3.71.43 \(\int \frac {4 e x+e (160 x-40 x^2) \log (4)+e (-16+4 x) \log (4-x)+e^{\frac {e^{\frac {x}{e}}}{4}} (4 e+e (80-20 x) \log (4)+e^{\frac {x}{e}} (20 x-5 x^2) \log (4)+e^{\frac {x}{e}} (-4+x) \log (4-x))}{e (-16+4 x)} \, dx\) [7043]

Optimal. Leaf size=30 \[ 25+\left (e^{\frac {e^{\frac {x}{e}}}{4}}+x\right ) (-5 x \log (4)+\log (4-x)) \]

[Out]

(ln(4-x)-10*x*ln(2))*(x+exp(1/4*exp(x/exp(1))))+25

________________________________________________________________________________________

Rubi [F]
time = 1.13, 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 e x+e \left (160 x-40 x^2\right ) \log (4)+e (-16+4 x) \log (4-x)+e^{\frac {e^{\frac {x}{e}}}{4}} \left (4 e+e (80-20 x) \log (4)+e^{\frac {x}{e}} \left (20 x-5 x^2\right ) \log (4)+e^{\frac {x}{e}} (-4+x) \log (4-x)\right )}{e (-16+4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*E*x + E*(160*x - 40*x^2)*Log[4] + E*(-16 + 4*x)*Log[4 - x] + E^(E^(x/E)/4)*(4*E + E*(80 - 20*x)*Log[4]
+ E^(x/E)*(20*x - 5*x^2)*Log[4] + E^(x/E)*(-4 + x)*Log[4 - x]))/(E*(-16 + 4*x)),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {4 e x+e \left (160 x-40 x^2\right ) \log (4)+e (-16+4 x) \log (4-x)+e^{\frac {e^{\frac {x}{e}}}{4}} \left (4 e+e (80-20 x) \log (4)+e^{\frac {x}{e}} \left (20 x-5 x^2\right ) \log (4)+e^{\frac {x}{e}} (-4+x) \log (4-x)\right )}{-16+4 x} \, dx}{e}\\ &=\frac {\int \left (-\frac {1}{4} e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} (5 x \log (4)-\log (4-x))+\frac {e \left (5 e^{\frac {e^{\frac {x}{e}}}{4}} x \log (4)+10 x^2 \log (4)-e^{\frac {e^{\frac {x}{e}}}{4}} (1+20 \log (4))-x (1+40 \log (4))+4 \log (4-x)-x \log (4-x)\right )}{4-x}\right ) \, dx}{e}\\ &=-\frac {\int e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} (5 x \log (4)-\log (4-x)) \, dx}{4 e}+\int \frac {5 e^{\frac {e^{\frac {x}{e}}}{4}} x \log (4)+10 x^2 \log (4)-e^{\frac {e^{\frac {x}{e}}}{4}} (1+20 \log (4))-x (1+40 \log (4))+4 \log (4-x)-x \log (4-x)}{4-x} \, dx\\ &=-\frac {\int \left (5 e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} x \log (4)-e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} \log (4-x)\right ) \, dx}{4 e}+\int \frac {-x (1+40 \log (4)-10 x \log (4))-e^{\frac {e^{\frac {x}{e}}}{4}} (1+20 \log (4)-5 x \log (4))-(-4+x) \log (4-x)}{4-x} \, dx\\ &=\frac {\int e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} \log (4-x) \, dx}{4 e}-\frac {(5 \log (4)) \int e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} x \, dx}{4 e}+\int \left (-\frac {e^{\frac {e^{\frac {x}{e}}}{4}} (-1-20 \log (4)+5 x \log (4))}{-4+x}+\frac {10 x^2 \log (4)-x (1+40 \log (4))+4 \log (4-x)-x \log (4-x)}{4-x}\right ) \, dx\\ &=e^{\frac {e^{\frac {x}{e}}}{4}} \log (4-x)-\frac {\int \frac {4 e^{\frac {1}{4} \left (4+e^{\frac {x}{e}}\right )}}{-4+x} \, dx}{4 e}-\frac {(5 \log (4)) \int e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} x \, dx}{4 e}-\int \frac {e^{\frac {e^{\frac {x}{e}}}{4}} (-1-20 \log (4)+5 x \log (4))}{-4+x} \, dx+\int \frac {10 x^2 \log (4)-x (1+40 \log (4))+4 \log (4-x)-x \log (4-x)}{4-x} \, dx\\ &=e^{\frac {e^{\frac {x}{e}}}{4}} \log (4-x)-\frac {\int \frac {e^{\frac {1}{4} \left (4+e^{\frac {x}{e}}\right )}}{-4+x} \, dx}{e}-\frac {(5 \log (4)) \int e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} x \, dx}{4 e}-\int \left (\frac {e^{\frac {e^{\frac {x}{e}}}{4}}}{4-x}+5 e^{\frac {e^{\frac {x}{e}}}{4}} \log (4)\right ) \, dx+\int \left (\frac {x (1+40 \log (4)-10 x \log (4))}{-4+x}+\log (4-x)\right ) \, dx\\ &=e^{\frac {e^{\frac {x}{e}}}{4}} \log (4-x)-\frac {\int \frac {e^{\frac {1}{4} \left (4+e^{\frac {x}{e}}\right )}}{-4+x} \, dx}{e}-(5 \log (4)) \int e^{\frac {e^{\frac {x}{e}}}{4}} \, dx-\frac {(5 \log (4)) \int e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} x \, dx}{4 e}-\int \frac {e^{\frac {e^{\frac {x}{e}}}{4}}}{4-x} \, dx+\int \frac {x (1+40 \log (4)-10 x \log (4))}{-4+x} \, dx+\int \log (4-x) \, dx\\ &=e^{\frac {e^{\frac {x}{e}}}{4}} \log (4-x)-\frac {\int \frac {e^{\frac {1}{4} \left (4+e^{\frac {x}{e}}\right )}}{-4+x} \, dx}{e}-\frac {(5 \log (4)) \int e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} x \, dx}{4 e}-(5 e \log (4)) \text {Subst}\left (\int \frac {e^{x/4}}{x} \, dx,x,e^{\frac {x}{e}}\right )-\int \frac {e^{\frac {e^{\frac {x}{e}}}{4}}}{4-x} \, dx+\int \left (1+\frac {4}{-4+x}-10 x \log (4)\right ) \, dx-\text {Subst}(\int \log (x) \, dx,x,4-x)\\ &=-5 x^2 \log (4)-5 e \text {Ei}\left (\frac {e^{\frac {x}{e}}}{4}\right ) \log (4)+4 \log (4-x)+e^{\frac {e^{\frac {x}{e}}}{4}} \log (4-x)-(4-x) \log (4-x)-\frac {\int \frac {e^{\frac {1}{4} \left (4+e^{\frac {x}{e}}\right )}}{-4+x} \, dx}{e}-\frac {(5 \log (4)) \int e^{\frac {e^{\frac {x}{e}}}{4}+\frac {x}{e}} x \, dx}{4 e}-\int \frac {e^{\frac {e^{\frac {x}{e}}}{4}}}{4-x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.27, size = 50, normalized size = 1.67 \begin {gather*} -5 x \left (e^{\frac {e^{\frac {x}{e}}}{4}}+x\right ) \log (4)+\left (-4+e^{\frac {e^{\frac {x}{e}}}{4}}+x\right ) \log (4-x)+4 \log (-4+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E*x + E*(160*x - 40*x^2)*Log[4] + E*(-16 + 4*x)*Log[4 - x] + E^(E^(x/E)/4)*(4*E + E*(80 - 20*x)*L
og[4] + E^(x/E)*(20*x - 5*x^2)*Log[4] + E^(x/E)*(-4 + x)*Log[4 - x]))/(E*(-16 + 4*x)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.32, size = 45, normalized size = 1.50

method	result	size
risch	\(-10 x^{2} \ln \left (2\right )+\ln \left (-x +4\right ) x +\left (-10 x \,{\mathrm e} \ln \left (2\right )+{\mathrm e} \ln \left (-x +4\right )\right ) {\mathrm e}^{-1+\frac {{\mathrm e}^{{\mathrm e}^{-1} x}}{4}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-4)*exp(x/exp(1))*ln(-x+4)+2*(-5*x^2+20*x)*ln(2)*exp(x/exp(1))+2*(-20*x+80)*exp(1)*ln(2)+4*exp(1))*exp
(1/4*exp(x/exp(1)))+(4*x-16)*exp(1)*ln(-x+4)+2*(-40*x^2+160*x)*exp(1)*ln(2)+4*x*exp(1))/(4*x-16)/exp(1),x,meth
od=_RETURNVERBOSE)

[Out]

-10*x^2*ln(2)+ln(-x+4)*x+(-10*x*exp(1)*ln(2)+exp(1)*ln(-x+4))*exp(-1+1/4*exp(exp(-1)*x))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (28) = 56\).
time = 0.56, size = 155, normalized size = 5.17 \begin {gather*} -{\left (10 \, {\left (x^{2} + 8 \, x + 32 \, \log \left (x - 4\right )\right )} e \log \left (2\right ) - 80 \, {\left (x + 4 \, \log \left (x - 4\right )\right )} e \log \left (2\right ) - {\left (x + 4 \, \log \left (x - 4\right )\right )} e \log \left (-x + 4\right ) + 4 \, e \log \left (x - 4\right ) \log \left (-x + 4\right ) + {\left (2 \, \log \left (x - 4\right )^{2} + x + 4 \, \log \left (x - 4\right )\right )} e - 2 \, {\left (2 \, \log \left (x - 4\right ) \log \left (-x + 4\right ) - \log \left (-x + 4\right )^{2}\right )} e - {\left (x + 4 \, \log \left (x - 4\right )\right )} e + {\left (10 \, x e \log \left (2\right ) - e \log \left (-x + 4\right )\right )} e^{\left (\frac {1}{4} \, e^{\left (x e^{\left (-1\right )}\right )}\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-4)*exp(x/exp(1))*log(-x+4)+2*(-5*x^2+20*x)*log(2)*exp(x/exp(1))+2*(-20*x+80)*exp(1)*log(2)+4*ex
p(1))*exp(1/4*exp(x/exp(1)))+(4*x-16)*exp(1)*log(-x+4)+2*(-40*x^2+160*x)*exp(1)*log(2)+4*x*exp(1))/(4*x-16)/ex
p(1),x, algorithm="maxima")

[Out]

-(10*(x^2 + 8*x + 32*log(x - 4))*e*log(2) - 80*(x + 4*log(x - 4))*e*log(2) - (x + 4*log(x - 4))*e*log(-x + 4)
+ 4*e*log(x - 4)*log(-x + 4) + (2*log(x - 4)^2 + x + 4*log(x - 4))*e - 2*(2*log(x - 4)*log(-x + 4) - log(-x +
4)^2)*e - (x + 4*log(x - 4))*e + (10*x*e*log(2) - e*log(-x + 4))*e^(1/4*e^(x*e^(-1))))*e^(-1)

________________________________________________________________________________________

Fricas [A]
time = 0.40, size = 40, normalized size = 1.33 \begin {gather*} -10 \, x^{2} \log \left (2\right ) - {\left (10 \, x \log \left (2\right ) - \log \left (-x + 4\right )\right )} e^{\left (\frac {1}{4} \, e^{\left (x e^{\left (-1\right )}\right )}\right )} + x \log \left (-x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-4)*exp(x/exp(1))*log(-x+4)+2*(-5*x^2+20*x)*log(2)*exp(x/exp(1))+2*(-20*x+80)*exp(1)*log(2)+4*ex
p(1))*exp(1/4*exp(x/exp(1)))+(4*x-16)*exp(1)*log(-x+4)+2*(-40*x^2+160*x)*exp(1)*log(2)+4*x*exp(1))/(4*x-16)/ex
p(1),x, algorithm="fricas")

[Out]

-10*x^2*log(2) - (10*x*log(2) - log(-x + 4))*e^(1/4*e^(x*e^(-1))) + x*log(-x + 4)

________________________________________________________________________________________

Sympy [A]
time = 23.34, size = 36, normalized size = 1.20 \begin {gather*} - 10 x^{2} \log {\left (2 \right )} + x \log {\left (4 - x \right )} + \left (- 10 x \log {\left (2 \right )} + \log {\left (4 - x \right )}\right ) e^{\frac {e^{\frac {x}{e}}}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-4)*exp(x/exp(1))*ln(-x+4)+2*(-5*x**2+20*x)*ln(2)*exp(x/exp(1))+2*(-20*x+80)*exp(1)*ln(2)+4*exp(
1))*exp(1/4*exp(x/exp(1)))+(4*x-16)*exp(1)*ln(-x+4)+2*(-40*x**2+160*x)*exp(1)*ln(2)+4*x*exp(1))/(4*x-16)/exp(1
),x)

[Out]

-10*x**2*log(2) + x*log(4 - x) + (-10*x*log(2) + log(4 - x))*exp(exp(x*exp(-1))/4)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-4)*exp(x/exp(1))*log(-x+4)+2*(-5*x^2+20*x)*log(2)*exp(x/exp(1))+2*(-20*x+80)*exp(1)*log(2)+4*ex
p(1))*exp(1/4*exp(x/exp(1)))+(4*x-16)*exp(1)*log(-x+4)+2*(-40*x^2+160*x)*exp(1)*log(2)+4*x*exp(1))/(4*x-16)/ex
p(1),x, algorithm="giac")

[Out]

integrate(-1/4*(80*(x^2 - 4*x)*e*log(2) - 4*(x - 4)*e*log(-x + 4) - 4*x*e + (40*(x - 4)*e*log(2) + 10*(x^2 - 4
*x)*e^(x*e^(-1))*log(2) - (x - 4)*e^(x*e^(-1))*log(-x + 4) - 4*e)*e^(1/4*e^(x*e^(-1))))*e^(-1)/(x - 4), x)

________________________________________________________________________________________

Mupad [B]
time = 0.51, size = 23, normalized size = 0.77 \begin {gather*} \left (x+{\mathrm {e}}^{\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^{-1}}}{4}}\right )\,\left (\ln \left (4-x\right )-10\,x\,\ln \left (2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-1)*(4*x*exp(1) + exp(exp(x*exp(-1))/4)*(4*exp(1) - 2*exp(1)*log(2)*(20*x - 80) + exp(x*exp(-1))*log(
4 - x)*(x - 4) + 2*exp(x*exp(-1))*log(2)*(20*x - 5*x^2)) + exp(1)*log(4 - x)*(4*x - 16) + 2*exp(1)*log(2)*(160
*x - 40*x^2)))/(4*x - 16),x)

[Out]

(x + exp(exp(x*exp(-1))/4))*(log(4 - x) - 10*x*log(2))

________________________________________________________________________________________