3.29.38 \(\int \frac {3+e^x (-10-15 x)+e^x (5+5 x) \log (x)}{(1+3 x-15 e^x x+5 e^x x \log (x)) \log (\frac {1}{3} (-1-3 x+15 e^x x-5 e^x x \log (x)))} \, dx\)

Optimal. Leaf size=31 \[ \log \left (\log \left (-x+\frac {-x+5 e^x x^2 (3-\log (x))}{3 x}\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 9.20, 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 {3+e^x (-10-15 x)+e^x (5+5 x) \log (x)}{\left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3 + E^x*(-10 - 15*x) + E^x*(5 + 5*x)*Log[x])/((1 + 3*x - 15*E^x*x + 5*E^x*x*Log[x])*Log[(-1 - 3*x + 15*E^
x*x - 5*E^x*x*Log[x])/3]),x]

[Out]

3*Defer[Int][1/((3 - Log[x])*Log[(-1 - 3*x + 15*E^x*x - 5*E^x*x*Log[x])/3]), x] + 2*Defer[Int][1/(x*(3 - Log[x
])*Log[(-1 - 3*x + 15*E^x*x - 5*E^x*x*Log[x])/3]), x] + Defer[Int][Log[x]/((-3 + Log[x])*Log[(-1 - 3*x + 15*E^
x*x - 5*E^x*x*Log[x])/3]), x] + Defer[Int][Log[x]/(x*(-3 + Log[x])*Log[(-1 - 3*x + 15*E^x*x - 5*E^x*x*Log[x])/
3]), x] + 2*Defer[Int][1/(x*(-3 + Log[x])*(1 + 3*x - 15*E^x*x + 5*E^x*x*Log[x])*Log[(-1 - 3*x + 15*E^x*x - 5*E
^x*x*Log[x])/3]), x] + 9*Defer[Int][x/((-3 + Log[x])*(1 + 3*x - 15*E^x*x + 5*E^x*x*Log[x])*Log[(-1 - 3*x + 15*
E^x*x - 5*E^x*x*Log[x])/3]), x] + Defer[Int][Log[x]/((3 - Log[x])*(1 + 3*x - 15*E^x*x + 5*E^x*x*Log[x])*Log[(-
1 - 3*x + 15*E^x*x - 5*E^x*x*Log[x])/3]), x] + Defer[Int][Log[x]/(x*(3 - Log[x])*(1 + 3*x - 15*E^x*x + 5*E^x*x
*Log[x])*Log[(-1 - 3*x + 15*E^x*x - 5*E^x*x*Log[x])/3]), x] + 3*Defer[Int][(x*Log[x])/((3 - Log[x])*(1 + 3*x -
15*E^x*x + 5*E^x*x*Log[x])*Log[(-1 - 3*x + 15*E^x*x - 5*E^x*x*Log[x])/3]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2+3 x-\log (x)-x \log (x)}{x (3-\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}+\frac {-2-9 x^2+\log (x)+x \log (x)+3 x^2 \log (x)}{x (3-\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}\right ) \, dx\\ &=\int \frac {2+3 x-\log (x)-x \log (x)}{x (3-\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+\int \frac {-2-9 x^2+\log (x)+x \log (x)+3 x^2 \log (x)}{x (3-\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx\\ &=\int \left (\frac {3}{(3-\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}+\frac {2}{x (3-\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}+\frac {\log (x)}{(-3+\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}+\frac {\log (x)}{x (-3+\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}\right ) \, dx+\int \left (\frac {2}{x (-3+\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}+\frac {9 x}{(-3+\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}+\frac {\log (x)}{(3-\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}+\frac {\log (x)}{x (3-\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}+\frac {3 x \log (x)}{(3-\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )}\right ) \, dx\\ &=2 \int \frac {1}{x (3-\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+2 \int \frac {1}{x (-3+\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+3 \int \frac {1}{(3-\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+3 \int \frac {x \log (x)}{(3-\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+9 \int \frac {x}{(-3+\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+\int \frac {\log (x)}{(-3+\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+\int \frac {\log (x)}{x (-3+\log (x)) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+\int \frac {\log (x)}{(3-\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx+\int \frac {\log (x)}{x (3-\log (x)) \left (1+3 x-15 e^x x+5 e^x x \log (x)\right ) \log \left (\frac {1}{3} \left (-1-3 x+15 e^x x-5 e^x x \log (x)\right )\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.85, size = 25, normalized size = 0.81 \begin {gather*} \log \left (\log \left (-\frac {1}{3}+\left (-1+5 e^x\right ) x-\frac {5}{3} e^x x \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + E^x*(-10 - 15*x) + E^x*(5 + 5*x)*Log[x])/((1 + 3*x - 15*E^x*x + 5*E^x*x*Log[x])*Log[(-1 - 3*x +
15*E^x*x - 5*E^x*x*Log[x])/3]),x]

[Out]

Log[Log[-1/3 + (-1 + 5*E^x)*x - (5*E^x*x*Log[x])/3]]

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 19, normalized size = 0.61 \begin {gather*} \log \left (\log \left (-\frac {5}{3} \, x e^{x} \log \relax (x) + 5 \, x e^{x} - x - \frac {1}{3}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*exp(x)*log(x)+(-15*x-10)*exp(x)+3)/(5*x*exp(x)*log(x)-15*exp(x)*x+3*x+1)/log(-5/3*x*exp(x)*
log(x)+5*exp(x)*x-x-1/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.34, size = 24, normalized size = 0.77 \begin {gather*} \log \left (\log \relax (3) - \log \left (-5 \, x e^{x} \log \relax (x) + 15 \, x e^{x} - 3 \, x - 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*exp(x)*log(x)+(-15*x-10)*exp(x)+3)/(5*x*exp(x)*log(x)-15*exp(x)*x+3*x+1)/log(-5/3*x*exp(x)*
log(x)+5*exp(x)*x-x-1/3),x, algorithm="giac")

[Out]

log(log(3) - log(-5*x*e^x*log(x) + 15*x*e^x - 3*x - 1))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 20, normalized size = 0.65

method	result	size
risch	\(\ln \left (\ln \left (-\frac {5 x \,{\mathrm e}^{x} \ln \relax (x )}{3}+5 \,{\mathrm e}^{x} x -x -\frac {1}{3}\right )\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x+5)*exp(x)*ln(x)+(-15*x-10)*exp(x)+3)/(5*x*exp(x)*ln(x)-15*exp(x)*x+3*x+1)/ln(-5/3*x*exp(x)*ln(x)+5*e
xp(x)*x-x-1/3),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-5/3*x*exp(x)*ln(x)+5*exp(x)*x-x-1/3))

________________________________________________________________________________________

maxima [A]  time = 0.95, size = 24, normalized size = 0.77 \begin {gather*} \log \left (-\log \relax (3) + \log \left (-5 \, x e^{x} \log \relax (x) + 15 \, x e^{x} - 3 \, x - 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*exp(x)*log(x)+(-15*x-10)*exp(x)+3)/(5*x*exp(x)*log(x)-15*exp(x)*x+3*x+1)/log(-5/3*x*exp(x)*
log(x)+5*exp(x)*x-x-1/3),x, algorithm="maxima")

[Out]

log(-log(3) + log(-5*x*e^x*log(x) + 15*x*e^x - 3*x - 1))

________________________________________________________________________________________

mupad [B]  time = 2.05, size = 19, normalized size = 0.61 \begin {gather*} \ln \left (\ln \left (5\,x\,{\mathrm {e}}^x-x-\frac {5\,x\,{\mathrm {e}}^x\,\ln \relax (x)}{3}-\frac {1}{3}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*log(x)*(5*x + 5) - exp(x)*(15*x + 10) + 3)/(log(5*x*exp(x) - x - (5*x*exp(x)*log(x))/3 - 1/3)*(3*x
- 15*x*exp(x) + 5*x*exp(x)*log(x) + 1)),x)

[Out]

log(log(5*x*exp(x) - x - (5*x*exp(x)*log(x))/3 - 1/3))

________________________________________________________________________________________

sympy [A]  time = 1.32, size = 26, normalized size = 0.84 \begin {gather*} \log {\left (\log {\left (- \frac {5 x e^{x} \log {\relax (x )}}{3} + 5 x e^{x} - x - \frac {1}{3} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*exp(x)*ln(x)+(-15*x-10)*exp(x)+3)/(5*x*exp(x)*ln(x)-15*exp(x)*x+3*x+1)/ln(-5/3*x*exp(x)*ln(
x)+5*exp(x)*x-x-1/3),x)

[Out]

log(log(-5*x*exp(x)*log(x)/3 + 5*x*exp(x) - x - 1/3))

________________________________________________________________________________________