3.71.81 \(\int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+(e^{4-x} (-1+x)+x+(-1+x^2) \log (-1+x)) \log (x)}{5 x-5 x^2+(e^{4-x} (-x+x^2)+(-x+x^2) \log (-1+x)) \log (x)} \, dx\)

Optimal. Leaf size=25 \[ x+\log \left (x \left (-1+\frac {1}{5} \left (e^{4-x}+\log (-1+x)\right ) \log (x)\right )\right ) \]

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Rubi [F]  time = 60.57, 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 {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 26, normalized size = 1.04 \begin {gather*} \log (x)+\log \left (-5 e^x+e^4 \log (x)+e^x \log (-1+x) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + E^(4 - x)*(-1 + x) - 5*x^2 + (-1 + x)*Log[-1 + x] + (E^(4 - x)*(-1 + x) + x + (-1 + x^2)*Log[-1
+ x])*Log[x])/(5*x - 5*x^2 + (E^(4 - x)*(-x + x^2) + (-x + x^2)*Log[-1 + x])*Log[x]),x]

[Out]

Log[x] + Log[-5*E^x + E^4*Log[x] + E^x*Log[-1 + x]*Log[x]]

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fricas [B]  time = 0.47, size = 47, normalized size = 1.88 \begin {gather*} x + \log \relax (x) + \log \left (\frac {{\left (e^{\left (-x + 4\right )} + \log \left (x - 1\right )\right )} \log \relax (x) - 5}{e^{\left (-x + 4\right )} + \log \left (x - 1\right )}\right ) + \log \left (e^{\left (-x + 4\right )} + \log \left (x - 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-1)*log(x-1)+(x-1)*exp(-x+4)+x)*log(x)+(x-1)*log(x-1)+(x-1)*exp(-x+4)-5*x^2+5)/(((x^2-x)*log(x
-1)+(x^2-x)*exp(-x+4))*log(x)-5*x^2+5*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 23, normalized size = 0.92 \begin {gather*} x + \log \left (e^{\left (-x + 4\right )} \log \relax (x) + \log \left (x - 1\right ) \log \relax (x) - 5\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-1)*log(x-1)+(x-1)*exp(-x+4)+x)*log(x)+(x-1)*log(x-1)+(x-1)*exp(-x+4)-5*x^2+5)/(((x^2-x)*log(x
-1)+(x^2-x)*exp(-x+4))*log(x)-5*x^2+5*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 30, normalized size = 1.20

method	result	size
risch	\(x +\ln \relax (x )+\ln \left (\ln \relax (x )\right )+\ln \left (\ln \left (x -1\right )+\frac {\ln \relax (x ) {\mathrm e}^{-x +4}-5}{\ln \relax (x )}\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-1)*ln(x-1)+(x-1)*exp(-x+4)+x)*ln(x)+(x-1)*ln(x-1)+(x-1)*exp(-x+4)-5*x^2+5)/(((x^2-x)*ln(x-1)+(x^2-x
)*exp(-x+4))*ln(x)-5*x^2+5*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 36, normalized size = 1.44 \begin {gather*} x + \log \relax (x) + \log \left (\frac {{\left (e^{x} \log \left (x - 1\right ) \log \relax (x) + e^{4} \log \relax (x) - 5 \, e^{x}\right )} e^{\left (-x\right )}}{\log \relax (x)}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-1)*log(x-1)+(x-1)*exp(-x+4)+x)*log(x)+(x-1)*log(x-1)+(x-1)*exp(-x+4)-5*x^2+5)/(((x^2-x)*log(x
-1)+(x^2-x)*exp(-x+4))*log(x)-5*x^2+5*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.32, size = 31, normalized size = 1.24 \begin {gather*} x+\ln \left (\ln \relax (x)\right )+\ln \left (\frac {\ln \left (x-1\right )\,\ln \relax (x)+{\mathrm {e}}^{4-x}\,\ln \relax (x)-5}{\ln \relax (x)}\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.91, size = 29, normalized size = 1.16 \begin {gather*} x + \log {\relax (x )} + \log {\left (\frac {\log {\relax (x )} \log {\left (x - 1 \right )} - 5}{\log {\relax (x )}} + e^{4 - x} \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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