Integrand size = 135, antiderivative size = 28 \[ \int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+\left (52 x^2+36 x^3-2 x^4+e^3 \left (52 x+36 x^2\right )+\left (-52 x-36 x^2\right ) \log (3)\right ) \log \left (-e^3-x+\log (3)\right )+\left (-4 e^3 x^3-4 x^4+4 x^3 \log (3)\right ) \log ^2\left (-e^3-x+\log (3)\right )}{-e^3-x+\log (3)} \, dx=\left (7-x-5 (4+x)+x^2 \log \left (-e^3-x+\log (3)\right )\right )^2 \] Output:
(ln(ln(3)-exp(3)-x)*x^2-13-6*x)^2
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+\left (52 x^2+36 x^3-2 x^4+e^3 \left (52 x+36 x^2\right )+\left (-52 x-36 x^2\right ) \log (3)\right ) \log \left (-e^3-x+\log (3)\right )+\left (-4 e^3 x^3-4 x^4+4 x^3 \log (3)\right ) \log ^2\left (-e^3-x+\log (3)\right )}{-e^3-x+\log (3)} \, dx=\left (-13-6 x+x^2 \log \left (-e^3-x+\log (3)\right )\right )^2 \] Input:
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]
Output:
(-13 - 6*x + x^2*Log[-E^3 - x + Log[3]])^2
Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {7292, 27, 7237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {12 x^3-46 x^2+\left (-4 x^4-4 e^3 x^3+4 x^3 \log (3)\right ) \log ^2\left (-x-e^3+\log (3)\right )+\left (-2 x^4+36 x^3+52 x^2+e^3 \left (36 x^2+52 x\right )+\left (-36 x^2-52 x\right ) \log (3)\right ) \log \left (-x-e^3+\log (3)\right )-156 x+e^3 (-72 x-156)+(72 x+156) \log (3)}{-x-e^3+\log (3)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \left (x^2 \left (-\log \left (-x-e^3+\log (3)\right )\right )+6 x+13\right ) \left (-x^2-2 x^2 \log \left (-x-e^3+\log (3)\right )+6 x-2 e^3 x \left (1-\frac {\log (3)}{e^3}\right ) \log \left (-x-e^3+\log (3)\right )+6 e^3 \left (1-\frac {\log (729)}{6 e^3}\right )\right )}{x+e^3-\log (3)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\left (-\log \left (-x+\log (3)-e^3\right ) x^2+6 x+13\right ) \left (-2 \log \left (-x+\log (3)-e^3\right ) x^2-x^2-2 \left (e^3-\log (3)\right ) \log \left (-x+\log (3)-e^3\right ) x+6 x-\log (729)+6 e^3\right )}{x-\log (3)+e^3}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \left (x^2 \left (-\log \left (-x-e^3+\log (3)\right )\right )+6 x+13\right )^2\) |
Input:
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])*L og[-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]
Output:
(13 + 6*x - x^2*Log[-E^3 - x + Log[3]])^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(22)=44\).
Time = 204.87 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79
method | result | size |
risch | \(\ln \left (\ln \left (3\right )-{\mathrm e}^{3}-x \right )^{2} x^{4}+\left (-12 x^{3}-26 x^{2}\right ) \ln \left (\ln \left (3\right )-{\mathrm e}^{3}-x \right )+36 x^{2}+156 x\) | \(50\) |
norman | \(\ln \left (\ln \left (3\right )-{\mathrm e}^{3}-x \right )^{2} x^{4}+156 x +36 x^{2}-12 x^{3} \ln \left (\ln \left (3\right )-{\mathrm e}^{3}-x \right )-26 \ln \left (\ln \left (3\right )-{\mathrm e}^{3}-x \right ) x^{2}\) | \(59\) |
parallelrisch | \(\ln \left (\ln \left (3\right )-{\mathrm e}^{3}-x \right )^{2} x^{4}-12 x^{3} \ln \left (\ln \left (3\right )-{\mathrm e}^{3}-x \right )-26 \ln \left (\ln \left (3\right )-{\mathrm e}^{3}-x \right ) x^{2}-36 \,{\mathrm e}^{6}+72 \,{\mathrm e}^{3} \ln \left (3\right )-36 \ln \left (3\right )^{2}+36 x^{2}-312 \,{\mathrm e}^{3}+312 \ln \left (3\right )+156 x\) | \(85\) |
parts | \(\text {Expression too large to display}\) | \(1580\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1698\) |
default | \(\text {Expression too large to display}\) | \(1698\) |
Input:
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,meth od=_RETURNVERBOSE)
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+\left (52 x^2+36 x^3-2 x^4+e^3 \left (52 x+36 x^2\right )+\left (-52 x-36 x^2\right ) \log (3)\right ) \log \left (-e^3-x+\log (3)\right )+\left (-4 e^3 x^3-4 x^4+4 x^3 \log (3)\right ) \log ^2\left (-e^3-x+\log (3)\right )}{-e^3-x+\log (3)} \, dx=x^{4} \log \left (-x - e^{3} + \log \left (3\right )\right )^{2} + 36 \, x^{2} - 2 \, {\left (6 \, x^{3} + 13 \, x^{2}\right )} \log \left (-x - e^{3} + \log \left (3\right )\right ) + 156 \, x \] Input:
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)-ex p(3)-x),x, algorithm="fricas")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+\left (52 x^2+36 x^3-2 x^4+e^3 \left (52 x+36 x^2\right )+\left (-52 x-36 x^2\right ) \log (3)\right ) \log \left (-e^3-x+\log (3)\right )+\left (-4 e^3 x^3-4 x^4+4 x^3 \log (3)\right ) \log ^2\left (-e^3-x+\log (3)\right )}{-e^3-x+\log (3)} \, dx=x^{4} \log {\left (- x - e^{3} + \log {\left (3 \right )} \right )}^{2} + 36 x^{2} + 156 x + \left (- 12 x^{3} - 26 x^{2}\right ) \log {\left (- x - e^{3} + \log {\left (3 \right )} \right )} \] Input:
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)-e xp(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)
Output:
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))
Leaf count of result is larger than twice the leaf count of optimal. 1942 vs. \(2 (22) = 44\).
Time = 0.14 (sec) , antiderivative size = 1942, normalized size of antiderivative = 69.36 \[ \int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+\left (52 x^2+36 x^3-2 x^4+e^3 \left (52 x+36 x^2\right )+\left (-52 x-36 x^2\right ) \log (3)\right ) \log \left (-e^3-x+\log (3)\right )+\left (-4 e^3 x^3-4 x^4+4 x^3 \log (3)\right ) \log ^2\left (-e^3-x+\log (3)\right )}{-e^3-x+\log (3)} \, dx=\text {Too large to display} \] Input:
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)-ex p(3)-x),x, algorithm="maxima")
Output:
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 - l og(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)*log(-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 + log(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(...
Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (22) = 44\).
Time = 0.14 (sec) , antiderivative size = 746, normalized size of antiderivative = 26.64 \[ \int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+\left (52 x^2+36 x^3-2 x^4+e^3 \left (52 x+36 x^2\right )+\left (-52 x-36 x^2\right ) \log (3)\right ) \log \left (-e^3-x+\log (3)\right )+\left (-4 e^3 x^3-4 x^4+4 x^3 \log (3)\right ) \log ^2\left (-e^3-x+\log (3)\right )}{-e^3-x+\log (3)} \, dx=\text {Too large to display} \] Input:
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)-ex p(3)-x),x, algorithm="giac")
Output:
(x + e^3 - log(3))^4*log(-x - e^3 + log(3))^2 - 4*(x + e^3 - log(3))^3*e^3 *log(-x - e^3 + log(3))^2 + 4*(x + e^3 - log(3))^3*log(3)*log(-x - e^3 + l og(3))^2 - 12*(x + e^3 - log(3))^2*e^3*log(3)*log(-x - e^3 + log(3))^2 + 6 *(x + e^3 - log(3))^2*log(3)^2*log(-x - e^3 + log(3))^2 - 12*(x + e^3 - lo g(3))*e^3*log(3)^2*log(-x - e^3 + log(3))^2 + 4*(x + e^3 - log(3))*log(3)^ 3*log(-x - e^3 + log(3))^2 - 4*e^3*log(3)^3*log(-x - e^3 + log(3))^2 + log (3)^4*log(-x - e^3 + log(3))^2 + 6*(x + e^3 - log(3))^2*e^6*log(-x - e^3 + log(3))^2 + 12*(x + e^3 - log(3))*e^6*log(3)*log(-x - e^3 + log(3))^2 + 6 *e^6*log(3)^2*log(-x - e^3 + log(3))^2 - 12*(x + e^3 - log(3))^3*log(-x - e^3 + log(3)) + 36*(x + e^3 - log(3))^2*e^3*log(-x - e^3 + log(3)) - 36*(x + e^3 - log(3))^2*log(3)*log(-x - e^3 + log(3)) + 72*(x + e^3 - log(3))*e ^3*log(3)*log(-x - e^3 + log(3)) - 36*(x + e^3 - log(3))*log(3)^2*log(-x - e^3 + log(3)) + 36*e^3*log(3)^2*log(-x - e^3 + log(3)) - 12*log(3)^3*log( -x - e^3 + log(3)) - 4*(x + e^3 - log(3))*e^9*log(-x - e^3 + log(3))^2 - 4 *e^9*log(3)*log(-x - e^3 + log(3))^2 - 26*(x + e^3 - log(3))^2*log(-x - e^ 3 + log(3)) - 36*(x + e^3 - log(3))*e^6*log(-x - e^3 + log(3)) + 52*(x + e ^3 - log(3))*e^3*log(-x - e^3 + log(3)) - 52*(x + e^3 - log(3))*log(3)*log (-x - e^3 + log(3)) - 36*e^6*log(3)*log(-x - e^3 + log(3)) + 52*e^3*log(3) *log(-x - e^3 + log(3)) - 26*log(3)^2*log(-x - e^3 + log(3)) + e^12*log(-x - e^3 + log(3))^2 + 36*(x + e^3 - log(3))^2 - 72*(x + e^3 - log(3))*e^...
Time = 21.38 (sec) , antiderivative size = 1018, normalized size of antiderivative = 36.36 \[ \int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+\left (52 x^2+36 x^3-2 x^4+e^3 \left (52 x+36 x^2\right )+\left (-52 x-36 x^2\right ) \log (3)\right ) \log \left (-e^3-x+\log (3)\right )+\left (-4 e^3 x^3-4 x^4+4 x^3 \log (3)\right ) \log ^2\left (-e^3-x+\log (3)\right )}{-e^3-x+\log (3)} \, dx=\text {Too large to display} \] Input:
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)
Output:
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) + (e xp(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 + e xp(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 - lo g(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) - exp(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*ex p(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)*l og(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) - ...
Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {e^3 (-156-72 x)-156 x-46 x^2+12 x^3+(156+72 x) \log (3)+\left (52 x^2+36 x^3-2 x^4+e^3 \left (52 x+36 x^2\right )+\left (-52 x-36 x^2\right ) \log (3)\right ) \log \left (-e^3-x+\log (3)\right )+\left (-4 e^3 x^3-4 x^4+4 x^3 \log (3)\right ) \log ^2\left (-e^3-x+\log (3)\right )}{-e^3-x+\log (3)} \, dx=x \left (\mathrm {log}\left (\mathrm {log}\left (3\right )-e^{3}-x \right )^{2} x^{3}-12 \,\mathrm {log}\left (\mathrm {log}\left (3\right )-e^{3}-x \right ) x^{2}-26 \,\mathrm {log}\left (\mathrm {log}\left (3\right )-e^{3}-x \right ) x +36 x +156\right ) \] Input:
int(((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)
Output:
x*(log(log(3) - e**3 - x)**2*x**3 - 12*log(log(3) - e**3 - x)*x**2 - 26*lo g(log(3) - e**3 - x)*x + 36*x + 156)