3.48.77 \(\int \frac {e^{2 x} (15 x^2+20 x^3-4 x^4)+e^{2 x} (24 x+19 x^2-2 x^3) \log (4-x)+e^{2 x} (2 x+2 x^2) \log ^2(4-x)}{-2 x^2+e^{2 x} (12 x^3-2 x^4)+(-4 x+e^{2 x} (12 x^2-x^3)) \log (4-x)+(-2+e^{2 x} x^2) \log ^2(4-x)} \, dx\)

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

________________________________________________________________________________________

Rubi [F]  time = 14.64, 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 {e^{2 x} \left (15 x^2+20 x^3-4 x^4\right )+e^{2 x} \left (24 x+19 x^2-2 x^3\right ) \log (4-x)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(4-x)}{-2 x^2+e^{2 x} \left (12 x^3-2 x^4\right )+\left (-4 x+e^{2 x} \left (12 x^2-x^3\right )\right ) \log (4-x)+\left (-2+e^{2 x} x^2\right ) \log ^2(4-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(15*x^2 + 20*x^3 - 4*x^4) + E^(2*x)*(24*x + 19*x^2 - 2*x^3)*Log[4 - x] + E^(2*x)*(2*x + 2*x^2)*Lo
g[4 - x]^2)/(-2*x^2 + E^(2*x)*(12*x^3 - 2*x^4) + (-4*x + E^(2*x)*(12*x^2 - x^3))*Log[4 - x] + (-2 + E^(2*x)*x^
2)*Log[4 - x]^2),x]

[Out]

-15*Defer[Int][(E^(2*x)*x^2)/((x + Log[4 - x])*(2*x*(1 + E^(2*x)*(-6 + x)*x) + (2 - E^(2*x)*x^2)*Log[4 - x])),
x] - 20*Defer[Int][(E^(2*x)*x^3)/((x + Log[4 - x])*(2*x*(1 + E^(2*x)*(-6 + x)*x) + (2 - E^(2*x)*x^2)*Log[4 -
x])), x] + 4*Defer[Int][(E^(2*x)*x^4)/((x + Log[4 - x])*(2*x*(1 + E^(2*x)*(-6 + x)*x) + (2 - E^(2*x)*x^2)*Log[
4 - x])), x] - 24*Defer[Int][(E^(2*x)*x*Log[4 - x])/((x + Log[4 - x])*(2*x*(1 + E^(2*x)*(-6 + x)*x) + (2 - E^(
2*x)*x^2)*Log[4 - x])), x] - 19*Defer[Int][(E^(2*x)*x^2*Log[4 - x])/((x + Log[4 - x])*(2*x*(1 + E^(2*x)*(-6 +
x)*x) + (2 - E^(2*x)*x^2)*Log[4 - x])), x] + 2*Defer[Int][(E^(2*x)*x^3*Log[4 - x])/((x + Log[4 - x])*(2*x*(1 +
E^(2*x)*(-6 + x)*x) + (2 - E^(2*x)*x^2)*Log[4 - x])), x] - 2*Defer[Int][(E^(2*x)*x*Log[4 - x]^2)/((x + Log[4
- x])*(2*x*(1 + E^(2*x)*(-6 + x)*x) + (2 - E^(2*x)*x^2)*Log[4 - x])), x] - 2*Defer[Int][(E^(2*x)*x^2*Log[4 - x
]^2)/((x + Log[4 - x])*(2*x*(1 + E^(2*x)*(-6 + x)*x) + (2 - E^(2*x)*x^2)*Log[4 - x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} x \left (x \left (-15-20 x+4 x^2\right )+\left (-24-19 x+2 x^2\right ) \log (4-x)-2 (1+x) \log ^2(4-x)\right )}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx\\ &=\int \left (-\frac {15 e^{2 x} x^2}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )}-\frac {20 e^{2 x} x^3}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )}+\frac {4 e^{2 x} x^4}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )}-\frac {24 e^{2 x} x \log (4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )}-\frac {19 e^{2 x} x^2 \log (4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )}+\frac {2 e^{2 x} x^3 \log (4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )}-\frac {2 e^{2 x} x \log ^2(4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )}-\frac {2 e^{2 x} x^2 \log ^2(4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )}\right ) \, dx\\ &=2 \int \frac {e^{2 x} x^3 \log (4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )} \, dx-2 \int \frac {e^{2 x} x \log ^2(4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )} \, dx-2 \int \frac {e^{2 x} x^2 \log ^2(4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )} \, dx+4 \int \frac {e^{2 x} x^4}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )} \, dx-15 \int \frac {e^{2 x} x^2}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )} \, dx-19 \int \frac {e^{2 x} x^2 \log (4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )} \, dx-20 \int \frac {e^{2 x} x^3}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )} \, dx-24 \int \frac {e^{2 x} x \log (4-x)}{(x+\log (4-x)) \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right )} \, dx\\ &=2 \int \frac {e^{2 x} x^3 \log (4-x)}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx-2 \int \frac {e^{2 x} x \log ^2(4-x)}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx-2 \int \frac {e^{2 x} x^2 \log ^2(4-x)}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx+4 \int \frac {e^{2 x} x^4}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx-15 \int \frac {e^{2 x} x^2}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx-19 \int \frac {e^{2 x} x^2 \log (4-x)}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx-20 \int \frac {e^{2 x} x^3}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx-24 \int \frac {e^{2 x} x \log (4-x)}{(x+\log (4-x)) \left (2 x \left (1+e^{2 x} (-6+x) x\right )+\left (2-e^{2 x} x^2\right ) \log (4-x)\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 61, normalized size = 1.85 \begin {gather*} -\log (x+\log (4-x))+\log \left (2 x-12 e^{2 x} x^2+2 e^{2 x} x^3+2 \log (4-x)-e^{2 x} x^2 \log (4-x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(15*x^2 + 20*x^3 - 4*x^4) + E^(2*x)*(24*x + 19*x^2 - 2*x^3)*Log[4 - x] + E^(2*x)*(2*x + 2*x
^2)*Log[4 - x]^2)/(-2*x^2 + E^(2*x)*(12*x^3 - 2*x^4) + (-4*x + E^(2*x)*(12*x^2 - x^3))*Log[4 - x] + (-2 + E^(2
*x)*x^2)*Log[4 - x]^2),x]

[Out]

-Log[x + Log[4 - x]] + Log[2*x - 12*E^(2*x)*x^2 + 2*E^(2*x)*x^3 + 2*Log[4 - x] - E^(2*x)*x^2*Log[4 - x]]

________________________________________________________________________________________

fricas [B]  time = 0.95, size = 83, normalized size = 2.52 \begin {gather*} -\log \left (x + \log \left (-x + 4\right )\right ) + 2 \, \log \relax (x) + \log \left (-\frac {2 \, {\left (x^{3} - 6 \, x^{2}\right )} e^{\left (2 \, x\right )} - {\left (x^{2} e^{\left (2 \, x\right )} - 2\right )} \log \left (-x + 4\right ) + 2 \, x}{x^{2} e^{\left (2 \, x\right )} - 2}\right ) + \log \left (\frac {x^{2} e^{\left (2 \, x\right )} - 2}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(x)^2*log(-x+4)^2+(-2*x^3+19*x^2+24*x)*exp(x)^2*log(-x+4)+(-4*x^4+20*x^3+15*x^2)*exp
(x)^2)/((exp(x)^2*x^2-2)*log(-x+4)^2+((-x^3+12*x^2)*exp(x)^2-4*x)*log(-x+4)+(-2*x^4+12*x^3)*exp(x)^2-2*x^2),x,
algorithm="fricas")

[Out]

-log(x + log(-x + 4)) + 2*log(x) + log(-(2*(x^3 - 6*x^2)*e^(2*x) - (x^2*e^(2*x) - 2)*log(-x + 4) + 2*x)/(x^2*e
^(2*x) - 2)) + log((x^2*e^(2*x) - 2)/x^2)

________________________________________________________________________________________

giac [A]  time = 0.51, size = 57, normalized size = 1.73 \begin {gather*} \log \left (-2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} e^{\left (2 \, x\right )} \log \left (-x + 4\right ) + 12 \, x^{2} e^{\left (2 \, x\right )} - 2 \, x - 2 \, \log \left (-x + 4\right )\right ) - \log \left (x + \log \left (-x + 4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(x)^2*log(-x+4)^2+(-2*x^3+19*x^2+24*x)*exp(x)^2*log(-x+4)+(-4*x^4+20*x^3+15*x^2)*exp
(x)^2)/((exp(x)^2*x^2-2)*log(-x+4)^2+((-x^3+12*x^2)*exp(x)^2-4*x)*log(-x+4)+(-2*x^4+12*x^3)*exp(x)^2-2*x^2),x,
algorithm="giac")

[Out]

log(-2*x^3*e^(2*x) + x^2*e^(2*x)*log(-x + 4) + 12*x^2*e^(2*x) - 2*x - 2*log(-x + 4)) - log(x + log(-x + 4))

________________________________________________________________________________________

maple [B]  time = 0.07, size = 68, normalized size = 2.06

method	result	size
risch	\(2 \ln \relax (x )+\ln \left ({\mathrm e}^{2 x}-\frac {2}{x^{2}}\right )+\ln \left (\ln \left (-x +4\right )-\frac {2 x \left ({\mathrm e}^{2 x} x^{2}-6 x \,{\mathrm e}^{2 x}+1\right )}{{\mathrm e}^{2 x} x^{2}-2}\right )-\ln \left (x +\ln \left (-x +4\right )\right )\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+2*x)*exp(x)^2*ln(-x+4)^2+(-2*x^3+19*x^2+24*x)*exp(x)^2*ln(-x+4)+(-4*x^4+20*x^3+15*x^2)*exp(x)^2)/(
(exp(x)^2*x^2-2)*ln(-x+4)^2+((-x^3+12*x^2)*exp(x)^2-4*x)*ln(-x+4)+(-2*x^4+12*x^3)*exp(x)^2-2*x^2),x,method=_RE
TURNVERBOSE)

[Out]

2*ln(x)+ln(exp(2*x)-2/x^2)+ln(ln(-x+4)-2*x*(exp(2*x)*x^2-6*x*exp(2*x)+1)/(exp(2*x)*x^2-2))-ln(x+ln(-x+4))

________________________________________________________________________________________

maxima [B]  time = 0.43, size = 83, normalized size = 2.52 \begin {gather*} -\log \left (x + \log \left (-x + 4\right )\right ) + 2 \, \log \relax (x) + \log \left (-\frac {2 \, {\left (x^{3} - 6 \, x^{2}\right )} e^{\left (2 \, x\right )} - {\left (x^{2} e^{\left (2 \, x\right )} - 2\right )} \log \left (-x + 4\right ) + 2 \, x}{x^{2} e^{\left (2 \, x\right )} - 2}\right ) + \log \left (\frac {x^{2} e^{\left (2 \, x\right )} - 2}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(x)^2*log(-x+4)^2+(-2*x^3+19*x^2+24*x)*exp(x)^2*log(-x+4)+(-4*x^4+20*x^3+15*x^2)*exp
(x)^2)/((exp(x)^2*x^2-2)*log(-x+4)^2+((-x^3+12*x^2)*exp(x)^2-4*x)*log(-x+4)+(-2*x^4+12*x^3)*exp(x)^2-2*x^2),x,
algorithm="maxima")

[Out]

-log(x + log(-x + 4)) + 2*log(x) + log(-(2*(x^3 - 6*x^2)*e^(2*x) - (x^2*e^(2*x) - 2)*log(-x + 4) + 2*x)/(x^2*e
^(2*x) - 2)) + log((x^2*e^(2*x) - 2)/x^2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,x}\,\left (2\,x^2+2\,x\right )\,{\ln \left (4-x\right )}^2+{\mathrm {e}}^{2\,x}\,\left (-2\,x^3+19\,x^2+24\,x\right )\,\ln \left (4-x\right )+{\mathrm {e}}^{2\,x}\,\left (-4\,x^4+20\,x^3+15\,x^2\right )}{{\ln \left (4-x\right )}^2\,\left (x^2\,{\mathrm {e}}^{2\,x}-2\right )+{\mathrm {e}}^{2\,x}\,\left (12\,x^3-2\,x^4\right )-\ln \left (4-x\right )\,\left (4\,x-{\mathrm {e}}^{2\,x}\,\left (12\,x^2-x^3\right )\right )-2\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(15*x^2 + 20*x^3 - 4*x^4) + exp(2*x)*log(4 - x)^2*(2*x + 2*x^2) + exp(2*x)*log(4 - x)*(24*x + 19
*x^2 - 2*x^3))/(log(4 - x)^2*(x^2*exp(2*x) - 2) + exp(2*x)*(12*x^3 - 2*x^4) - log(4 - x)*(4*x - exp(2*x)*(12*x
^2 - x^3)) - 2*x^2),x)

[Out]

int((exp(2*x)*(15*x^2 + 20*x^3 - 4*x^4) + exp(2*x)*log(4 - x)^2*(2*x + 2*x^2) + exp(2*x)*log(4 - x)*(24*x + 19
*x^2 - 2*x^3))/(log(4 - x)^2*(x^2*exp(2*x) - 2) + exp(2*x)*(12*x^3 - 2*x^4) - log(4 - x)*(4*x - exp(2*x)*(12*x
^2 - x^3)) - 2*x^2), x)

________________________________________________________________________________________

sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+2*x)*exp(x)**2*ln(-x+4)**2+(-2*x**3+19*x**2+24*x)*exp(x)**2*ln(-x+4)+(-4*x**4+20*x**3+15*x*
*2)*exp(x)**2)/((exp(x)**2*x**2-2)*ln(-x+4)**2+((-x**3+12*x**2)*exp(x)**2-4*x)*ln(-x+4)+(-2*x**4+12*x**3)*exp(
x)**2-2*x**2),x)

[Out]

Exception raised: PolynomialError

________________________________________________________________________________________