3.98.6 \(\int \frac {x \log ^2(\frac {x}{2})+e^{\frac {(27-9 x) \log (2)+(6+15 \log (2)) \log (\frac {x}{2})}{\log (2) \log (\frac {x}{2})}} (-27+9 x-9 x \log (\frac {x}{2}))}{e^{\frac {(27-9 x) \log (2)+(6+15 \log (2)) \log (\frac {x}{2})}{\log (2) \log (\frac {x}{2})}} x \log ^2(\frac {x}{2})+x^2 \log ^2(\frac {x}{2})} \, dx\)

Optimal. Leaf size=33 \[ \log \left (\left (e^{3 \left (5+\frac {2}{\log (2)}+\frac {3 (3-x)}{\log \left (\frac {x}{2}\right )}\right )}+x\right ) \log (4)\right ) \]

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Rubi [F]  time = 13.38, 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 \log ^2\left (\frac {x}{2}\right )+\exp \left (\frac {(27-9 x) \log (2)+(6+15 \log (2)) \log \left (\frac {x}{2}\right )}{\log (2) \log \left (\frac {x}{2}\right )}\right ) \left (-27+9 x-9 x \log \left (\frac {x}{2}\right )\right )}{\exp \left (\frac {(27-9 x) \log (2)+(6+15 \log (2)) \log \left (\frac {x}{2}\right )}{\log (2) \log \left (\frac {x}{2}\right )}\right ) x \log ^2\left (\frac {x}{2}\right )+x^2 \log ^2\left (\frac {x}{2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x*Log[x/2]^2 + E^(((27 - 9*x)*Log[2] + (6 + 15*Log[2])*Log[x/2])/(Log[2]*Log[x/2]))*(-27 + 9*x - 9*x*Log[
x/2]))/(E^(((27 - 9*x)*Log[2] + (6 + 15*Log[2])*Log[x/2])/(Log[2]*Log[x/2]))*x*Log[x/2]^2 + x^2*Log[x/2]^2),x]

[Out]

Log[x] - Defer[Int][E^(15*(1 + 2/Log[32]) + 27/Log[x/2])/(x*(E^(15 + 6/Log[2] + 27/Log[x/2]) + E^((9*x)/Log[x/
2])*x)), x] + 9*Defer[Int][E^(15*(1 + 2/Log[32]) + 27/Log[x/2])/((E^(15 + 6/Log[2] + 27/Log[x/2]) + E^((9*x)/L
og[x/2])*x)*Log[x/2]^2), x] - 27*Defer[Int][E^(15*(1 + 2/Log[32]) + 27/Log[x/2])/(x*(E^(15 + 6/Log[2] + 27/Log
[x/2]) + E^((9*x)/Log[x/2])*x)*Log[x/2]^2), x] - 9*Defer[Int][E^(15*(1 + 2/Log[32]) + 27/Log[x/2])/((E^(15 + 6
/Log[2] + 27/Log[x/2]) + E^((9*x)/Log[x/2])*x)*Log[x/2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \log ^2\left (\frac {x}{2}\right )-9 e^{15+\frac {6}{\log (2)}-\frac {9 (-3+x)}{\log \left (\frac {x}{2}\right )}} \left (3-x+x \log \left (\frac {x}{2}\right )\right )}{x \left (e^{15+\frac {6}{\log (2)}-\frac {9 (-3+x)}{\log \left (\frac {x}{2}\right )}}+x\right ) \log ^2\left (\frac {x}{2}\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}} \left (-27+9 x-9 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )}\right ) \, dx\\ &=\log (x)+\int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}} \left (-27+9 x-9 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )} \, dx\\ &=\log (x)+\int \left (-\frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right )}+\frac {9 e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{\left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )}-\frac {27 e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )}-\frac {9 e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{\left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log \left (\frac {x}{2}\right )}\right ) \, dx\\ &=\log (x)+9 \int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{\left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )} \, dx-9 \int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{\left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log \left (\frac {x}{2}\right )} \, dx-27 \int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )} \, dx-\int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 26, normalized size = 0.79 \begin {gather*} \log \left (e^{15+\frac {6}{\log (2)}-\frac {9 (-3+x)}{\log \left (\frac {x}{2}\right )}}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*Log[x/2]^2 + E^(((27 - 9*x)*Log[2] + (6 + 15*Log[2])*Log[x/2])/(Log[2]*Log[x/2]))*(-27 + 9*x - 9*
x*Log[x/2]))/(E^(((27 - 9*x)*Log[2] + (6 + 15*Log[2])*Log[x/2])/(Log[2]*Log[x/2]))*x*Log[x/2]^2 + x^2*Log[x/2]
^2),x]

[Out]

Log[E^(15 + 6/Log[2] - (9*(-3 + x))/Log[x/2]) + x]

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fricas [A]  time = 0.84, size = 36, normalized size = 1.09 \begin {gather*} \log \left (x + e^{\left (-\frac {3 \, {\left (3 \, {\left (x - 3\right )} \log \relax (2) - {\left (5 \, \log \relax (2) + 2\right )} \log \left (\frac {1}{2} \, x\right )\right )}}{\log \relax (2) \log \left (\frac {1}{2} \, x\right )}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x*log(1/2*x)+9*x-27)*exp(((15*log(2)+6)*log(1/2*x)+(-9*x+27)*log(2))/log(2)/log(1/2*x))+x*log(1
/2*x)^2)/(x*log(1/2*x)^2*exp(((15*log(2)+6)*log(1/2*x)+(-9*x+27)*log(2))/log(2)/log(1/2*x))+x^2*log(1/2*x)^2),
x, algorithm="fricas")

[Out]

log(x + e^(-3*(3*(x - 3)*log(2) - (5*log(2) + 2)*log(1/2*x))/(log(2)*log(1/2*x))))

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giac [A]  time = 2.25, size = 41, normalized size = 1.24 \begin {gather*} \log \left (x + e^{\left (\frac {9 \, {\left (x \log \relax (2) - 3 \, \log \relax (x)\right )}}{\log \relax (2)^{2} - \log \relax (2) \log \relax (x)} + \frac {3 \, {\left (5 \, \log \relax (2) - 7\right )}}{\log \relax (2)}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x*log(1/2*x)+9*x-27)*exp(((15*log(2)+6)*log(1/2*x)+(-9*x+27)*log(2))/log(2)/log(1/2*x))+x*log(1
/2*x)^2)/(x*log(1/2*x)^2*exp(((15*log(2)+6)*log(1/2*x)+(-9*x+27)*log(2))/log(2)/log(1/2*x))+x^2*log(1/2*x)^2),
x, algorithm="giac")

[Out]

log(x + e^(9*(x*log(2) - 3*log(x))/(log(2)^2 - log(2)*log(x)) + 3*(5*log(2) - 7)/log(2)))

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maple [A]  time = 0.14, size = 36, normalized size = 1.09

method	result	size
norman	\(\ln \left ({\mathrm e}^{\frac {\left (15 \ln \relax (2)+6\right ) \ln \left (\frac {x}{2}\right )+\left (-9 x +27\right ) \ln \relax (2)}{\ln \relax (2) \ln \left (\frac {x}{2}\right )}}+x \right )\)	\(36\)
risch	\(-\frac {9 \left (x -3\right )}{\ln \left (\frac {x}{2}\right )}-\frac {\left (15 \ln \relax (2)+6\right ) \ln \left (\frac {x}{2}\right )+\left (-9 x +27\right ) \ln \relax (2)}{\ln \relax (2) \ln \left (\frac {x}{2}\right )}+\ln \left ({\mathrm e}^{-\frac {3 \left (-5 \ln \left (\frac {x}{2}\right ) \ln \relax (2)+3 x \ln \relax (2)-9 \ln \relax (2)-2 \ln \left (\frac {x}{2}\right )\right )}{\ln \left (\frac {x}{2}\right ) \ln \relax (2)}}+x \right )\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*x*ln(1/2*x)+9*x-27)*exp(((15*ln(2)+6)*ln(1/2*x)+(-9*x+27)*ln(2))/ln(2)/ln(1/2*x))+x*ln(1/2*x)^2)/(x*l
n(1/2*x)^2*exp(((15*ln(2)+6)*ln(1/2*x)+(-9*x+27)*ln(2))/ln(2)/ln(1/2*x))+x^2*ln(1/2*x)^2),x,method=_RETURNVERB
OSE)

[Out]

ln(exp(((15*ln(2)+6)*ln(1/2*x)+(-9*x+27)*ln(2))/ln(2)/ln(1/2*x))+x)

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maxima [B]  time = 0.59, size = 56, normalized size = 1.70 \begin {gather*} \frac {9 \, x}{\log \relax (2) - \log \relax (x)} + \log \relax (x) + \log \left (\frac {x e^{\left (-\frac {9 \, x}{\log \relax (2) - \log \relax (x)}\right )} + e^{\left (-\frac {27}{\log \relax (2) - \log \relax (x)} + \frac {6}{\log \relax (2)} + 15\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x*log(1/2*x)+9*x-27)*exp(((15*log(2)+6)*log(1/2*x)+(-9*x+27)*log(2))/log(2)/log(1/2*x))+x*log(1
/2*x)^2)/(x*log(1/2*x)^2*exp(((15*log(2)+6)*log(1/2*x)+(-9*x+27)*log(2))/log(2)/log(1/2*x))+x^2*log(1/2*x)^2),
x, algorithm="maxima")

[Out]

9*x/(log(2) - log(x)) + log(x) + log((x*e^(-9*x/(log(2) - log(x))) + e^(-27/(log(2) - log(x)) + 6/log(2) + 15)
)/x)

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mupad [B]  time = 6.50, size = 64, normalized size = 1.94 \begin {gather*} \ln \left (x+\frac {{\mathrm {e}}^{\frac {3\,\ln \relax (x)\,\left (2\,\ln \left (\frac {x}{2}\right )+5\,\ln \relax (2)\,\ln \relax (x)-5\,{\ln \relax (2)}^2\right )}{{\ln \left (\frac {x}{2}\right )}^2\,\ln \relax (2)}-\frac {9\,x-21}{\ln \left (\frac {x}{2}\right )}}}{2^{\frac {15}{\ln \left (\frac {x}{2}\right )}}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(log(2)*(9*x - 27) - log(x/2)*(15*log(2) + 6))/(log(x/2)*log(2)))*(9*x*log(x/2) - 9*x + 27) - x*log
(x/2)^2)/(x^2*log(x/2)^2 + x*log(x/2)^2*exp(-(log(2)*(9*x - 27) - log(x/2)*(15*log(2) + 6))/(log(x/2)*log(2)))
),x)

[Out]

log(x + exp((3*log(x)*(2*log(x/2) + 5*log(2)*log(x) - 5*log(2)^2))/(log(x/2)^2*log(2)) - (9*x - 21)/log(x/2))/
2^(15/log(x/2)))

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sympy [A]  time = 0.54, size = 32, normalized size = 0.97 \begin {gather*} \log {\left (x + e^{\frac {\left (27 - 9 x\right ) \log {\relax (2 )} + \left (6 + 15 \log {\relax (2 )}\right ) \log {\left (\frac {x}{2} \right )}}{\log {\relax (2 )} \log {\left (\frac {x}{2} \right )}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x*ln(1/2*x)+9*x-27)*exp(((15*ln(2)+6)*ln(1/2*x)+(-9*x+27)*ln(2))/ln(2)/ln(1/2*x))+x*ln(1/2*x)**
2)/(x*ln(1/2*x)**2*exp(((15*ln(2)+6)*ln(1/2*x)+(-9*x+27)*ln(2))/ln(2)/ln(1/2*x))+x**2*ln(1/2*x)**2),x)

[Out]

log(x + exp(((27 - 9*x)*log(2) + (6 + 15*log(2))*log(x/2))/(log(2)*log(x/2))))

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