3.56.35 \(\int \frac {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+(-148+16 e^{1-x}) \log (x) \log (-\frac {4 \log (x)}{-37+4 e^{1-x}})+(74-8 e^{1-x}) \log (x) \log (-\frac {4 \log (x)}{-37+4 e^{1-x}}) \log (\log (-\frac {4 \log (x)}{-37+4 e^{1-x}}))}{(-185 x^2+20 e^{1-x} x^2) \log (x) \log (-\frac {4 \log (x)}{-37+4 e^{1-x}})} \, dx\)

Optimal. Leaf size=29 \[ \frac {2 \left (-2+\log \left (\log \left (\frac {\log (x)}{\frac {37}{4}-e^{1-x}}\right )\right )\right )}{5 x} \]

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Rubi [F]  time = 3.62, 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 {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+\left (-148+16 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )+\left (74-8 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right ) \log \left (\log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )\right )}{\left (-185 x^2+20 e^{1-x} x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-74 + 8*E^(1 - x) + 8*E^(1 - x)*x*Log[x] + (-148 + 16*E^(1 - x))*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - x
))] + (74 - 8*E^(1 - x))*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - x))]*Log[Log[(-4*Log[x])/(-37 + 4*E^(1 - x))]]
)/((-185*x^2 + 20*E^(1 - x)*x^2)*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - x))]),x]

[Out]

-4/(5*x) - (8*E*Defer[Int][1/((-4*E + 37*E^x)*x*Log[(4*E^x*Log[x])/(-4*E + 37*E^x)]), x])/5 + (2*Defer[Int][1/
(x^2*Log[x]*Log[(4*E^x*Log[x])/(-4*E + 37*E^x)]), x])/5 - (2*Defer[Int][Log[Log[(4*E^x*Log[x])/(-4*E + 37*E^x)
]]/x^2, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e-74 e^x+2 \log (x) \left (4 e x+\left (-4 e+37 e^x\right ) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right ) \left (-2+\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )\right )\right )}{5 \left (4 e-37 e^x\right ) x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ &=\frac {1}{5} \int \frac {8 e-74 e^x+2 \log (x) \left (4 e x+\left (-4 e+37 e^x\right ) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right ) \left (-2+\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )\right )\right )}{\left (4 e-37 e^x\right ) x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ &=\frac {1}{5} \int \left (-\frac {8 e}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}-\frac {2 \left (-1-2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )+\log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right ) \log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )\right )}{x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}\right ) \, dx\\ &=-\left (\frac {2}{5} \int \frac {-1-2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )+\log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right ) \log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\right )-\frac {1}{5} (8 e) \int \frac {1}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ &=-\left (\frac {2}{5} \int \frac {-2-\frac {1}{\log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}+\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x^2} \, dx\right )-\frac {1}{5} (8 e) \int \frac {1}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ &=-\left (\frac {2}{5} \int \left (\frac {-1-2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}{x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}+\frac {\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x^2}\right ) \, dx\right )-\frac {1}{5} (8 e) \int \frac {1}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ &=-\left (\frac {2}{5} \int \frac {-1-2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}{x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\right )-\frac {2}{5} \int \frac {\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x^2} \, dx-\frac {1}{5} (8 e) \int \frac {1}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ &=-\left (\frac {2}{5} \int \frac {-2-\frac {1}{\log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}}{x^2} \, dx\right )-\frac {2}{5} \int \frac {\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x^2} \, dx-\frac {1}{5} (8 e) \int \frac {1}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ &=-\left (\frac {2}{5} \int \left (-\frac {2}{x^2}-\frac {1}{x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}\right ) \, dx\right )-\frac {2}{5} \int \frac {\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x^2} \, dx-\frac {1}{5} (8 e) \int \frac {1}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ &=-\frac {4}{5 x}+\frac {2}{5} \int \frac {1}{x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx-\frac {2}{5} \int \frac {\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x^2} \, dx-\frac {1}{5} (8 e) \int \frac {1}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 35, normalized size = 1.21 \begin {gather*} \frac {1}{5} \left (-\frac {4}{x}+\frac {2 \log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-74 + 8*E^(1 - x) + 8*E^(1 - x)*x*Log[x] + (-148 + 16*E^(1 - x))*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^
(1 - x))] + (74 - 8*E^(1 - x))*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - x))]*Log[Log[(-4*Log[x])/(-37 + 4*E^(1 -
x))]])/((-185*x^2 + 20*E^(1 - x)*x^2)*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - x))]),x]

[Out]

(-4/x + (2*Log[Log[(4*E^x*Log[x])/(-4*E + 37*E^x)]])/x)/5

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fricas [A]  time = 0.82, size = 25, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (\log \left (\log \left (-\frac {4 \, \log \relax (x)}{4 \, e^{\left (-x + 1\right )} - 37}\right )\right ) - 2\right )}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*exp(-x+1)+74)*log(x)*log(-4*log(x)/(4*exp(-x+1)-37))*log(log(-4*log(x)/(4*exp(-x+1)-37)))+(16*e
xp(-x+1)-148)*log(x)*log(-4*log(x)/(4*exp(-x+1)-37))+8*x*exp(-x+1)*log(x)+8*exp(-x+1)-74)/(20*x^2*exp(-x+1)-18
5*x^2)/log(x)/log(-4*log(x)/(4*exp(-x+1)-37)),x, algorithm="fricas")

[Out]

2/5*(log(log(-4*log(x)/(4*e^(-x + 1) - 37))) - 2)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left ({\left (4 \, e^{\left (-x + 1\right )} - 37\right )} \log \relax (x) \log \left (-\frac {4 \, \log \relax (x)}{4 \, e^{\left (-x + 1\right )} - 37}\right ) \log \left (\log \left (-\frac {4 \, \log \relax (x)}{4 \, e^{\left (-x + 1\right )} - 37}\right )\right ) - 4 \, x e^{\left (-x + 1\right )} \log \relax (x) - 2 \, {\left (4 \, e^{\left (-x + 1\right )} - 37\right )} \log \relax (x) \log \left (-\frac {4 \, \log \relax (x)}{4 \, e^{\left (-x + 1\right )} - 37}\right ) - 4 \, e^{\left (-x + 1\right )} + 37\right )}}{5 \, {\left (4 \, x^{2} e^{\left (-x + 1\right )} - 37 \, x^{2}\right )} \log \relax (x) \log \left (-\frac {4 \, \log \relax (x)}{4 \, e^{\left (-x + 1\right )} - 37}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*exp(-x+1)+74)*log(x)*log(-4*log(x)/(4*exp(-x+1)-37))*log(log(-4*log(x)/(4*exp(-x+1)-37)))+(16*e
xp(-x+1)-148)*log(x)*log(-4*log(x)/(4*exp(-x+1)-37))+8*x*exp(-x+1)*log(x)+8*exp(-x+1)-74)/(20*x^2*exp(-x+1)-18
5*x^2)/log(x)/log(-4*log(x)/(4*exp(-x+1)-37)),x, algorithm="giac")

[Out]

integrate(-2/5*((4*e^(-x + 1) - 37)*log(x)*log(-4*log(x)/(4*e^(-x + 1) - 37))*log(log(-4*log(x)/(4*e^(-x + 1)
- 37))) - 4*x*e^(-x + 1)*log(x) - 2*(4*e^(-x + 1) - 37)*log(x)*log(-4*log(x)/(4*e^(-x + 1) - 37)) - 4*e^(-x +
1) + 37)/((4*x^2*e^(-x + 1) - 37*x^2)*log(x)*log(-4*log(x)/(4*e^(-x + 1) - 37))), x)

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maple [C]  time = 0.18, size = 150, normalized size = 5.17

method	result	size
risch	\(\frac {2 \ln \left (i \pi +\ln \left (\ln \relax (x )\right )-\ln \left ({\mathrm e}^{1-x}-\frac {37}{4}\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \ln \relax (x )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right ) \left (-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )+\mathrm {csgn}\left (i \ln \relax (x )\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )+\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )\right )}{2}+i \pi \mathrm {csgn}\left (\frac {i \ln \relax (x )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )^{2} \left (\mathrm {csgn}\left (\frac {i \ln \relax (x )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )-1\right )\right )}{5 x}-\frac {4}{5 x}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*exp(1-x)+74)*ln(x)*ln(-4*ln(x)/(4*exp(1-x)-37))*ln(ln(-4*ln(x)/(4*exp(1-x)-37)))+(16*exp(1-x)-148)*ln
(x)*ln(-4*ln(x)/(4*exp(1-x)-37))+8*x*exp(1-x)*ln(x)+8*exp(1-x)-74)/(20*x^2*exp(1-x)-185*x^2)/ln(x)/ln(-4*ln(x)
/(4*exp(1-x)-37)),x,method=_RETURNVERBOSE)

[Out]

2/5/x*ln(I*Pi+ln(ln(x))-ln(exp(1-x)-37/4)-1/2*I*Pi*csgn(I*ln(x)/(exp(1-x)-37/4))*(-csgn(I*ln(x)/(exp(1-x)-37/4
))+csgn(I*ln(x)))*(-csgn(I*ln(x)/(exp(1-x)-37/4))+csgn(I/(exp(1-x)-37/4)))+I*Pi*csgn(I*ln(x)/(exp(1-x)-37/4))^
2*(csgn(I*ln(x)/(exp(1-x)-37/4))-1))-4/5/x

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maxima [A]  time = 0.54, size = 29, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (\log \left (x + 2 \, \log \relax (2) - \log \left (-4 \, e + 37 \, e^{x}\right ) + \log \left (\log \relax (x)\right )\right ) - 2\right )}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*exp(-x+1)+74)*log(x)*log(-4*log(x)/(4*exp(-x+1)-37))*log(log(-4*log(x)/(4*exp(-x+1)-37)))+(16*e
xp(-x+1)-148)*log(x)*log(-4*log(x)/(4*exp(-x+1)-37))+8*x*exp(-x+1)*log(x)+8*exp(-x+1)-74)/(20*x^2*exp(-x+1)-18
5*x^2)/log(x)/log(-4*log(x)/(4*exp(-x+1)-37)),x, algorithm="maxima")

[Out]

2/5*(log(x + 2*log(2) - log(-4*e + 37*e^x) + log(log(x))) - 2)/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {8\,{\mathrm {e}}^{1-x}+\ln \left (-\frac {4\,\ln \relax (x)}{4\,{\mathrm {e}}^{1-x}-37}\right )\,\ln \relax (x)\,\left (16\,{\mathrm {e}}^{1-x}-148\right )+8\,x\,{\mathrm {e}}^{1-x}\,\ln \relax (x)-\ln \left (-\frac {4\,\ln \relax (x)}{4\,{\mathrm {e}}^{1-x}-37}\right )\,\ln \left (\ln \left (-\frac {4\,\ln \relax (x)}{4\,{\mathrm {e}}^{1-x}-37}\right )\right )\,\ln \relax (x)\,\left (8\,{\mathrm {e}}^{1-x}-74\right )-74}{\ln \left (-\frac {4\,\ln \relax (x)}{4\,{\mathrm {e}}^{1-x}-37}\right )\,\ln \relax (x)\,\left (20\,x^2\,{\mathrm {e}}^{1-x}-185\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*exp(1 - x) + log(-(4*log(x))/(4*exp(1 - x) - 37))*log(x)*(16*exp(1 - x) - 148) + 8*x*exp(1 - x)*log(x)
- log(-(4*log(x))/(4*exp(1 - x) - 37))*log(log(-(4*log(x))/(4*exp(1 - x) - 37)))*log(x)*(8*exp(1 - x) - 74) -
74)/(log(-(4*log(x))/(4*exp(1 - x) - 37))*log(x)*(20*x^2*exp(1 - x) - 185*x^2)),x)

[Out]

int((8*exp(1 - x) + log(-(4*log(x))/(4*exp(1 - x) - 37))*log(x)*(16*exp(1 - x) - 148) + 8*x*exp(1 - x)*log(x)
- log(-(4*log(x))/(4*exp(1 - x) - 37))*log(log(-(4*log(x))/(4*exp(1 - x) - 37)))*log(x)*(8*exp(1 - x) - 74) -
74)/(log(-(4*log(x))/(4*exp(1 - x) - 37))*log(x)*(20*x^2*exp(1 - x) - 185*x^2)), x)

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sympy [A]  time = 134.53, size = 27, normalized size = 0.93 \begin {gather*} \frac {2 \log {\left (\log {\left (- \frac {4 \log {\relax (x )}}{4 e^{1 - x} - 37} \right )} \right )}}{5 x} - \frac {4}{5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*exp(-x+1)+74)*ln(x)*ln(-4*ln(x)/(4*exp(-x+1)-37))*ln(ln(-4*ln(x)/(4*exp(-x+1)-37)))+(16*exp(-x+
1)-148)*ln(x)*ln(-4*ln(x)/(4*exp(-x+1)-37))+8*x*exp(-x+1)*ln(x)+8*exp(-x+1)-74)/(20*x**2*exp(-x+1)-185*x**2)/l
n(x)/ln(-4*ln(x)/(4*exp(-x+1)-37)),x)

[Out]

2*log(log(-4*log(x)/(4*exp(1 - x) - 37)))/(5*x) - 4/(5*x)

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