3.48.44 \(\int \frac {\frac {5 (12-6 e^x-18 x)}{e^2}+\frac {25 (-200+297 x+18 x^2+e^x (109+6 x))}{e^4}+\frac {25 (8-4 e^x-12 x) \log (e^x+x)}{e^4}}{9 e^x+9 x+\frac {5 (e^x (-300-18 x)-300 x-18 x^2)}{e^2}+\frac {25 (2500 x+309 x^2+9 x^3+e^x (2500+309 x+9 x^2))}{e^4}+(\frac {5 (12 e^x+12 x)}{e^2}+\frac {25 (e^x (-200-12 x)-200 x-12 x^2)}{e^4}) \log (e^x+x)+\frac {25 (4 e^x+4 x) \log ^2(e^x+x)}{e^4}} \, dx\)

Optimal. Leaf size=28 \[ \log \left (x+\left (-\frac {e^2}{5}+x+\frac {2}{3} \left (25-\log \left (e^x+x\right )\right )\right )^2\right ) \]

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Rubi [F]  time = 7.18, 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 {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((5*(12 - 6*E^x - 18*x))/E^2 + (25*(-200 + 297*x + 18*x^2 + E^x*(109 + 6*x)))/E^4 + (25*(8 - 4*E^x - 12*x)
*Log[E^x + x])/E^4)/(9*E^x + 9*x + (5*(E^x*(-300 - 18*x) - 300*x - 18*x^2))/E^2 + (25*(2500*x + 309*x^2 + 9*x^
3 + E^x*(2500 + 309*x + 9*x^2)))/E^4 + ((5*(12*E^x + 12*x))/E^2 + (25*(E^x*(-200 - 12*x) - 200*x - 12*x^2))/E^
4)*Log[E^x + x] + (25*(4*E^x + 4*x)*Log[E^x + x]^2)/E^4),x]

[Out]

100*Defer[Int][Log[E^x + x]/(-62500*(1 + (3*E^2*(-500 + 3*E^2))/62500) - 7725*(1 - (6*E^2)/515)*x - 225*x^2 +
5000*(1 - (3*E^2)/250)*Log[E^x + x] + 300*x*Log[E^x + x] - 100*Log[E^x + x]^2), x] + 5*(545 - 6*E^2)*Defer[Int
][(62500*(1 + (3*E^2*(-500 + 3*E^2))/62500) + 7725*(1 - (6*E^2)/515)*x + 225*x^2 - 5000*(1 - (3*E^2)/250)*Log[
E^x + x] - 300*x*Log[E^x + x] + 100*Log[E^x + x]^2)^(-1), x] + 150*Defer[Int][x/(62500*(1 + (3*E^2*(-500 + 3*E
^2))/62500) + 7725*(1 - (6*E^2)/515)*x + 225*x^2 - 5000*(1 - (3*E^2)/250)*Log[E^x + x] - 300*x*Log[E^x + x] +
100*Log[E^x + x]^2), x] + 300*Defer[Int][x/((-E^x - x)*(62500*(1 + (3*E^2*(-500 + 3*E^2))/62500) + 7725*(1 - (
6*E^2)/515)*x + 225*x^2 - 5000*(1 - (3*E^2)/250)*Log[E^x + x] - 300*x*Log[E^x + x] + 100*Log[E^x + x]^2)), x]
- 20*(250 - 3*E^2)*Defer[Int][1/((E^x + x)*(62500*(1 + (3*E^2*(-500 + 3*E^2))/62500) + 7725*(1 - (6*E^2)/515)*
x + 225*x^2 - 5000*(1 - (3*E^2)/250)*Log[E^x + x] - 300*x*Log[E^x + x] + 100*Log[E^x + x]^2)), x] + 20*(250 -
3*E^2)*Defer[Int][x/((E^x + x)*(62500*(1 + (3*E^2*(-500 + 3*E^2))/62500) + 7725*(1 - (6*E^2)/515)*x + 225*x^2
- 5000*(1 - (3*E^2)/250)*Log[E^x + x] - 300*x*Log[E^x + x] + 100*Log[E^x + x]^2)), x] + 300*Defer[Int][x^2/((E
^x + x)*(62500*(1 + (3*E^2*(-500 + 3*E^2))/62500) + 7725*(1 - (6*E^2)/515)*x + 225*x^2 - 5000*(1 - (3*E^2)/250
)*Log[E^x + x] - 300*x*Log[E^x + x] + 100*Log[E^x + x]^2)), x] + 200*Defer[Int][(x*Log[E^x + x])/((-E^x - x)*(
62500*(1 + (3*E^2*(-500 + 3*E^2))/62500) + 7725*(1 - (6*E^2)/515)*x + 225*x^2 - 5000*(1 - (3*E^2)/250)*Log[E^x
+ x] - 300*x*Log[E^x + x] + 100*Log[E^x + x]^2)), x] + 200*Defer[Int][Log[E^x + x]/((E^x + x)*(62500*(1 + (3*
E^2*(-500 + 3*E^2))/62500) + 7725*(1 - (6*E^2)/515)*x + 225*x^2 - 5000*(1 - (3*E^2)/250)*Log[E^x + x] - 300*x*
Log[E^x + x] + 100*Log[E^x + x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-6 e^{2+x}-6 e^2 (-2+3 x)+5 e^x (109+6 x)+5 \left (-200+297 x+18 x^2\right )-20 \left (-2+e^x+3 x\right ) \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (9 e^4-30 e^2 (50+3 x)+25 \left (2500+309 x+9 x^2\right )+20 \left (3 e^2-5 (50+3 x)\right ) \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx\\ &=5 \int \frac {-6 e^{2+x}-6 e^2 (-2+3 x)+5 e^x (109+6 x)+5 \left (-200+297 x+18 x^2\right )-20 \left (-2+e^x+3 x\right ) \log \left (e^x+x\right )}{\left (e^x+x\right ) \left (9 e^4-30 e^2 (50+3 x)+25 \left (2500+309 x+9 x^2\right )+20 \left (3 e^2-5 (50+3 x)\right ) \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx\\ &=5 \int \left (\frac {545 \left (1-\frac {6 e^2}{545}\right )+30 x-20 \log \left (e^x+x\right )}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )}+\frac {4 (1-x) \left (-250 \left (1-\frac {3 e^2}{250}\right )-15 x+10 \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}\right ) \, dx\\ &=5 \int \frac {545 \left (1-\frac {6 e^2}{545}\right )+30 x-20 \log \left (e^x+x\right )}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx+20 \int \frac {(1-x) \left (-250 \left (1-\frac {3 e^2}{250}\right )-15 x+10 \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx\\ &=5 \int \left (\frac {20 \log \left (e^x+x\right )}{-62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )-7725 \left (1-\frac {6 e^2}{515}\right ) x-225 x^2+5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )+300 x \log \left (e^x+x\right )-100 \log ^2\left (e^x+x\right )}+\frac {545-6 e^2}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )}+\frac {30 x}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )}\right ) \, dx+20 \int \left (\frac {x \left (250 \left (1-\frac {3 e^2}{250}\right )+15 x-10 \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}+\frac {-250 \left (1-\frac {3 e^2}{250}\right )-15 x+10 \log \left (e^x+x\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}\right ) \, dx\\ &=20 \int \frac {x \left (250 \left (1-\frac {3 e^2}{250}\right )+15 x-10 \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+20 \int \frac {-250 \left (1-\frac {3 e^2}{250}\right )-15 x+10 \log \left (e^x+x\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+100 \int \frac {\log \left (e^x+x\right )}{-62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )-7725 \left (1-\frac {6 e^2}{515}\right ) x-225 x^2+5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )+300 x \log \left (e^x+x\right )-100 \log ^2\left (e^x+x\right )} \, dx+150 \int \frac {x}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx+\left (5 \left (545-6 e^2\right )\right ) \int \frac {1}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.14, size = 65, normalized size = 2.32 \begin {gather*} \log \left (62500-1500 e^2+9 e^4+7725 x-90 e^2 x+225 x^2-5000 \log \left (e^x+x\right )+60 e^2 \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5*(12 - 6*E^x - 18*x))/E^2 + (25*(-200 + 297*x + 18*x^2 + E^x*(109 + 6*x)))/E^4 + (25*(8 - 4*E^x -
12*x)*Log[E^x + x])/E^4)/(9*E^x + 9*x + (5*(E^x*(-300 - 18*x) - 300*x - 18*x^2))/E^2 + (25*(2500*x + 309*x^2
+ 9*x^3 + E^x*(2500 + 309*x + 9*x^2)))/E^4 + ((5*(12*E^x + 12*x))/E^2 + (25*(E^x*(-200 - 12*x) - 200*x - 12*x^
2))/E^4)*Log[E^x + x] + (25*(4*E^x + 4*x)*Log[E^x + x]^2)/E^4),x]

[Out]

Log[62500 - 1500*E^2 + 9*E^4 + 7725*x - 90*E^2*x + 225*x^2 - 5000*Log[E^x + x] + 60*E^2*Log[E^x + x] - 300*x*L
og[E^x + x] + 100*Log[E^x + x]^2]

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fricas [B]  time = 0.63, size = 77, normalized size = 2.75 \begin {gather*} \log \left (4 \, e^{\left (2 \, \log \relax (5) - 4\right )} \log \left (x + e^{x}\right )^{2} + {\left (9 \, x^{2} + 309 \, x + 2500\right )} e^{\left (2 \, \log \relax (5) - 4\right )} - 6 \, {\left (3 \, x + 50\right )} e^{\left (\log \relax (5) - 2\right )} - 4 \, {\left ({\left (3 \, x + 50\right )} e^{\left (2 \, \log \relax (5) - 4\right )} - 3 \, e^{\left (\log \relax (5) - 2\right )}\right )} \log \left (x + e^{x}\right ) + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)-12*x+8)*exp(log(5)-2)^2*log(exp(x)+x)+((6*x+109)*exp(x)+18*x^2+297*x-200)*exp(log(5)-2)^
2+(-6*exp(x)-18*x+12)*exp(log(5)-2))/((4*exp(x)+4*x)*exp(log(5)-2)^2*log(exp(x)+x)^2+(((-12*x-200)*exp(x)-12*x
^2-200*x)*exp(log(5)-2)^2+(12*exp(x)+12*x)*exp(log(5)-2))*log(exp(x)+x)+((9*x^2+309*x+2500)*exp(x)+9*x^3+309*x
^2+2500*x)*exp(log(5)-2)^2+((-18*x-300)*exp(x)-18*x^2-300*x)*exp(log(5)-2)+9*exp(x)+9*x),x, algorithm="fricas"
)

[Out]

log(4*e^(2*log(5) - 4)*log(x + e^x)^2 + (9*x^2 + 309*x + 2500)*e^(2*log(5) - 4) - 6*(3*x + 50)*e^(log(5) - 2)
- 4*((3*x + 50)*e^(2*log(5) - 4) - 3*e^(log(5) - 2))*log(x + e^x) + 9)

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giac [B]  time = 0.34, size = 57, normalized size = 2.04 \begin {gather*} \log \left (225 \, x^{2} - 90 \, x e^{2} - 300 \, x \log \left (x + e^{x}\right ) + 60 \, e^{2} \log \left (x + e^{x}\right ) + 100 \, \log \left (x + e^{x}\right )^{2} + 7725 \, x + 9 \, e^{4} - 1500 \, e^{2} - 5000 \, \log \left (x + e^{x}\right ) + 62500\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)-12*x+8)*exp(log(5)-2)^2*log(exp(x)+x)+((6*x+109)*exp(x)+18*x^2+297*x-200)*exp(log(5)-2)^
2+(-6*exp(x)-18*x+12)*exp(log(5)-2))/((4*exp(x)+4*x)*exp(log(5)-2)^2*log(exp(x)+x)^2+(((-12*x-200)*exp(x)-12*x
^2-200*x)*exp(log(5)-2)^2+(12*exp(x)+12*x)*exp(log(5)-2))*log(exp(x)+x)+((9*x^2+309*x+2500)*exp(x)+9*x^3+309*x
^2+2500*x)*exp(log(5)-2)^2+((-18*x-300)*exp(x)-18*x^2-300*x)*exp(log(5)-2)+9*exp(x)+9*x),x, algorithm="giac")

[Out]

log(225*x^2 - 90*x*e^2 - 300*x*log(x + e^x) + 60*e^2*log(x + e^x) + 100*log(x + e^x)^2 + 7725*x + 9*e^4 - 1500
*e^2 - 5000*log(x + e^x) + 62500)

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maple [B]  time = 0.07, size = 61, normalized size = 2.18

method	result	size
risch	\(\ln \left (\ln \left ({\mathrm e}^{x}+x \right )^{2}-\frac {{\mathrm e}^{2} \left (15 \,{\mathrm e}^{-2} x +250 \,{\mathrm e}^{-2}-3\right ) \ln \left ({\mathrm e}^{x}+x \right )}{5}+\frac {\left (225 \,{\mathrm e}^{-4} x^{2}+7725 \,{\mathrm e}^{-4} x -90 \,{\mathrm e}^{-2} x +62500 \,{\mathrm e}^{-4}-1500 \,{\mathrm e}^{-2}+9\right ) {\mathrm e}^{4}}{100}\right )\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*exp(x)-12*x+8)*exp(ln(5)-2)^2*ln(exp(x)+x)+((6*x+109)*exp(x)+18*x^2+297*x-200)*exp(ln(5)-2)^2+(-6*exp
(x)-18*x+12)*exp(ln(5)-2))/((4*exp(x)+4*x)*exp(ln(5)-2)^2*ln(exp(x)+x)^2+(((-12*x-200)*exp(x)-12*x^2-200*x)*ex
p(ln(5)-2)^2+(12*exp(x)+12*x)*exp(ln(5)-2))*ln(exp(x)+x)+((9*x^2+309*x+2500)*exp(x)+9*x^3+309*x^2+2500*x)*exp(
ln(5)-2)^2+((-18*x-300)*exp(x)-18*x^2-300*x)*exp(ln(5)-2)+9*exp(x)+9*x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(exp(x)+x)^2-1/5*exp(2)*(15*exp(-2)*x+250*exp(-2)-3)*ln(exp(x)+x)+1/100*(225*exp(-4)*x^2+7725*exp(-4)*x-9
0*exp(-2)*x+62500*exp(-4)-1500*exp(-2)+9)*exp(4))

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maxima [A]  time = 0.45, size = 48, normalized size = 1.71 \begin {gather*} \log \left (\frac {9}{4} \, x^{2} - \frac {3}{20} \, x {\left (6 \, e^{2} - 515\right )} - \frac {1}{5} \, {\left (15 \, x - 3 \, e^{2} + 250\right )} \log \left (x + e^{x}\right ) + \log \left (x + e^{x}\right )^{2} + \frac {9}{100} \, e^{4} - 15 \, e^{2} + 625\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)-12*x+8)*exp(log(5)-2)^2*log(exp(x)+x)+((6*x+109)*exp(x)+18*x^2+297*x-200)*exp(log(5)-2)^
2+(-6*exp(x)-18*x+12)*exp(log(5)-2))/((4*exp(x)+4*x)*exp(log(5)-2)^2*log(exp(x)+x)^2+(((-12*x-200)*exp(x)-12*x
^2-200*x)*exp(log(5)-2)^2+(12*exp(x)+12*x)*exp(log(5)-2))*log(exp(x)+x)+((9*x^2+309*x+2500)*exp(x)+9*x^3+309*x
^2+2500*x)*exp(log(5)-2)^2+((-18*x-300)*exp(x)-18*x^2-300*x)*exp(log(5)-2)+9*exp(x)+9*x),x, algorithm="maxima"
)

[Out]

log(9/4*x^2 - 3/20*x*(6*e^2 - 515) - 1/5*(15*x - 3*e^2 + 250)*log(x + e^x) + log(x + e^x)^2 + 9/100*e^4 - 15*e
^2 + 625)

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mupad [B]  time = 4.51, size = 55, normalized size = 1.96 \begin {gather*} \ln \left (\frac {309\,x}{4}-50\,\ln \left (x+{\mathrm {e}}^x\right )-15\,{\mathrm {e}}^2+\frac {9\,{\mathrm {e}}^4}{100}-\frac {9\,x\,{\mathrm {e}}^2}{10}+{\ln \left (x+{\mathrm {e}}^x\right )}^2+\frac {9\,x^2}{4}+\frac {3\,\ln \left (x+{\mathrm {e}}^x\right )\,{\mathrm {e}}^2}{5}-3\,x\,\ln \left (x+{\mathrm {e}}^x\right )+625\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(5) - 2)*(18*x + 6*exp(x) - 12) - exp(2*log(5) - 4)*(297*x + exp(x)*(6*x + 109) + 18*x^2 - 200) +
log(x + exp(x))*exp(2*log(5) - 4)*(12*x + 4*exp(x) - 8))/(9*x + 9*exp(x) + exp(2*log(5) - 4)*(2500*x + exp(x)
*(309*x + 9*x^2 + 2500) + 309*x^2 + 9*x^3) - exp(log(5) - 2)*(300*x + exp(x)*(18*x + 300) + 18*x^2) + log(x +
exp(x))*(exp(log(5) - 2)*(12*x + 12*exp(x)) - exp(2*log(5) - 4)*(200*x + exp(x)*(12*x + 200) + 12*x^2)) + log(
x + exp(x))^2*exp(2*log(5) - 4)*(4*x + 4*exp(x))),x)

[Out]

log((309*x)/4 - 50*log(x + exp(x)) - 15*exp(2) + (9*exp(4))/100 - (9*x*exp(2))/10 + log(x + exp(x))^2 + (9*x^2
)/4 + (3*log(x + exp(x))*exp(2))/5 - 3*x*log(x + exp(x)) + 625)

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sympy [B]  time = 0.72, size = 61, normalized size = 2.18 \begin {gather*} \log {\left (\frac {9 x^{2}}{4} - \frac {9 x e^{2}}{10} + \frac {309 x}{4} + \left (- 3 x - 50 + \frac {3 e^{2}}{5}\right ) \log {\left (x + e^{x} \right )} + \log {\left (x + e^{x} \right )}^{2} - 15 e^{2} + \frac {9 e^{4}}{100} + 625 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)-12*x+8)*exp(ln(5)-2)**2*ln(exp(x)+x)+((6*x+109)*exp(x)+18*x**2+297*x-200)*exp(ln(5)-2)**
2+(-6*exp(x)-18*x+12)*exp(ln(5)-2))/((4*exp(x)+4*x)*exp(ln(5)-2)**2*ln(exp(x)+x)**2+(((-12*x-200)*exp(x)-12*x*
*2-200*x)*exp(ln(5)-2)**2+(12*exp(x)+12*x)*exp(ln(5)-2))*ln(exp(x)+x)+((9*x**2+309*x+2500)*exp(x)+9*x**3+309*x
**2+2500*x)*exp(ln(5)-2)**2+((-18*x-300)*exp(x)-18*x**2-300*x)*exp(ln(5)-2)+9*exp(x)+9*x),x)

[Out]

log(9*x**2/4 - 9*x*exp(2)/10 + 309*x/4 + (-3*x - 50 + 3*exp(2)/5)*log(x + exp(x)) + log(x + exp(x))**2 - 15*ex
p(2) + 9*exp(4)/100 + 625)

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