3.63.63 \(\int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} (-750+295 x-30 x^2)}{(150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} (750 x-300 x^2+30 x^3)) \log (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x})} \, dx\)

Optimal. Leaf size=24 \[ \log \left (\log \left (\frac {1}{\frac {1}{5}+x+e^{\frac {1}{6 (5-x)}} x}\right )\right ) \]

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Rubi [A]  time = 0.22, antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps used = 1, number of rules used = 1, integrand size = 102, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {6684} \begin {gather*} \log \left (\log \left (\frac {5}{5 e^{\frac {1}{6 (5-x)}} x+5 x+1}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-750 + 300*x - 30*x^2 + (-750 + 295*x - 30*x^2)/E^(-30 + 6*x)^(-1))/((150 + 690*x - 294*x^2 + 30*x^3 + (7
50*x - 300*x^2 + 30*x^3)/E^(-30 + 6*x)^(-1))*Log[5/(1 + 5*x + (5*x)/E^(-30 + 6*x)^(-1))]),x]

[Out]

Log[Log[5/(1 + 5*x + 5*E^(1/(6*(5 - x)))*x)]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (\log \left (\frac {5}{1+5 x+5 e^{\frac {1}{6 (5-x)}} x}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.56, size = 22, normalized size = 0.92 \begin {gather*} \log \left (\log \left (\frac {5}{1+5 \left (1+e^{\frac {1}{30-6 x}}\right ) x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-750 + 300*x - 30*x^2 + (-750 + 295*x - 30*x^2)/E^(-30 + 6*x)^(-1))/((150 + 690*x - 294*x^2 + 30*x^
3 + (750*x - 300*x^2 + 30*x^3)/E^(-30 + 6*x)^(-1))*Log[5/(1 + 5*x + (5*x)/E^(-30 + 6*x)^(-1))]),x]

[Out]

Log[Log[5/(1 + 5*(1 + E^(30 - 6*x)^(-1))*x)]]

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fricas [A]  time = 0.92, size = 22, normalized size = 0.92 \begin {gather*} \log \left (\log \left (\frac {5}{5 \, x e^{\left (-\frac {1}{6 \, {\left (x - 5\right )}}\right )} + 5 \, x + 1}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3-300*x^2+750*x)*exp(-1/(6*x-30))+30*
x^3-294*x^2+690*x+150)/log(5/(5*x*exp(-1/(6*x-30))+1+5*x)),x, algorithm="fricas")

[Out]

log(log(5/(5*x*e^(-1/6/(x - 5)) + 5*x + 1)))

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giac [A]  time = 0.37, size = 25, normalized size = 1.04 \begin {gather*} \log \left (\log \left (\frac {5}{5 \, x e^{\left (-\frac {x}{30 \, {\left (x - 5\right )}} + \frac {1}{30}\right )} + 5 \, x + 1}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3-300*x^2+750*x)*exp(-1/(6*x-30))+30*
x^3-294*x^2+690*x+150)/log(5/(5*x*exp(-1/(6*x-30))+1+5*x)),x, algorithm="giac")

[Out]

log(log(5/(5*x*e^(-1/30*x/(x - 5) + 1/30) + 5*x + 1)))

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maple [A]  time = 0.51, size = 17, normalized size = 0.71

method	result	size
risch	\(\ln \left (\ln \left (\frac {1}{5}+x \left ({\mathrm e}^{-\frac {1}{6 \left (x -5\right )}}+1\right )\right )\right )\)	\(17\)
norman	\(\ln \left (\ln \left (\frac {5}{5 x \,{\mathrm e}^{-\frac {1}{6 x -30}}+1+5 x}\right )\right )\)	\(25\)
default	\(\ln \left (-\ln \left (\frac {5}{5 x \,{\mathrm e}^{-\frac {1}{6 x -30}}+1+5 x}\right )\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3-300*x^2+750*x)*exp(-1/(6*x-30))+30*x^3-29
4*x^2+690*x+150)/ln(5/(5*x*exp(-1/(6*x-30))+1+5*x)),x,method=_RETURNVERBOSE)

[Out]

ln(ln(1/5+x*(exp(-1/6/(x-5))+1)))

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maxima [A]  time = 0.58, size = 43, normalized size = 1.79 \begin {gather*} \log \left (-\frac {6 \, x \log \relax (5) - 6 \, {\left (x - 5\right )} \log \left ({\left (5 \, x + 1\right )} e^{\left (\frac {1}{6 \, {\left (x - 5\right )}}\right )} + 5 \, x\right ) - 30 \, \log \relax (5) + 1}{6 \, {\left (x - 5\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3-300*x^2+750*x)*exp(-1/(6*x-30))+30*
x^3-294*x^2+690*x+150)/log(5/(5*x*exp(-1/(6*x-30))+1+5*x)),x, algorithm="maxima")

[Out]

log(-1/6*(6*x*log(5) - 6*(x - 5)*log((5*x + 1)*e^(1/6/(x - 5)) + 5*x) - 30*log(5) + 1)/(x - 5))

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mupad [B]  time = 4.81, size = 24, normalized size = 1.00 \begin {gather*} \ln \left (\ln \left (\frac {5}{5\,x+5\,x\,{\mathrm {e}}^{-\frac {1}{6\,x-30}}+1}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-1/(6*x - 30))*(30*x^2 - 295*x + 750) - 300*x + 30*x^2 + 750)/(log(5/(5*x + 5*x*exp(-1/(6*x - 30)) +
1))*(690*x + exp(-1/(6*x - 30))*(750*x - 300*x^2 + 30*x^3) - 294*x^2 + 30*x^3 + 150)),x)

[Out]

log(log(5/(5*x + 5*x*exp(-1/(6*x - 30)) + 1)))

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sympy [A]  time = 1.15, size = 20, normalized size = 0.83 \begin {gather*} \log {\left (\log {\left (\frac {5}{5 x + 5 x e^{- \frac {1}{6 x - 30}} + 1} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-30*x**2+295*x-750)*exp(-1/(6*x-30))-30*x**2+300*x-750)/((30*x**3-300*x**2+750*x)*exp(-1/(6*x-30))
+30*x**3-294*x**2+690*x+150)/ln(5/(5*x*exp(-1/(6*x-30))+1+5*x)),x)

[Out]

log(log(5/(5*x + 5*x*exp(-1/(6*x - 30)) + 1)))

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