Integrand size = 95, antiderivative size = 27 \[ \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{81 x^3+72 x^4+16 x^5} \, dx=\left (3+\frac {e^{\frac {4-\log (\log (4))}{3+\frac {4 x}{3}}}}{x}\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(27)=54\).
Time = 2.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26 \[ \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{81 x^3+72 x^4+16 x^5} \, dx=-2 \left (-\frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{2 x^2}-\frac {3 e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x}\right ) \]
Integrate[(E^((2*(12 - 3*Log[Log[4]]))/(9 + 4*x))*(-162 - 240*x - 32*x^2 + 24*x*Log[Log[4]]) + E^((12 - 3*Log[Log[4]])/(9 + 4*x))*(-486*x - 720*x^2 - 96*x^3 + 72*x^2*Log[Log[4]]))/(81*x^3 + 72*x^4 + 16*x^5),x]
-2*(-1/2*E^(24/(9 + 4*x))/(x^2*Log[4]^(6/(9 + 4*x))) - (3*E^(12/(9 + 4*x)) )/(x*Log[4]^(3/(9 + 4*x))))
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.
\(\displaystyle \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{4 x+9}} \left (-32 x^2-240 x+24 x \log (\log (4))-162\right )+e^{\frac {12-3 \log (\log (4))}{4 x+9}} \left (-96 x^3-720 x^2+72 x^2 \log (\log (4))-486 x\right )}{16 x^5+72 x^4+81 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{4 x+9}} \left (-32 x^2-240 x+24 x \log (\log (4))-162\right )+e^{\frac {12-3 \log (\log (4))}{4 x+9}} \left (-96 x^3-720 x^2+72 x^2 \log (\log (4))-486 x\right )}{x^3 \left (16 x^2+72 x+81\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{4 x+9}} \left (-32 x^2-240 x+24 x \log (\log (4))-162\right )+e^{\frac {12-3 \log (\log (4))}{4 x+9}} \left (-96 x^3-720 x^2+72 x^2 \log (\log (4))-486 x\right )}{x^3 (4 x+9)^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 e^{\frac {12}{4 x+9}} \log ^{-\frac {6}{4 x+9}}(4) \left (e^{\frac {12}{4 x+9}}+3 x \log ^{\frac {3}{4 x+9}}(4)\right ) \left (-16 x^2-12 x (10-\log (\log (4)))-81\right )}{x^3 (4 x+9)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {e^{\frac {12}{4 x+9}} \log ^{-\frac {6}{4 x+9}}(4) \left (3 \log ^{\frac {3}{4 x+9}}(4) x+e^{\frac {12}{4 x+9}}\right ) \left (16 x^2+12 (10-\log (\log (4))) x+81\right )}{x^3 (4 x+9)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {e^{\frac {12}{4 x+9}} \log ^{-\frac {6}{4 x+9}}(4) \left (3 \log ^{\frac {3}{4 x+9}}(4) x+e^{\frac {12}{4 x+9}}\right ) \left (16 x^2+12 (10-\log (\log (4))) x+81\right )}{x^3 (4 x+9)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {e^{\frac {24}{4 x+9}} \log ^{-\frac {6}{4 x+9}}(4) \left (16 x^2+12 (10-\log (\log (4))) x+81\right )}{x^3 (4 x+9)^2}+\frac {3 e^{\frac {12}{4 x+9}} \log ^{-\frac {3}{4 x+9}}(4) \left (16 x^2+12 (10-\log (\log (4))) x+81\right )}{x^2 (4 x+9)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\int \frac {e^{\frac {24}{4 x+9}} \log ^{-\frac {6}{4 x+9}}(4)}{x^3}dx+\frac {4}{27} (4-\log (\log (4))) \int \frac {e^{\frac {24}{4 x+9}} \log ^{-\frac {6}{4 x+9}}(4)}{x^2}dx+3 \int \frac {e^{\frac {12}{4 x+9}} \log ^{-\frac {3}{4 x+9}}(4)}{x^2}dx-\frac {32}{243} (4-\log (\log (4))) \int \frac {e^{\frac {24}{4 x+9}} \log ^{-\frac {6}{4 x+9}}(4)}{x}dx+\frac {4}{9} (4-\log (\log (4))) \int \frac {e^{\frac {12}{4 x+9}} \log ^{-\frac {3}{4 x+9}}(4)}{x}dx+\frac {4}{9} (4-\log (\log (4))) \operatorname {ExpIntegralEi}\left (\frac {3 (4-\log (\log (4)))}{4 x+9}\right )-\frac {32}{243} (4-\log (\log (4))) \operatorname {ExpIntegralEi}\left (\frac {6 (4-\log (\log (4)))}{4 x+9}\right )-\frac {8}{81} e^{\frac {24}{4 x+9}} \log ^{-\frac {6}{4 x+9}}(4)+\frac {4}{3} e^{\frac {12}{4 x+9}} \log ^{-\frac {3}{4 x+9}}(4)\right )\) |
Int[(E^((2*(12 - 3*Log[Log[4]]))/(9 + 4*x))*(-162 - 240*x - 32*x^2 + 24*x* Log[Log[4]]) + E^((12 - 3*Log[Log[4]])/(9 + 4*x))*(-486*x - 720*x^2 - 96*x ^3 + 72*x^2*Log[Log[4]]))/(81*x^3 + 72*x^4 + 16*x^5),x]
3.18.53.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 0.59 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67
method | result | size |
risch | \(\frac {{\mathrm e}^{-\frac {6 \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )-4\right )}{4 x +9}}}{x^{2}}+\frac {6 \,{\mathrm e}^{-\frac {3 \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )-4\right )}{4 x +9}}}{x}\) | \(45\) |
parallelrisch | \(\frac {39366 \,{\mathrm e}^{-\frac {3 \left (\ln \left (2 \ln \left (2\right )\right )-4\right )}{4 x +9}} x +6561 \,{\mathrm e}^{-\frac {6 \left (\ln \left (2 \ln \left (2\right )\right )-4\right )}{4 x +9}}}{6561 x^{2}}\) | \(48\) |
norman | \(\frac {9 \,{\mathrm e}^{\frac {-6 \ln \left (2 \ln \left (2\right )\right )+24}{4 x +9}}+54 x \,{\mathrm e}^{\frac {-3 \ln \left (2 \ln \left (2\right )\right )+12}{4 x +9}}+4 x \,{\mathrm e}^{\frac {-6 \ln \left (2 \ln \left (2\right )\right )+24}{4 x +9}}+24 x^{2} {\mathrm e}^{\frac {-3 \ln \left (2 \ln \left (2\right )\right )+12}{4 x +9}}}{x^{2} \left (4 x +9\right )}\) | \(102\) |
parts | \(\text {Expression too large to display}\) | \(6940\) |
derivativedivides | \(\text {Expression too large to display}\) | \(11858\) |
default | \(\text {Expression too large to display}\) | \(11858\) |
int(((24*x*ln(2*ln(2))-32*x^2-240*x-162)*exp((-3*ln(2*ln(2))+12)/(4*x+9))^ 2+(72*x^2*ln(2*ln(2))-96*x^3-720*x^2-486*x)*exp((-3*ln(2*ln(2))+12)/(4*x+9 )))/(16*x^5+72*x^4+81*x^3),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{81 x^3+72 x^4+16 x^5} \, dx=\frac {6 \, x e^{\left (-\frac {3 \, {\left (\log \left (2 \, \log \left (2\right )\right ) - 4\right )}}{4 \, x + 9}\right )} + e^{\left (-\frac {6 \, {\left (\log \left (2 \, \log \left (2\right )\right ) - 4\right )}}{4 \, x + 9}\right )}}{x^{2}} \]
integrate(((24*x*log(2*log(2))-32*x^2-240*x-162)*exp((-3*log(2*log(2))+12) /(4*x+9))^2+(72*x^2*log(2*log(2))-96*x^3-720*x^2-486*x)*exp((-3*log(2*log( 2))+12)/(4*x+9)))/(16*x^5+72*x^4+81*x^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{81 x^3+72 x^4+16 x^5} \, dx=\frac {6 x^{2} e^{\frac {12 - 3 \log {\left (2 \log {\left (2 \right )} \right )}}{4 x + 9}} + x e^{\frac {2 \cdot \left (12 - 3 \log {\left (2 \log {\left (2 \right )} \right )}\right )}{4 x + 9}}}{x^{3}} \]
integrate(((24*x*ln(2*ln(2))-32*x**2-240*x-162)*exp((-3*ln(2*ln(2))+12)/(4 *x+9))**2+(72*x**2*ln(2*ln(2))-96*x**3-720*x**2-486*x)*exp((-3*ln(2*ln(2)) +12)/(4*x+9)))/(16*x**5+72*x**4+81*x**3),x)
(6*x**2*exp((12 - 3*log(2*log(2)))/(4*x + 9)) + x*exp(2*(12 - 3*log(2*log( 2)))/(4*x + 9)))/x**3
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (25) = 50\).
Time = 0.35 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.89 \[ \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{81 x^3+72 x^4+16 x^5} \, dx=\frac {6 \, x e^{\left (\frac {3 \, \log \left (2\right )}{4 \, x + 9} - \frac {3 \, \log \left (\log \left (2\right )\right )}{4 \, x + 9} + \frac {12}{4 \, x + 9}\right )} + e^{\left (-\frac {6 \, \log \left (\log \left (2\right )\right )}{4 \, x + 9} + \frac {24}{4 \, x + 9}\right )}}{2^{\frac {6}{4 \, x + 9}} x^{2}} \]
integrate(((24*x*log(2*log(2))-32*x^2-240*x-162)*exp((-3*log(2*log(2))+12) /(4*x+9))^2+(72*x^2*log(2*log(2))-96*x^3-720*x^2-486*x)*exp((-3*log(2*log( 2))+12)/(4*x+9)))/(16*x^5+72*x^4+81*x^3),x, algorithm=\
(6*x*e^(3*log(2)/(4*x + 9) - 3*log(log(2))/(4*x + 9) + 12/(4*x + 9)) + e^( -6*log(log(2))/(4*x + 9) + 24/(4*x + 9)))/(2^(6/(4*x + 9))*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (25) = 50\).
Time = 1.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.74 \[ \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{81 x^3+72 x^4+16 x^5} \, dx=\frac {6 \, x e^{\left (\frac {4 \, {\left (x \log \left (2\right ) + x \log \left (\log \left (2\right )\right ) - 4 \, x\right )}}{3 \, {\left (4 \, x + 9\right )}} - \frac {1}{3} \, \log \left (2\right ) + \frac {4}{3}\right )} \log \left (2\right )^{\frac {1}{3}} + e^{\left (\frac {8 \, {\left (x \log \left (2\right ) + x \log \left (\log \left (2\right )\right ) - 4 \, x\right )}}{3 \, {\left (4 \, x + 9\right )}} - \frac {2}{3} \, \log \left (2\right ) + \frac {8}{3}\right )}}{x^{2} \log \left (2\right )^{\frac {2}{3}}} \]
integrate(((24*x*log(2*log(2))-32*x^2-240*x-162)*exp((-3*log(2*log(2))+12) /(4*x+9))^2+(72*x^2*log(2*log(2))-96*x^3-720*x^2-486*x)*exp((-3*log(2*log( 2))+12)/(4*x+9)))/(16*x^5+72*x^4+81*x^3),x, algorithm=\
(6*x*e^(4/3*(x*log(2) + x*log(log(2)) - 4*x)/(4*x + 9) - 1/3*log(2) + 4/3) *log(2)^(1/3) + e^(8/3*(x*log(2) + x*log(log(2)) - 4*x)/(4*x + 9) - 2/3*lo g(2) + 8/3))/(x^2*log(2)^(2/3))
Time = 0.84 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.48 \[ \int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{81 x^3+72 x^4+16 x^5} \, dx=\frac {{\mathrm {e}}^{\frac {12}{4\,x+9}}\,\left ({\mathrm {e}}^{\frac {12}{4\,x+9}}+3\,2^{\frac {3}{4\,x+9}+1}\,x\,{\ln \left (2\right )}^{\frac {3}{4\,x+9}}\right )\,{\left (\frac {1}{64\,{\ln \left (2\right )}^6}\right )}^{\frac {1}{4\,x+9}}}{x^2} \]
int(-(exp(-(2*(3*log(2*log(2)) - 12))/(4*x + 9))*(240*x - 24*x*log(2*log(2 )) + 32*x^2 + 162) + exp(-(3*log(2*log(2)) - 12)/(4*x + 9))*(486*x - 72*x^ 2*log(2*log(2)) + 720*x^2 + 96*x^3))/(81*x^3 + 72*x^4 + 16*x^5),x)