∫ e3(−156−72x)−156x−46x2+12x3+(156+72x) log(3)+(52x2+36x3−2x4+e3(52x+36x2)+(−52x−36x2) log(3)) log(−e3−x+log(3))+(−4e3x3−4x4+4x3 log(3)) log2(−e3−x+log(3)) ______________________________________________________________________________________________________________________________

3.93.81 \(\int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+(52 x^2+36 x^3-2 x^4+e^3 (52 x+36 x^2)+(-52 x-36 x^2) \log (3)) \log (-e^3-x+\log (3))+(-4 e^3 x^3-4 x^4+4 x^3 \log (3)) \log ^2(-e^3-x+\log (3))}{-e^3-x+\log (3)} \, dx\)

Optimal. Leaf size=28 \[ \left (7-x-5 (4+x)+x^2 \log \left (-e^3-x+\log (3)\right )\right )^2 \]

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Rubi [A] time = 0.28, antiderivative size = 24, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, integrand size = 135, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6688, 12, 6686} \begin {gather*} \left (x^2 \left (-\log \left (-x-e^3+\log (3)\right )\right )+6 x+13\right )^2 \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(E^3*(-156 - 72*x) - 156*x - 46*x^2 + 12*x^3 + (156 + 72*x)*Log[3] + (52*x^2 + 36*x^3 - 2*x^4 + E^3*(52*x

+ 36*x^2) + (-52*x - 36*x^2)*Log[3])*Log[-E^3 - x + Log[3]] + (-4*E^3*x^3 - 4*x^4 + 4*x^3*Log[3])*Log[-E^3 - x

+ Log[3]]^2)/(-E^3 - x + Log[3]),x]

[Out]

(13 + 6*x - x^2*Log[-E^3 - x + Log[3]])^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /; !F

alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (13+6 x-x^2 \log \left (-e^3-x+\log (3)\right )\right ) \left (6 x-x^2+6 e^3 \left (1-\frac {\log (729)}{6 e^3}\right )-2 x \left (e^3+x-\log (3)\right ) \log \left (-e^3-x+\log (3)\right )\right )}{e^3+x-\log (3)} \, dx\\ &=2 \int \frac {\left (13+6 x-x^2 \log \left (-e^3-x+\log (3)\right )\right ) \left (6 x-x^2+6 e^3 \left (1-\frac {\log (729)}{6 e^3}\right )-2 x \left (e^3+x-\log (3)\right ) \log \left (-e^3-x+\log (3)\right )\right )}{e^3+x-\log (3)} \, dx\\ &=\left (13+6 x-x^2 \log \left (-e^3-x+\log (3)\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.05, size = 23, normalized size = 0.82 \begin {gather*} \left (-13-6 x+x^2 \log \left (-e^3-x+\log (3)\right )\right )^2 \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^3*(-156 - 72*x) - 156*x - 46*x^2 + 12*x^3 + (156 + 72*x)*Log[3] + (52*x^2 + 36*x^3 - 2*x^4 + E^3*

(52*x + 36*x^2) + (-52*x - 36*x^2)*Log[3])*Log[-E^3 - x + Log[3]] + (-4*E^3*x^3 - 4*x^4 + 4*x^3*Log[3])*Log[-E

^3 - x + Log[3]]^2)/(-E^3 - x + Log[3]),x]

[Out]

(-13 - 6*x + x^2*Log[-E^3 - x + Log[3]])^2

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fricas [B] time = 0.79, size = 50, normalized size = 1.79 \begin {gather*} x^{4} \log \left (-x - e^{3} + \log \relax (3)\right )^{2} + 36 \, x^{2} - 2 \, {\left (6 \, x^{3} + 13 \, x^{2}\right )} \log \left (-x - e^{3} + \log \relax (3)\right ) + 156 \, x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*log(3)-4*x^3*exp(3)-4*x^4)*log(log(3)-exp(3)-x)^2+((-36*x^2-52*x)*log(3)+(36*x^2+52*x)*exp(3

)-2*x^4+36*x^3+52*x^2)*log(log(3)-exp(3)-x)+(72*x+156)*log(3)+(-72*x-156)*exp(3)+12*x^3-46*x^2-156*x)/(log(3)-

exp(3)-x),x, algorithm="fricas")

[Out]

x^4*log(-x - e^3 + log(3))^2 + 36*x^2 - 2*(6*x^3 + 13*x^2)*log(-x - e^3 + log(3)) + 156*x

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giac [B] time = 0.17, size = 58, normalized size = 2.07 \begin {gather*} x^{4} \log \left (-x - e^{3} + \log \relax (3)\right )^{2} - 12 \, x^{3} \log \left (-x - e^{3} + \log \relax (3)\right ) - 26 \, x^{2} \log \left (-x - e^{3} + \log \relax (3)\right ) + 36 \, x^{2} + 156 \, x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*log(3)-4*x^3*exp(3)-4*x^4)*log(log(3)-exp(3)-x)^2+((-36*x^2-52*x)*log(3)+(36*x^2+52*x)*exp(3

)-2*x^4+36*x^3+52*x^2)*log(log(3)-exp(3)-x)+(72*x+156)*log(3)+(-72*x-156)*exp(3)+12*x^3-46*x^2-156*x)/(log(3)-

exp(3)-x),x, algorithm="giac")

[Out]

x^4*log(-x - e^3 + log(3))^2 - 12*x^3*log(-x - e^3 + log(3)) - 26*x^2*log(-x - e^3 + log(3)) + 36*x^2 + 156*x

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maple [B] time = 0.53, size = 50, normalized size = 1.79


method	result	size

risch	\(\ln \left (\ln \relax (3)-{\mathrm e}^{3}-x \right )^{2} x^{4}+\left (-12 x^{3}-26 x^{2}\right ) \ln \left (\ln \relax (3)-{\mathrm e}^{3}-x \right )+36 x^{2}+156 x\)	\(50\)
norman	\(\ln \left (\ln \relax (3)-{\mathrm e}^{3}-x \right )^{2} x^{4}+156 x +36 x^{2}-12 x^{3} \ln \left (\ln \relax (3)-{\mathrm e}^{3}-x \right )-26 \ln \left (\ln \relax (3)-{\mathrm e}^{3}-x \right ) x^{2}\)	\(59\)
derivativedivides	\(\text {Expression too large to display}\)	\(1698\)
default	\(\text {Expression too large to display}\)	\(1698\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3*ln(3)-4*x^3*exp(3)-4*x^4)*ln(ln(3)-exp(3)-x)^2+((-36*x^2-52*x)*ln(3)+(36*x^2+52*x)*exp(3)-2*x^4+36

*x^3+52*x^2)*ln(ln(3)-exp(3)-x)+(72*x+156)*ln(3)+(-72*x-156)*exp(3)+12*x^3-46*x^2-156*x)/(ln(3)-exp(3)-x),x,me

thod=_RETURNVERBOSE)

[Out]

ln(ln(3)-exp(3)-x)^2*x^4+(-12*x^3-26*x^2)*ln(ln(3)-exp(3)-x)+36*x^2+156*x

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maxima [B] time = 1.17, size = 1942, normalized size = 69.36 result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*log(3)-4*x^3*exp(3)-4*x^4)*log(log(3)-exp(3)-x)^2+((-36*x^2-52*x)*log(3)+(36*x^2+52*x)*exp(3

)-2*x^4+36*x^3+52*x^2)*log(log(3)-exp(3)-x)+(72*x+156)*log(3)+(-72*x-156)*exp(3)+12*x^3-46*x^2-156*x)/(log(3)-

exp(3)-x),x, algorithm="maxima")

[Out]

1/8*(8*log(-x - e^3 + log(3))^2 - 4*log(-x - e^3 + log(3)) + 1)*(x + e^3 - log(3))^4 - 16/27*(9*(e^3 - log(3))

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

1/8*x^4 + 7/18*x^3*(e^3 - log(3)) - 4/3*(4*e^3*log(3)^3 - log(3)^4 - 6*e^6*log(3)^2 + 4*e^9*log(3) - e^12)*lo

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

^2 - 2*(2*e^3*log(3) - log(3)^2 - e^6)*log(-x - e^3 + log(3)) - e^6)*(x + e^3 - log(3))^2 + 13/12*(2*e^3*log(3

) - log(3)^2 - e^6)*x^2 - 9*x^2*(e^3 - log(3)) + (4*e^3*log(3)^3 - log(3)^4 - 6*e^6*log(3)^2 + 4*e^9*log(3) -

e^12)*log(x + e^3 - log(3))^2 - 18*(3*e^3*log(3)^2 - log(3)^3 - 3*e^6*log(3) + e^9)*log(x + e^3 - log(3))^2 -

26*(2*e^3*log(3) - log(3)^2 - e^6)*log(x + e^3 - log(3))^2 - 18*(x^2 - 2*x*(e^3 - log(3)) - 2*(2*e^3*log(3) -

log(3)^2 - e^6)*log(x + e^3 - log(3)))*e^3*log(-x - e^3 + log(3)) + 52*((e^3 - log(3))*log(x + e^3 - log(3)) -

x)*e^3*log(-x - e^3 + log(3)) + 18*(x^2 - 2*x*(e^3 - log(3)) - 2*(2*e^3*log(3) - log(3)^2 - e^6)*log(x + e^3

- log(3)))*log(3)*log(-x - e^3 + log(3)) - 52*((e^3 - log(3))*log(x + e^3 - log(3)) - x)*log(3)*log(-x - e^3 +

log(3)) - 16*(6*e^3*log(3)^2 - 2*log(3)^3 + (3*e^3*log(3)^2 - log(3)^3 - 3*e^6*log(3) + e^9)*log(-x - e^3 + l

og(3))^2 - 6*e^6*log(3) - 2*(3*e^3*log(3)^2 - log(3)^3 - 3*e^6*log(3) + e^9)*log(-x - e^3 + log(3)) + 2*e^9)*(

x + e^3 - log(3)) + 25/6*(3*e^3*log(3)^2 - log(3)^3 - 3*e^6*log(3) + e^9)*x - 54*(2*e^3*log(3) - log(3)^2 - e^

6)*x + 36*x^2 - 124*x*(e^3 - log(3)) + 1/27*(4*(9*log(-x - e^3 + log(3))^2 - 6*log(-x - e^3 + log(3)) + 2)*(x

+ e^3 - log(3))^3 - 36*(3*e^3*log(3)^2 - log(3)^3 - 3*e^6*log(3) + e^9)*log(-x - e^3 + log(3))^3 - 81*(2*(e^3

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

)^2 - 324*((2*e^3*log(3) - log(3)^2 - e^6)*log(-x - e^3 + log(3))^2 + 4*e^3*log(3) - 2*log(3)^2 - 2*(2*e^3*log

(3) - log(3)^2 - e^6)*log(-x - e^3 + log(3)) - 2*e^6)*(x + e^3 - log(3)))*e^3 - 9*(2*(2*e^3*log(3) - log(3)^2

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

g(3)))*e^3 - 26*((e^3 - log(3))*log(x + e^3 - log(3))^2 + 2*(e^3 - log(3))*log(x + e^3 - log(3)) - 2*x)*e^3 -

72*((e^3 - log(3))*log(x + e^3 - log(3)) - x)*e^3 - 1/27*(4*(9*log(-x - e^3 + log(3))^2 - 6*log(-x - e^3 + log

(3)) + 2)*(x + e^3 - log(3))^3 - 36*(3*e^3*log(3)^2 - log(3)^3 - 3*e^6*log(3) + e^9)*log(-x - e^3 + log(3))^3

- 81*(2*(e^3 - log(3))*log(-x - e^3 + log(3))^2 - 2*(e^3 - log(3))*log(-x - e^3 + log(3)) + e^3 - log(3))*(x +

e^3 - log(3))^2 - 324*((2*e^3*log(3) - log(3)^2 - e^6)*log(-x - e^3 + log(3))^2 + 4*e^3*log(3) - 2*log(3)^2 -

2*(2*e^3*log(3) - log(3)^2 - e^6)*log(-x - e^3 + log(3)) - 2*e^6)*(x + e^3 - log(3)))*log(3) + 9*(2*(2*e^3*lo

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

log(x + e^3 - log(3)))*log(3) + 26*((e^3 - log(3))*log(x + e^3 - log(3))^2 + 2*(e^3 - log(3))*log(x + e^3 - lo

g(3)) - 2*x)*log(3) + 72*((e^3 - log(3))*log(x + e^3 - log(3)) - x)*log(3) + 25/6*(4*e^3*log(3)^3 - log(3)^4 -

6*e^6*log(3)^2 + 4*e^9*log(3) - e^12)*log(x + e^3 - log(3)) - 54*(3*e^3*log(3)^2 - log(3)^3 - 3*e^6*log(3) +

e^9)*log(x + e^3 - log(3)) - 124*(2*e^3*log(3) - log(3)^2 - e^6)*log(x + e^3 - log(3)) - 156*(e^3 - log(3))*lo

g(x + e^3 - log(3)) + 156*e^3*log(x + e^3 - log(3)) - 156*log(3)*log(x + e^3 - log(3)) + 1/6*(3*x^4 - 4*x^3*(e

^3 - log(3)) - 6*(2*e^3*log(3) - log(3)^2 - e^6)*x^2 - 12*(3*e^3*log(3)^2 - log(3)^3 - 3*e^6*log(3) + e^9)*x -

12*(4*e^3*log(3)^3 - log(3)^4 - 6*e^6*log(3)^2 + 4*e^9*log(3) - e^12)*log(x + e^3 - log(3)))*log(-x - e^3 + l

og(3)) - 6*(2*x^3 - 3*x^2*(e^3 - log(3)) - 6*(2*e^3*log(3) - log(3)^2 - e^6)*x - 6*(3*e^3*log(3)^2 - log(3)^3

- 3*e^6*log(3) + e^9)*log(x + e^3 - log(3)))*log(-x - e^3 + log(3)) - 26*(x^2 - 2*x*(e^3 - log(3)) - 2*(2*e^3*

log(3) - log(3)^2 - e^6)*log(x + e^3 - log(3)))*log(-x - e^3 + log(3)) + 156*x

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mupad [B] time = 57.16, size = 1018, normalized size = 36.36 result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((156*x - log(3)*(72*x + 156) + log(log(3) - exp(3) - x)^2*(4*x^3*exp(3) - 4*x^3*log(3) + 4*x^4) - log(log(

3) - exp(3) - x)*(exp(3)*(52*x + 36*x^2) - log(3)*(52*x + 36*x^2) + 52*x^2 + 36*x^3 - 2*x^4) + 46*x^2 - 12*x^3

+ exp(3)*(72*x + 156))/(x + exp(3) - log(3)),x)

[Out]

156*x + x^2*(6*exp(3) - 6*log(3)) + 72*x*exp(3) - 72*x*log(3) + x*(2*exp(9) - 6*exp(6)*log(3) + (exp(3) - log(

729)/6)*(exp(6) - 2*exp(3)*log(3) + (exp(3) - log(729)/6)*((7*exp(3))/6 - log(81)/6 - log(729)/12 + 12) + log(

3)^2 - 26) + 6*exp(3)*log(3)^2 - 2*log(3)^3) + log(x + exp(3) - log(3))*(26*exp(6) - 12*exp(9) - (25*exp(12))/

6 - 52*exp(3)*log(3) + 36*exp(6)*log(3) + (50*exp(9)*log(3))/3 - 36*exp(3)*log(3)^2 + (50*exp(3)*log(3)^3)/3 -

25*exp(6)*log(3)^2 + 26*log(3)^2 + 12*log(3)^3 - (25*log(3)^4)/6) + log(x + exp(3) - log(3))*(46*exp(6) - 92*

exp(3)*log(3) + 46*log(3)^2) - x^2*(exp(6)/2 - exp(3)*log(3) + ((exp(3) - log(729)/6)*((7*exp(3))/6 - log(81)/

6 - log(729)/12 + 12))/2 + log(3)^2/2 - 13) - log(x + exp(3) - log(3))*(156*log(3) - 72*exp(3)*log(3) + 12*log

(3)*log(729)) + ((25*x^3*(exp(3) - log(3))^2)/36 - x^4*((19*exp(3))/72 - (19*log(3))/72) - (x^5*log(log(3) - e

xp(3) - x))/2 - x^2*((37*exp(9))/12 - (37*exp(6)*log(3))/4 + (37*exp(3)*log(3)^2)/4 - (37*log(3)^3)/12) + log(

log(3) - exp(3) - x)*((25*exp(15))/6 - (125*exp(12)*log(3))/6 + (125*exp(3)*log(3)^4)/6 - (125*exp(6)*log(3)^3

)/3 + (125*exp(9)*log(3)^2)/3 - (25*log(3)^5)/6) + x^5*log(log(3) - exp(3) - x)^2 + x^5/8 - log(log(3) - exp(3

) - x)^2*(exp(3) - log(3))*(exp(12) - 4*exp(9)*log(3) - 4*exp(3)*log(3)^3 + 6*exp(6)*log(3)^2 + log(3)^4) + x*

log(log(3) - exp(3) - x)*((37*exp(12))/6 - (74*exp(9)*log(3))/3 - (74*exp(3)*log(3)^3)/3 + 37*exp(6)*log(3)^2

+ (37*log(3)^4)/6) + x^2*log(log(3) - exp(3) - x)*(exp(9) - 3*exp(6)*log(3) + 3*exp(3)*log(3)^2 - log(3)^3) -

x*log(log(3) - exp(3) - x)^2*(exp(12) - 4*exp(9)*log(3) - 4*exp(3)*log(3)^3 + 6*exp(6)*log(3)^2 + log(3)^4) -

(x*(25*exp(15) - 125*exp(12)*log(3) + 125*exp(3)*log(3)^4 - 250*exp(6)*log(3)^3 + 250*exp(9)*log(3)^2 - 25*log

(3)^5))/(6*(exp(3) - log(3))) + x^4*log(log(3) - exp(3) - x)*(exp(3)/6 - log(3)/6) - (x^3*log(log(3) - exp(3)

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

+ exp(3) - log(3))*(156*exp(3) - 156*log(3)) - x*(46*exp(3) - 46*log(3)) - log(log(3) - exp(3) - x)*(x^3*((2*e

xp(3))/3 - (2*log(3))/3 + 12) - x^4/2 + x*(2*exp(9) - 6*exp(6)*log(3) + 6*exp(3)*log(3)^2 - 2*log(3)^3) - x^2*

(exp(6) - 2*exp(3)*log(3) + log(3)^2 - 26)) + 23*x^2 - 4*x^3 - x^4/8 + log(x + exp(3) - log(3))*(12*exp(9) - 3

6*exp(6)*log(3) + 36*exp(3)*log(3)^2 - 12*log(3)^3) + log(x + exp(3) - log(3))*(156*exp(3) - 72*exp(6) + 12*ex

p(3)*log(729)) + x^3*((7*exp(3))/18 - log(81)/18 - log(729)/36 + 4) - x*(exp(3) - log(3))*(12*exp(3) - 12*log(

3)) + log(log(3) - exp(3) - x)^2*(exp(3) - log(3))*(exp(9) - 3*exp(6)*log(3) + 3*exp(3)*log(3)^2 - log(3)^3)

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sympy [B] time = 0.23, size = 44, normalized size = 1.57 \begin {gather*} x^{4} \log {\left (- x - e^{3} + \log {\relax (3 )} \right )}^{2} + 36 x^{2} + 156 x + \left (- 12 x^{3} - 26 x^{2}\right ) \log {\left (- x - e^{3} + \log {\relax (3 )} \right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3*ln(3)-4*x**3*exp(3)-4*x**4)*ln(ln(3)-exp(3)-x)**2+((-36*x**2-52*x)*ln(3)+(36*x**2+52*x)*exp

(3)-2*x**4+36*x**3+52*x**2)*ln(ln(3)-exp(3)-x)+(72*x+156)*ln(3)+(-72*x-156)*exp(3)+12*x**3-46*x**2-156*x)/(ln(

3)-exp(3)-x),x)

[Out]

x**4*log(-x - exp(3) + log(3))**2 + 36*x**2 + 156*x + (-12*x**3 - 26*x**2)*log(-x - exp(3) + log(3))

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