\(\int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log (e^{9+6 x^6+x^{12}} (12-4 x))}{-3+x} \, dx\) [2465]

Optimal result

Integrand size = 51, antiderivative size = 19 \[ \int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log \left (e^{9+6 x^6+x^{12}} (12-4 x)\right )}{-3+x} \, dx=x \log \left (4 e^{\left (3+x^6\right )^2} (3-x)\right ) \]

[Out]

ln(4*(-x+3)*exp((x^6+3)^2))*x

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6874, 1634, 2628} \[ \int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log \left (e^{9+6 x^6+x^{12}} (12-4 x)\right )}{-3+x} \, dx=x \log \left (4 e^{\left (x^6+3\right )^2} (3-x)\right ) \]

[In]

Int[(x - 108*x^6 + 36*x^7 - 36*x^12 + 12*x^13 + (-3 + x)*Log[E^(9 + 6*x^6 + x^12)*(12 - 4*x)])/(-3 + x),x]

[Out]

x*Log[4*E^(3 + x^6)^2*(3 - x)]

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2628

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x \left (-1+108 x^5-36 x^6+36 x^{11}-12 x^{12}\right )}{3-x}+\log \left (-4 e^{\left (3+x^6\right )^2} (-3+x)\right )\right ) \, dx \\ & = \int \frac {x \left (-1+108 x^5-36 x^6+36 x^{11}-12 x^{12}\right )}{3-x} \, dx+\int \log \left (-4 e^{\left (3+x^6\right )^2} (-3+x)\right ) \, dx \\ & = x \log \left (4 e^{\left (3+x^6\right )^2} (3-x)\right )-\int \frac {x \left (-1+108 x^5-36 x^6+36 x^{11}-12 x^{12}\right )}{3-x} \, dx+\int \left (1+\frac {3}{-3+x}+36 x^6+12 x^{12}\right ) \, dx \\ & = x+\frac {36 x^7}{7}+\frac {12 x^{13}}{13}+3 \log (3-x)+x \log \left (4 e^{\left (3+x^6\right )^2} (3-x)\right )-\int \left (1+\frac {3}{-3+x}+36 x^6+12 x^{12}\right ) \, dx \\ & = x \log \left (4 e^{\left (3+x^6\right )^2} (3-x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log \left (e^{9+6 x^6+x^{12}} (12-4 x)\right )}{-3+x} \, dx=x \log \left (-4 e^{\left (3+x^6\right )^2} (-3+x)\right ) \]

[In]

Integrate[(x - 108*x^6 + 36*x^7 - 36*x^12 + 12*x^13 + (-3 + x)*Log[E^(9 + 6*x^6 + x^12)*(12 - 4*x)])/(-3 + x),
x]

[Out]

x*Log[-4*E^(3 + x^6)^2*(-3 + x)]

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11

method	result	size
default	\(\ln \left (\left (-4 x +12\right ) {\mathrm e}^{x^{12}+6 x^{6}+9}\right ) x\)	\(21\)
parts	\(\ln \left (\left (-4 x +12\right ) {\mathrm e}^{x^{12}+6 x^{6}+9}\right ) x\)	\(21\)
parallelrisch	\(-6 x^{12}-36 x^{6}+\ln \left (\left (-4 x +12\right ) {\mathrm e}^{x^{12}+6 x^{6}+9}\right ) x -6 \ln \left (-3+x \right )+6 \ln \left (\left (-4 x +12\right ) {\mathrm e}^{x^{12}+6 x^{6}+9}\right )\)	\(58\)
risch	\(x \ln \left ({\mathrm e}^{\left (x^{6}+3\right )^{2}}\right )+\ln \left (-3+x \right ) x +\frac {i x \pi \,\operatorname {csgn}\left (i {\mathrm e}^{\left (x^{6}+3\right )^{2}}\right ) \operatorname {csgn}\left (i \left (-3+x \right ) {\mathrm e}^{\left (x^{6}+3\right )^{2}}\right )^{2}}{2}-\frac {i x \pi \,\operatorname {csgn}\left (i {\mathrm e}^{\left (x^{6}+3\right )^{2}}\right ) \operatorname {csgn}\left (i \left (-3+x \right ) {\mathrm e}^{\left (x^{6}+3\right )^{2}}\right ) \operatorname {csgn}\left (i \left (-3+x \right )\right )}{2}+\frac {i x \pi \operatorname {csgn}\left (i \left (-3+x \right ) {\mathrm e}^{\left (x^{6}+3\right )^{2}}\right )^{3}}{2}-i x \pi \operatorname {csgn}\left (i \left (-3+x \right ) {\mathrm e}^{\left (x^{6}+3\right )^{2}}\right )^{2}+\frac {i x \pi \operatorname {csgn}\left (i \left (-3+x \right ) {\mathrm e}^{\left (x^{6}+3\right )^{2}}\right )^{2} \operatorname {csgn}\left (i \left (-3+x \right )\right )}{2}+i \pi x +2 x \ln \left (2\right )\)	\(175\)

[In]

int(((-3+x)*ln((-4*x+12)*exp(x^12+6*x^6+9))+12*x^13-36*x^12+36*x^7-108*x^6+x)/(-3+x),x,method=_RETURNVERBOSE)

[Out]

ln((-4*x+12)*exp(x^12+6*x^6+9))*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log \left (e^{9+6 x^6+x^{12}} (12-4 x)\right )}{-3+x} \, dx=x \log \left (-4 \, {\left (x - 3\right )} e^{\left (x^{12} + 6 \, x^{6} + 9\right )}\right ) \]

[In]

integrate(((-3+x)*log((-4*x+12)*exp(x^12+6*x^6+9))+12*x^13-36*x^12+36*x^7-108*x^6+x)/(-3+x),x, algorithm="fric
as")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log \left (e^{9+6 x^6+x^{12}} (12-4 x)\right )}{-3+x} \, dx=\frac {3 x^{12}}{2} + 9 x^{6} + \left (x - \frac {3}{2}\right ) \log {\left (\left (12 - 4 x\right ) e^{x^{12} + 6 x^{6} + 9} \right )} + \frac {3 \log {\left (x - 3 \right )}}{2} \]

[In]

integrate(((-3+x)*ln((-4*x+12)*exp(x**12+6*x**6+9))+12*x**13-36*x**12+36*x**7-108*x**6+x)/(-3+x),x)

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 4.42 \[ \int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log \left (e^{9+6 x^6+x^{12}} (12-4 x)\right )}{-3+x} \, dx={\left (x + 3 \, \log \left (x - 3\right )\right )} \log \left (-4 \, x e^{\left (x^{12} + 6 \, x^{6} + 9\right )} + 12 \, e^{\left (x^{12} + 6 \, x^{6} + 9\right )}\right ) + 3 \, {\left (-i \, \pi - 2 \, \log \left (2\right ) - 535824\right )} \log \left (x - 3\right ) - 3 \, {\left (x^{12} + 6 \, x^{6} - 535814\right )} \log \left (x - 3\right ) - 3 \, \log \left (x - 3\right )^{2} + 3 \, \log \left (x - 3\right ) \]

[In]

integrate(((-3+x)*log((-4*x+12)*exp(x^12+6*x^6+9))+12*x^13-36*x^12+36*x^7-108*x^6+x)/(-3+x),x, algorithm="maxi
ma")

[Out]

(x + 3*log(x - 3))*log(-4*x*e^(x^12 + 6*x^6 + 9) + 12*e^(x^12 + 6*x^6 + 9)) + 3*(-I*pi - 2*log(2) - 535824)*lo
g(x - 3) - 3*(x^12 + 6*x^6 - 535814)*log(x - 3) - 3*log(x - 3)^2 + 3*log(x - 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log \left (e^{9+6 x^6+x^{12}} (12-4 x)\right )}{-3+x} \, dx=x^{13} + 6 \, x^{7} + x \log \left (-4 \, x + 12\right ) + 9 \, x \]

[In]

integrate(((-3+x)*log((-4*x+12)*exp(x^12+6*x^6+9))+12*x^13-36*x^12+36*x^7-108*x^6+x)/(-3+x),x, algorithm="giac
")

[Out]

x^13 + 6*x^7 + x*log(-4*x + 12) + 9*x

Mupad [B] (verification not implemented)

Time = 9.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {x-108 x^6+36 x^7-36 x^{12}+12 x^{13}+(-3+x) \log \left (e^{9+6 x^6+x^{12}} (12-4 x)\right )}{-3+x} \, dx=x\,\left (\ln \left (12-4\,x\right )+6\,x^6+x^{12}+9\right ) \]

[In]

int((x + log(-exp(6*x^6 + x^12 + 9)*(4*x - 12))*(x - 3) - 108*x^6 + 36*x^7 - 36*x^12 + 12*x^13)/(x - 3),x)

[Out]

x*(log(12 - 4*x) + 6*x^6 + x^12 + 9)