3.97.51 \(\int \frac {100+100 x+202 x^2+304 x^3+6 x^4+10 x^5+6 x^6+(200 x+300 x^2+4 x^3+10 x^4+6 x^5) \log (x)+(-102 x-2 x^2-6 x^3-8 x^4+(-100-6 x^2-8 x^3) \log (x)) \log (\frac {5}{x+\log (x)})+(2 x^2+2 x \log (x)) \log ^2(\frac {5}{x+\log (x)})}{625 x+625 \log (x)} \, dx\)

Optimal. Leaf size=30 \[ \left (2+\frac {1}{25} x^2 \left (x+\frac {x-\log \left (\frac {5}{x+\log (x)}\right )}{x}\right )\right )^2 \]

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Rubi [A]  time = 0.24, antiderivative size = 26, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 3, integrand size = 135, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6688, 12, 6686} \begin {gather*} \frac {1}{625} \left (x^3+x^2-x \log \left (\frac {5}{x+\log (x)}\right )+50\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(100 + 100*x + 202*x^2 + 304*x^3 + 6*x^4 + 10*x^5 + 6*x^6 + (200*x + 300*x^2 + 4*x^3 + 10*x^4 + 6*x^5)*Log
[x] + (-102*x - 2*x^2 - 6*x^3 - 8*x^4 + (-100 - 6*x^2 - 8*x^3)*Log[x])*Log[5/(x + Log[x])] + (2*x^2 + 2*x*Log[
x])*Log[5/(x + Log[x])]^2)/(625*x + 625*Log[x]),x]

[Out]

(50 + x^2 + x^3 - x*Log[5/(x + Log[x])])^2/625

Rule 12

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (50+x^2+x^3-x \log \left (\frac {5}{x+\log (x)}\right )\right ) \left (1+x+2 x^2+3 x^3+\log (x) \left (x (2+3 x)-\log \left (\frac {5}{x+\log (x)}\right )\right )-x \log \left (\frac {5}{x+\log (x)}\right )\right )}{625 (x+\log (x))} \, dx\\ &=\frac {2}{625} \int \frac {\left (50+x^2+x^3-x \log \left (\frac {5}{x+\log (x)}\right )\right ) \left (1+x+2 x^2+3 x^3+\log (x) \left (x (2+3 x)-\log \left (\frac {5}{x+\log (x)}\right )\right )-x \log \left (\frac {5}{x+\log (x)}\right )\right )}{x+\log (x)} \, dx\\ &=\frac {1}{625} \left (50+x^2+x^3-x \log \left (\frac {5}{x+\log (x)}\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 26, normalized size = 0.87 \begin {gather*} \frac {1}{625} \left (50+x^2+x^3-x \log \left (\frac {5}{x+\log (x)}\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100 + 100*x + 202*x^2 + 304*x^3 + 6*x^4 + 10*x^5 + 6*x^6 + (200*x + 300*x^2 + 4*x^3 + 10*x^4 + 6*x^
5)*Log[x] + (-102*x - 2*x^2 - 6*x^3 - 8*x^4 + (-100 - 6*x^2 - 8*x^3)*Log[x])*Log[5/(x + Log[x])] + (2*x^2 + 2*
x*Log[x])*Log[5/(x + Log[x])]^2)/(625*x + 625*Log[x]),x]

[Out]

(50 + x^2 + x^3 - x*Log[5/(x + Log[x])])^2/625

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fricas [B]  time = 0.53, size = 63, normalized size = 2.10 \begin {gather*} \frac {1}{625} \, x^{6} + \frac {2}{625} \, x^{5} + \frac {1}{625} \, x^{4} + \frac {1}{625} \, x^{2} \log \left (\frac {5}{x + \log \relax (x)}\right )^{2} + \frac {4}{25} \, x^{3} + \frac {4}{25} \, x^{2} - \frac {2}{625} \, {\left (x^{4} + x^{3} + 50 \, x\right )} \log \left (\frac {5}{x + \log \relax (x)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x^2)*log(5/(x+log(x)))^2+((-8*x^3-6*x^2-100)*log(x)-8*x^4-6*x^3-2*x^2-102*x)*log(5/(x
+log(x)))+(6*x^5+10*x^4+4*x^3+300*x^2+200*x)*log(x)+6*x^6+10*x^5+6*x^4+304*x^3+202*x^2+100*x+100)/(625*log(x)+
625*x),x, algorithm="fricas")

[Out]

1/625*x^6 + 2/625*x^5 + 1/625*x^4 + 1/625*x^2*log(5/(x + log(x)))^2 + 4/25*x^3 + 4/25*x^2 - 2/625*(x^4 + x^3 +
50*x)*log(5/(x + log(x)))

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giac [B]  time = 0.25, size = 83, normalized size = 2.77 \begin {gather*} \frac {1}{625} \, x^{6} + \frac {2}{625} \, x^{5} - \frac {1}{625} \, x^{4} {\left (2 \, \log \relax (5) - 1\right )} - \frac {2}{625} \, x^{3} {\left (\log \relax (5) - 50\right )} + \frac {1}{625} \, x^{2} \log \left (x + \log \relax (x)\right )^{2} + \frac {1}{625} \, {\left (\log \relax (5)^{2} + 100\right )} x^{2} - \frac {4}{25} \, x \log \relax (5) + \frac {2}{625} \, {\left (x^{4} + x^{3} - x^{2} \log \relax (5) + 50 \, x\right )} \log \left (x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x^2)*log(5/(x+log(x)))^2+((-8*x^3-6*x^2-100)*log(x)-8*x^4-6*x^3-2*x^2-102*x)*log(5/(x
+log(x)))+(6*x^5+10*x^4+4*x^3+300*x^2+200*x)*log(x)+6*x^6+10*x^5+6*x^4+304*x^3+202*x^2+100*x+100)/(625*log(x)+
625*x),x, algorithm="giac")

[Out]

1/625*x^6 + 2/625*x^5 - 1/625*x^4*(2*log(5) - 1) - 2/625*x^3*(log(5) - 50) + 1/625*x^2*log(x + log(x))^2 + 1/6
25*(log(5)^2 + 100)*x^2 - 4/25*x*log(5) + 2/625*(x^4 + x^3 - x^2*log(5) + 50*x)*log(x + log(x))

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maple [B]  time = 0.14, size = 94, normalized size = 3.13

method	result	size
risch	\(\frac {x^{6}}{625}+\frac {2 x^{5}}{625}+\frac {x^{4}}{625}+\frac {4 x^{3}}{25}+\frac {4 x^{2}}{25}-\frac {4 x \ln \relax (5)}{25}+\frac {x^{2} \ln \relax (5)^{2}}{625}-\frac {2 x^{3} \ln \relax (5)}{625}-\frac {2 x^{4} \ln \relax (5)}{625}+\frac {x^{2} \ln \left (x +\ln \relax (x )\right )^{2}}{625}+\left (\frac {2 x^{4}}{625}-\frac {2 x^{2} \ln \relax (5)}{625}+\frac {2 x^{3}}{625}+\frac {4 x}{25}\right ) \ln \left (x +\ln \relax (x )\right )\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(x)+2*x^2)*ln(5/(x+ln(x)))^2+((-8*x^3-6*x^2-100)*ln(x)-8*x^4-6*x^3-2*x^2-102*x)*ln(5/(x+ln(x)))+(6
*x^5+10*x^4+4*x^3+300*x^2+200*x)*ln(x)+6*x^6+10*x^5+6*x^4+304*x^3+202*x^2+100*x+100)/(625*ln(x)+625*x),x,metho
d=_RETURNVERBOSE)

[Out]

1/625*x^6+2/625*x^5+1/625*x^4+4/25*x^3+4/25*x^2-4/25*x*ln(5)+1/625*x^2*ln(5)^2-2/625*x^3*ln(5)-2/625*x^4*ln(5)
+1/625*x^2*ln(x+ln(x))^2+(2/625*x^4-2/625*x^2*ln(5)+2/625*x^3+4/25*x)*ln(x+ln(x))

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maxima [B]  time = 0.47, size = 83, normalized size = 2.77 \begin {gather*} \frac {1}{625} \, x^{6} + \frac {2}{625} \, x^{5} - \frac {1}{625} \, x^{4} {\left (2 \, \log \relax (5) - 1\right )} - \frac {2}{625} \, x^{3} {\left (\log \relax (5) - 50\right )} + \frac {1}{625} \, x^{2} \log \left (x + \log \relax (x)\right )^{2} + \frac {1}{625} \, {\left (\log \relax (5)^{2} + 100\right )} x^{2} - \frac {4}{25} \, x \log \relax (5) + \frac {2}{625} \, {\left (x^{4} + x^{3} - x^{2} \log \relax (5) + 50 \, x\right )} \log \left (x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x^2)*log(5/(x+log(x)))^2+((-8*x^3-6*x^2-100)*log(x)-8*x^4-6*x^3-2*x^2-102*x)*log(5/(x
+log(x)))+(6*x^5+10*x^4+4*x^3+300*x^2+200*x)*log(x)+6*x^6+10*x^5+6*x^4+304*x^3+202*x^2+100*x+100)/(625*log(x)+
625*x),x, algorithm="maxima")

[Out]

1/625*x^6 + 2/625*x^5 - 1/625*x^4*(2*log(5) - 1) - 2/625*x^3*(log(5) - 50) + 1/625*x^2*log(x + log(x))^2 + 1/6
25*(log(5)^2 + 100)*x^2 - 4/25*x*log(5) + 2/625*(x^4 + x^3 - x^2*log(5) + 50*x)*log(x + log(x))

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mupad [B]  time = 5.63, size = 67, normalized size = 2.23 \begin {gather*} \frac {x^2\,{\ln \left (\frac {5}{x+\ln \relax (x)}\right )}^2}{625}+\frac {4\,x^2}{25}+\frac {4\,x^3}{25}+\frac {x^4}{625}+\frac {2\,x^5}{625}+\frac {x^6}{625}-\ln \left (\frac {5}{x+\ln \relax (x)}\right )\,\left (\frac {2\,x^4}{625}+\frac {2\,x^3}{625}+\frac {4\,x}{25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*x - log(5/(x + log(x)))*(102*x + log(x)*(6*x^2 + 8*x^3 + 100) + 2*x^2 + 6*x^3 + 8*x^4) + log(x)*(200*
x + 300*x^2 + 4*x^3 + 10*x^4 + 6*x^5) + 202*x^2 + 304*x^3 + 6*x^4 + 10*x^5 + 6*x^6 + log(5/(x + log(x)))^2*(2*
x*log(x) + 2*x^2) + 100)/(625*x + 625*log(x)),x)

[Out]

(x^2*log(5/(x + log(x)))^2)/625 + (4*x^2)/25 + (4*x^3)/25 + x^4/625 + (2*x^5)/625 + x^6/625 - log(5/(x + log(x
)))*((4*x)/25 + (2*x^3)/625 + (2*x^4)/625)

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sympy [B]  time = 0.56, size = 73, normalized size = 2.43 \begin {gather*} \frac {x^{6}}{625} + \frac {2 x^{5}}{625} + \frac {x^{4}}{625} + \frac {4 x^{3}}{25} + \frac {x^{2} \log {\left (\frac {5}{x + \log {\relax (x )}} \right )}^{2}}{625} + \frac {4 x^{2}}{25} + \left (- \frac {2 x^{4}}{625} - \frac {2 x^{3}}{625} - \frac {4 x}{25}\right ) \log {\left (\frac {5}{x + \log {\relax (x )}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(x)+2*x**2)*ln(5/(x+ln(x)))**2+((-8*x**3-6*x**2-100)*ln(x)-8*x**4-6*x**3-2*x**2-102*x)*ln(5/
(x+ln(x)))+(6*x**5+10*x**4+4*x**3+300*x**2+200*x)*ln(x)+6*x**6+10*x**5+6*x**4+304*x**3+202*x**2+100*x+100)/(62
5*ln(x)+625*x),x)

[Out]

x**6/625 + 2*x**5/625 + x**4/625 + 4*x**3/25 + x**2*log(5/(x + log(x)))**2/625 + 4*x**2/25 + (-2*x**4/625 - 2*
x**3/625 - 4*x/25)*log(5/(x + log(x)))

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