3.85.87 \(\int \frac {4-10 x+4 x^2+e^x (-1-2 x+7 x^2-4 x^3)+(-8+9 x+e^x (2+5 x-6 x^2-x^3)) \log (x)+(4+e^x (-1-2 x-2 x^2)) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x (4 x-8 x^2+4 x^3)+(-4 x+7 x^2-4 x^3+e^x (-7 x+6 x^2+x^3)) \log (x)+(8 x-8 x^2+e^x (2 x+2 x^2)) \log ^2(x)+(-4 x+e^x x) \log ^3(x)} \, dx\)

Optimal. Leaf size=26 \[ \log \left (\frac {1}{-4 \log (x)+e^x (4+\log (x))-\frac {x}{-1+x+\log (x)}}\right ) \]

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Rubi [F]  time = 52.73, 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 {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 - 10*x + 4*x^2 + E^x*(-1 - 2*x + 7*x^2 - 4*x^3) + (-8 + 9*x + E^x*(2 + 5*x - 6*x^2 - x^3))*Log[x] + (4
+ E^x*(-1 - 2*x - 2*x^2))*Log[x]^2 - E^x*x*Log[x]^3)/(x^2 - x^3 + E^x*(4*x - 8*x^2 + 4*x^3) + (-4*x + 7*x^2 -
4*x^3 + E^x*(-7*x + 6*x^2 + x^3))*Log[x] + (8*x - 8*x^2 + E^x*(2*x + 2*x^2))*Log[x]^2 + (-4*x + E^x*x)*Log[x]^
3),x]

[Out]

-x - Log[4 + Log[x]] - 39*Defer[Int][1/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3
+ x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] + 16*Defer[Int][1/(x*(4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x)
- x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] + 19*Defer[Int][x/((4 + Log[x])*(-1 + x + Log
[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 4*Defer[Int][x^2/((4 +
Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] +
17*Defer[Int][Log[x]/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-
4 + E^x)*Log[x]^2)), x] - 32*Defer[Int][Log[x]/(x*(4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*
x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] + 29*Defer[Int][(x*Log[x])/((4 + Log[x])*(-1 + x + Log[x])
*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 17*Defer[Int][(x^2*Log[x])
/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)),
x] + 29*Defer[Int][Log[x]^2/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log
[x] + (-4 + E^x)*Log[x]^2)), x] + 16*Defer[Int][Log[x]^2/(x*(4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x
+ (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 25*Defer[Int][(x*Log[x]^2)/((4 + Log[x])*(-1 +
x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 4*Defer[Int][(
x^2*Log[x]^2)/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x
)*Log[x]^2)), x] - 8*Defer[Int][Log[x]^3/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*
(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 8*Defer[Int][(x*Log[x]^3)/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x
*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 4*Defer[Int][Log[x]^4/((4 + Log[x
])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4+10 x-4 x^2+e^x (-1+x)^2 (1+4 x)+\left (8-9 x+e^x \left (-2-5 x+6 x^2+x^3\right )\right ) \log (x)+\left (-4+e^x \left (1+2 x+2 x^2\right )\right ) \log ^2(x)+e^x x \log ^3(x)}{x (1-x-\log (x)) \left (4 e^x (-1+x)-x+\left (4-4 x+e^x (3+x)\right ) \log (x)+\left (-4+e^x\right ) \log ^2(x)\right )} \, dx\\ &=\int \left (\frac {-1-4 x-x \log (x)}{x (4+\log (x))}-\frac {-16+39 x-19 x^2+4 x^3+32 \log (x)-17 x \log (x)-29 x^2 \log (x)+17 x^3 \log (x)-16 \log ^2(x)-29 x \log ^2(x)+25 x^2 \log ^2(x)+4 x^3 \log ^2(x)+8 x \log ^3(x)+8 x^2 \log ^3(x)+4 x \log ^4(x)}{x (4+\log (x)) (-1+x+\log (x)) \left (-4 e^x-x+4 e^x x+4 \log (x)+3 e^x \log (x)-4 x \log (x)+e^x x \log (x)-4 \log ^2(x)+e^x \log ^2(x)\right )}\right ) \, dx\\ &=\int \frac {-1-4 x-x \log (x)}{x (4+\log (x))} \, dx-\int \frac {-16+39 x-19 x^2+4 x^3+32 \log (x)-17 x \log (x)-29 x^2 \log (x)+17 x^3 \log (x)-16 \log ^2(x)-29 x \log ^2(x)+25 x^2 \log ^2(x)+4 x^3 \log ^2(x)+8 x \log ^3(x)+8 x^2 \log ^3(x)+4 x \log ^4(x)}{x (4+\log (x)) (-1+x+\log (x)) \left (-4 e^x-x+4 e^x x+4 \log (x)+3 e^x \log (x)-4 x \log (x)+e^x x \log (x)-4 \log ^2(x)+e^x \log ^2(x)\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.23, size = 66, normalized size = 2.54 \begin {gather*} \log (1-x-\log (x))-\log \left (-4 e^x-x+4 e^x x+4 \log (x)+3 e^x \log (x)-4 x \log (x)+e^x x \log (x)-4 \log ^2(x)+e^x \log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 10*x + 4*x^2 + E^x*(-1 - 2*x + 7*x^2 - 4*x^3) + (-8 + 9*x + E^x*(2 + 5*x - 6*x^2 - x^3))*Log[x]
+ (4 + E^x*(-1 - 2*x - 2*x^2))*Log[x]^2 - E^x*x*Log[x]^3)/(x^2 - x^3 + E^x*(4*x - 8*x^2 + 4*x^3) + (-4*x + 7*
x^2 - 4*x^3 + E^x*(-7*x + 6*x^2 + x^3))*Log[x] + (8*x - 8*x^2 + E^x*(2*x + 2*x^2))*Log[x]^2 + (-4*x + E^x*x)*L
og[x]^3),x]

[Out]

Log[1 - x - Log[x]] - Log[-4*E^x - x + 4*E^x*x + 4*Log[x] + 3*E^x*Log[x] - 4*x*Log[x] + E^x*x*Log[x] - 4*Log[x
]^2 + E^x*Log[x]^2]

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fricas [B]  time = 0.82, size = 58, normalized size = 2.23 \begin {gather*} \log \left (x + \log \relax (x) - 1\right ) - \log \left (\frac {{\left (e^{x} - 4\right )} \log \relax (x)^{2} + 4 \, {\left (x - 1\right )} e^{x} + {\left ({\left (x + 3\right )} e^{x} - 4 \, x + 4\right )} \log \relax (x) - x}{e^{x} - 4}\right ) - \log \left (e^{x} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6*x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*
x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6
*x^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x, algorithm="fricas")

[Out]

log(x + log(x) - 1) - log(((e^x - 4)*log(x)^2 + 4*(x - 1)*e^x + ((x + 3)*e^x - 4*x + 4)*log(x) - x)/(e^x - 4))
- log(e^x - 4)

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giac [B]  time = 0.69, size = 57, normalized size = 2.19 \begin {gather*} -\log \left (x e^{x} \log \relax (x) + e^{x} \log \relax (x)^{2} + 4 \, x e^{x} - 4 \, x \log \relax (x) + 3 \, e^{x} \log \relax (x) - 4 \, \log \relax (x)^{2} - x - 4 \, e^{x} + 4 \, \log \relax (x)\right ) + \log \left (x + \log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6*x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*
x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6
*x^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x, algorithm="giac")

[Out]

-log(x*e^x*log(x) + e^x*log(x)^2 + 4*x*e^x - 4*x*log(x) + 3*e^x*log(x) - 4*log(x)^2 - x - 4*e^x + 4*log(x)) +
log(x + log(x) - 1)

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maple [B]  time = 0.06, size = 65, normalized size = 2.50

method	result	size
risch	\(-\ln \left ({\mathrm e}^{x}-4\right )+\ln \left (-1+\ln \relax (x )+x \right )-\ln \left (\ln \relax (x )^{2}+\frac {\left ({\mathrm e}^{x} x -4 x +3 \,{\mathrm e}^{x}+4\right ) \ln \relax (x )}{{\mathrm e}^{x}-4}+\frac {4 \,{\mathrm e}^{x} x -x -4 \,{\mathrm e}^{x}}{{\mathrm e}^{x}-4}\right )\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*exp(x)*ln(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*ln(x)^2+((-x^3-6*x^2+5*x+2)*exp(x)+9*x-8)*ln(x)+(-4*x^3+7*x^2
-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)*x-4*x)*ln(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*ln(x)^2+((x^3+6*x^2-7*x)*e
xp(x)-4*x^3+7*x^2-4*x)*ln(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(exp(x)-4)+ln(-1+ln(x)+x)-ln(ln(x)^2+(exp(x)*x-4*x+3*exp(x)+4)/(exp(x)-4)*ln(x)+(4*exp(x)*x-x-4*exp(x))/(ex
p(x)-4))

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maxima [B]  time = 0.45, size = 64, normalized size = 2.46 \begin {gather*} -\log \left (\frac {{\left ({\left (x + 3\right )} \log \relax (x) + \log \relax (x)^{2} + 4 \, x - 4\right )} e^{x} - 4 \, {\left (x - 1\right )} \log \relax (x) - 4 \, \log \relax (x)^{2} - x}{{\left (x + 3\right )} \log \relax (x) + \log \relax (x)^{2} + 4 \, x - 4}\right ) - \log \left (\log \relax (x) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6*x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*
x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6
*x^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x, algorithm="maxima")

[Out]

-log((((x + 3)*log(x) + log(x)^2 + 4*x - 4)*e^x - 4*(x - 1)*log(x) - 4*log(x)^2 - x)/((x + 3)*log(x) + log(x)^
2 + 4*x - 4)) - log(log(x) + 4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {10\,x-\ln \relax (x)\,\left (9\,x+{\mathrm {e}}^x\,\left (-x^3-6\,x^2+5\,x+2\right )-8\right )+{\ln \relax (x)}^2\,\left ({\mathrm {e}}^x\,\left (2\,x^2+2\,x+1\right )-4\right )-4\,x^2+{\mathrm {e}}^x\,\left (4\,x^3-7\,x^2+2\,x+1\right )+x\,{\mathrm {e}}^x\,{\ln \relax (x)}^3-4}{\ln \relax (x)\,\left (4\,x-{\mathrm {e}}^x\,\left (x^3+6\,x^2-7\,x\right )-7\,x^2+4\,x^3\right )+{\ln \relax (x)}^3\,\left (4\,x-x\,{\mathrm {e}}^x\right )-{\ln \relax (x)}^2\,\left (8\,x+{\mathrm {e}}^x\,\left (2\,x^2+2\,x\right )-8\,x^2\right )-x^2+x^3-{\mathrm {e}}^x\,\left (4\,x^3-8\,x^2+4\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x - log(x)*(9*x + exp(x)*(5*x - 6*x^2 - x^3 + 2) - 8) + log(x)^2*(exp(x)*(2*x + 2*x^2 + 1) - 4) - 4*x^
2 + exp(x)*(2*x - 7*x^2 + 4*x^3 + 1) + x*exp(x)*log(x)^3 - 4)/(log(x)*(4*x - exp(x)*(6*x^2 - 7*x + x^3) - 7*x^
2 + 4*x^3) + log(x)^3*(4*x - x*exp(x)) - log(x)^2*(8*x + exp(x)*(2*x + 2*x^2) - 8*x^2) - x^2 + x^3 - exp(x)*(4
*x - 8*x^2 + 4*x^3)),x)

[Out]

int((10*x - log(x)*(9*x + exp(x)*(5*x - 6*x^2 - x^3 + 2) - 8) + log(x)^2*(exp(x)*(2*x + 2*x^2 + 1) - 4) - 4*x^
2 + exp(x)*(2*x - 7*x^2 + 4*x^3 + 1) + x*exp(x)*log(x)^3 - 4)/(log(x)*(4*x - exp(x)*(6*x^2 - 7*x + x^3) - 7*x^
2 + 4*x^3) + log(x)^3*(4*x - x*exp(x)) - log(x)^2*(8*x + exp(x)*(2*x + 2*x^2) - 8*x^2) - x^2 + x^3 - exp(x)*(4
*x - 8*x^2 + 4*x^3)), x)

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sympy [B]  time = 4.01, size = 53, normalized size = 2.04 \begin {gather*} - \log {\left (\frac {- 4 x \log {\relax (x )} - x - 4 \log {\relax (x )}^{2} + 4 \log {\relax (x )}}{x \log {\relax (x )} + 4 x + \log {\relax (x )}^{2} + 3 \log {\relax (x )} - 4} + e^{x} \right )} - \log {\left (\log {\relax (x )} + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(x)*ln(x)**3+((-2*x**2-2*x-1)*exp(x)+4)*ln(x)**2+((-x**3-6*x**2+5*x+2)*exp(x)+9*x-8)*ln(x)+(-
4*x**3+7*x**2-2*x-1)*exp(x)+4*x**2-10*x+4)/((exp(x)*x-4*x)*ln(x)**3+((2*x**2+2*x)*exp(x)-8*x**2+8*x)*ln(x)**2+
((x**3+6*x**2-7*x)*exp(x)-4*x**3+7*x**2-4*x)*ln(x)+(4*x**3-8*x**2+4*x)*exp(x)-x**3+x**2),x)

[Out]

-log((-4*x*log(x) - x - 4*log(x)**2 + 4*log(x))/(x*log(x) + 4*x + log(x)**2 + 3*log(x) - 4) + exp(x)) - log(lo
g(x) + 4)

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