) \log \left (-\log \relax (5) + \log \left (e + 4\right )\right ) - \log \left (e + 4\right ) \log \left (-\log \relax (5) + \log \left (e + 4\right )\right )\right )} x - {\left (4 \, x {\left (\log \relax (5) - \log \left (e + 4\right )\right )} + 5\right )} \log \left (-4 \, x {\left (\log \relax (5) - \log \left (e + 4\right )\right )} - 5\right )}{8 \, {\left (\log \relax (5) - \log \left (e + 4\right )\right )}} - \frac {5 \, \log \left (4 \, x \log \left (\frac {5}{e + 4}\right ) + 5\right )}{8 \, \log \left (\frac {5}{e + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(5/(exp(1)+4))+5)*log((4*x*log(5/(exp(1)+4))+5)/x/log(5/(exp(1)+4)))-5)/(8*x*log(5/(exp(1)+
4))+10),x, algorithm="maxima")

[Out]

-1/8*(4*x*(log(5) - log(e + 4))*log(x) + 4*(log(5)*log(-log(5) + log(e + 4)) - log(e + 4)*log(-log(5) + log(e
+ 4)))*x - (4*x*(log(5) - log(e + 4)) + 5)*log(-4*x*(log(5) - log(e + 4)) - 5))/(log(5) - log(e + 4)) - 5/8*lo
g(4*x*log(5/(e + 4)) + 5)/log(5/(e + 4))

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mupad [B]  time = 8.06, size = 38, normalized size = 1.65 \begin {gather*} -\frac {x\,\left (\ln \left (-\ln \left (\frac {5}{\mathrm {e}+4}\right )\right )-\ln \left (-\frac {4\,x\,\ln \left (\frac {5}{\mathrm {e}+4}\right )+5}{x}\right )\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((4*x*log(5/(exp(1) + 4)) + 5)/(x*log(5/(exp(1) + 4))))*(4*x*log(5/(exp(1) + 4)) + 5) - 5)/(8*x*log(5/
(exp(1) + 4)) + 10),x)

[Out]

-(x*(log(-log(5/(exp(1) + 4))) - log(-(4*x*log(5/(exp(1) + 4)) + 5)/x)))/2

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sympy [A]  time = 0.21, size = 27, normalized size = 1.17 \begin {gather*} \frac {x \log {\left (\frac {4 x \log {\left (\frac {5}{e + 4} \right )} + 5}{x \log {\left (\frac {5}{e + 4} \right )}} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*ln(5/(exp(1)+4))+5)*ln((4*x*ln(5/(exp(1)+4))+5)/x/ln(5/(exp(1)+4)))-5)/(8*x*ln(5/(exp(1)+4))+1
0),x)

[Out]

x*log((4*x*log(5/(E + 4)) + 5)/(x*log(5/(E + 4))))/2

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