Integrand size = 95, antiderivative size = 23 \begin {dmath*} \int \frac {e^{2 x} \left (18+4 x-4 x^2+(3-2 x) \log (3)+2 \log (5)\right )}{81-36 x^2+4 x^4+\left (18-18 x-4 x^2+4 x^3\right ) \log (3)+\left (1-2 x+x^2\right ) \log ^2(3)+\left (18-4 x^2+(2-2 x) \log (3)\right ) \log (5)+\log ^2(5)} \, dx=\frac {e^{2 x}}{9+\log (3)-x (2 x+\log (3))+\log (5)} \end {dmath*}
Time = 0.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \begin {dmath*} \int \frac {e^{2 x} \left (18+4 x-4 x^2+(3-2 x) \log (3)+2 \log (5)\right )}{81-36 x^2+4 x^4+\left (18-18 x-4 x^2+4 x^3\right ) \log (3)+\left (1-2 x+x^2\right ) \log ^2(3)+\left (18-4 x^2+(2-2 x) \log (3)\right ) \log (5)+\log ^2(5)} \, dx=\frac {e^{2 x}}{9-2 x^2-x \log (3)+\log (15)} \end {dmath*}
Integrate[(E^(2*x)*(18 + 4*x - 4*x^2 + (3 - 2*x)*Log[3] + 2*Log[5]))/(81 - 36*x^2 + 4*x^4 + (18 - 18*x - 4*x^2 + 4*x^3)*Log[3] + (1 - 2*x + x^2)*Log [3]^2 + (18 - 4*x^2 + (2 - 2*x)*Log[3])*Log[5] + Log[5]^2),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.75 (sec) , antiderivative size = 730, normalized size of antiderivative = 31.74, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2463, 2009}
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.
| \(\displaystyle \int \frac {e^{2 x} \left (-4 x^2+4 x+(3-2 x) \log (3)+18+2 \log (5)\right )}{4 x^4-36 x^2+\left (x^2-2 x+1\right ) \log ^2(3)+\log (5) \left (-4 x^2+(2-2 x) \log (3)+18\right )+\left (4 x^3-4 x^2-18 x+18\right ) \log (3)+81+\log ^2(5)} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {16 e^{2 x} \left (-4 x^2+4 x+(3-2 x) \log (3)+18+2 \log (5)\right )}{\left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )^{3/2} \left (-4 x+\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}-\log (3)\right )}+\frac {16 e^{2 x} \left (-4 x^2+4 x+(3-2 x) \log (3)+18+2 \log (5)\right )}{\left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )^{3/2} \left (4 x+\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}+\log (3)\right )}+\frac {16 e^{2 x} \left (-4 x^2+4 x+(3-2 x) \log (3)+18+2 \log (5)\right )}{\left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right ) \left (-4 x+\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}-\log (3)\right )^2}+\frac {16 e^{2 x} \left (-4 x^2+4 x+(3-2 x) \log (3)+18+2 \log (5)\right )}{\left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right ) \left (4 x+\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}+\log (3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 e^{\frac {1}{2} \sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}} \left (2-\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (4 x-\sqrt {72+8 \log (3)+\log ^2(3)+8 \log (5)}+\log (3)\right )\right )}{\sqrt {3} \left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )}+\frac {2 e^{\frac {1}{2} \sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (4 x-\sqrt {72+8 \log (3)+\log ^2(3)+8 \log (5)}+\log (3)\right )\right )}{\sqrt {3 \left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )}}-\frac {4 e^{\frac {1}{2} \sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (4 x-\sqrt {72+8 \log (3)+\log ^2(3)+8 \log (5)}+\log (3)\right )\right )}{\sqrt {3} \left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )}+\frac {2 e^{-\frac {1}{2} \sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}} \left (2+\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (4 x+\sqrt {72+8 \log (3)+\log ^2(3)+8 \log (5)}+\log (3)\right )\right )}{\sqrt {3} \left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )}-\frac {2 e^{-\frac {1}{2} \sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (4 x+\sqrt {72+8 \log (3)+\log ^2(3)+8 \log (5)}+\log (3)\right )\right )}{\sqrt {3 \left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )}}-\frac {4 e^{-\frac {1}{2} \sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (4 x+\sqrt {72+8 \log (3)+\log ^2(3)+8 \log (5)}+\log (3)\right )\right )}{\sqrt {3} \left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )}-\frac {4 e^{2 x}}{\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)} \left (4 x-\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}+\log (3)\right )}+\frac {4 e^{2 x}}{\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)} \left (4 x+\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}+\log (3)\right )}+\frac {2 e^{2 x} \left (4+\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}-\log (3)\right )}{\left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )^{3/2}}-\frac {2 e^{2 x} \left (4-\sqrt {72+\log ^2(3)+8 \log (3)+8 \log (5)}-\log (3)\right )}{\left (72+\log ^2(3)+8 \log (3)+8 \log (5)\right )^{3/2}}-\frac {4 e^{2 x}}{72+\log ^2(3)+8 \log (3)+8 \log (5)}\) |
Int[(E^(2*x)*(18 + 4*x - 4*x^2 + (3 - 2*x)*Log[3] + 2*Log[5]))/(81 - 36*x^ 2 + 4*x^4 + (18 - 18*x - 4*x^2 + 4*x^3)*Log[3] + (1 - 2*x + x^2)*Log[3]^2 + (18 - 4*x^2 + (2 - 2*x)*Log[3])*Log[5] + Log[5]^2),x]
(-4*E^(2*x))/(72 + 8*Log[3] + Log[3]^2 + 8*Log[5]) - (4*E^(Sqrt[72 + 8*Log [3] + Log[3]^2 + 8*Log[5]]/2)*ExpIntegralEi[(4*x + Log[3] - Sqrt[72 + 8*Lo g[3] + Log[3]^2 + 8*Log[5]])/2])/(Sqrt[3]*(72 + 8*Log[3] + Log[3]^2 + 8*Lo g[5])) - (4*ExpIntegralEi[(4*x + Log[3] + Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]])/2])/(Sqrt[3]*E^(Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]]/2)*(7 2 + 8*Log[3] + Log[3]^2 + 8*Log[5])) + (2*E^(Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]]/2)*ExpIntegralEi[(4*x + Log[3] - Sqrt[72 + 8*Log[3] + Log[3]^ 2 + 8*Log[5]])/2])/Sqrt[3*(72 + 8*Log[3] + Log[3]^2 + 8*Log[5])] - (2*ExpI ntegralEi[(4*x + Log[3] + Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]])/2])/( E^(Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]]/2)*Sqrt[3*(72 + 8*Log[3] + Lo g[3]^2 + 8*Log[5])]) + (2*E^(Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]]/2)* ExpIntegralEi[(4*x + Log[3] - Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]])/2 ]*(2 - Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]]))/(Sqrt[3]*(72 + 8*Log[3] + Log[3]^2 + 8*Log[5])) - (2*E^(2*x)*(4 - Log[3] - Sqrt[72 + 8*Log[3] + L og[3]^2 + 8*Log[5]]))/(72 + 8*Log[3] + Log[3]^2 + 8*Log[5])^(3/2) - (4*E^( 2*x))/(Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]]*(4*x + Log[3] - Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]])) + (2*ExpIntegralEi[(4*x + Log[3] + Sqrt [72 + 8*Log[3] + Log[3]^2 + 8*Log[5]])/2]*(2 + Sqrt[72 + 8*Log[3] + Log[3] ^2 + 8*Log[5]]))/(Sqrt[3]*E^(Sqrt[72 + 8*Log[3] + Log[3]^2 + 8*Log[5]]/2)* (72 + 8*Log[3] + Log[3]^2 + 8*Log[5])) + (2*E^(2*x)*(4 - Log[3] + Sqrt[...
3.13.98.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04
| method | result | size |
| gosper | \(\frac {{\mathrm e}^{2 x}}{-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9}\) | \(24\) |
| norman | \(\frac {{\mathrm e}^{2 x}}{-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9}\) | \(24\) |
| parallelrisch | \(\frac {{\mathrm e}^{2 x}}{-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9}\) | \(24\) |
| default | \(-\frac {36 \,{\mathrm e}^{2 x} \ln \left (3\right )}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}-\frac {{\mathrm e}^{2 x} \ln \left (3\right )^{2}}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}+\frac {2 \,{\mathrm e}^{2 x} \left (-x \ln \left (3\right )^{2}+\ln \left (3\right ) \ln \left (5\right )-4 x \ln \left (5\right )+\ln \left (3\right )^{2}-4 x \ln \left (3\right )+9 \ln \left (3\right )-36 x \right )}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}+\frac {4 \,{\mathrm e}^{2 x} \left (-x \ln \left (3\right )+2 \ln \left (5\right )+2 \ln \left (3\right )+18\right )}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}+\frac {18 \,{\mathrm e}^{2 x} \left (\ln \left (3\right )+4 x \right )}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}+\frac {2 \ln \left (3\right )^{2} {\mathrm e}^{2 x} x}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}-\frac {2 \ln \left (5\right ) {\mathrm e}^{2 x} \ln \left (3\right )}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}+\frac {8 \ln \left (5\right ) {\mathrm e}^{2 x} x}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}+\frac {12 \ln \left (3\right ) {\mathrm e}^{2 x} x}{\left (\ln \left (3\right )^{2}+8 \ln \left (5\right )+8 \ln \left (3\right )+72\right ) \left (-x \ln \left (3\right )-2 x^{2}+\ln \left (5\right )+\ln \left (3\right )+9\right )}\) | \(438\) |
int((2*ln(5)+(3-2*x)*ln(3)-4*x^2+4*x+18)*exp(x)^2/(ln(5)^2+((2-2*x)*ln(3)- 4*x^2+18)*ln(5)+(x^2-2*x+1)*ln(3)^2+(4*x^3-4*x^2-18*x+18)*ln(3)+4*x^4-36*x ^2+81),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \begin {dmath*} \int \frac {e^{2 x} \left (18+4 x-4 x^2+(3-2 x) \log (3)+2 \log (5)\right )}{81-36 x^2+4 x^4+\left (18-18 x-4 x^2+4 x^3\right ) \log (3)+\left (1-2 x+x^2\right ) \log ^2(3)+\left (18-4 x^2+(2-2 x) \log (3)\right ) \log (5)+\log ^2(5)} \, dx=-\frac {e^{\left (2 \, x\right )}}{2 \, x^{2} + {\left (x - 1\right )} \log \left (3\right ) - \log \left (5\right ) - 9} \end {dmath*}
integrate((2*log(5)+(3-2*x)*log(3)-4*x^2+4*x+18)*exp(x)^2/(log(5)^2+((2-2* x)*log(3)-4*x^2+18)*log(5)+(x^2-2*x+1)*log(3)^2+(4*x^3-4*x^2-18*x+18)*log( 3)+4*x^4-36*x^2+81),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \begin {dmath*} \int \frac {e^{2 x} \left (18+4 x-4 x^2+(3-2 x) \log (3)+2 \log (5)\right )}{81-36 x^2+4 x^4+\left (18-18 x-4 x^2+4 x^3\right ) \log (3)+\left (1-2 x+x^2\right ) \log ^2(3)+\left (18-4 x^2+(2-2 x) \log (3)\right ) \log (5)+\log ^2(5)} \, dx=- \frac {e^{2 x}}{2 x^{2} + x \log {\left (3 \right )} - 9 - \log {\left (5 \right )} - \log {\left (3 \right )}} \end {dmath*}
integrate((2*ln(5)+(3-2*x)*ln(3)-4*x**2+4*x+18)*exp(x)**2/(ln(5)**2+((2-2* x)*ln(3)-4*x**2+18)*ln(5)+(x**2-2*x+1)*ln(3)**2+(4*x**3-4*x**2-18*x+18)*ln (3)+4*x**4-36*x**2+81),x)
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \begin {dmath*} \int \frac {e^{2 x} \left (18+4 x-4 x^2+(3-2 x) \log (3)+2 \log (5)\right )}{81-36 x^2+4 x^4+\left (18-18 x-4 x^2+4 x^3\right ) \log (3)+\left (1-2 x+x^2\right ) \log ^2(3)+\left (18-4 x^2+(2-2 x) \log (3)\right ) \log (5)+\log ^2(5)} \, dx=-\frac {e^{\left (2 \, x\right )}}{2 \, x^{2} + x \log \left (3\right ) - \log \left (5\right ) - \log \left (3\right ) - 9} \end {dmath*}
integrate((2*log(5)+(3-2*x)*log(3)-4*x^2+4*x+18)*exp(x)^2/(log(5)^2+((2-2* x)*log(3)-4*x^2+18)*log(5)+(x^2-2*x+1)*log(3)^2+(4*x^3-4*x^2-18*x+18)*log( 3)+4*x^4-36*x^2+81),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \begin {dmath*} \int \frac {e^{2 x} \left (18+4 x-4 x^2+(3-2 x) \log (3)+2 \log (5)\right )}{81-36 x^2+4 x^4+\left (18-18 x-4 x^2+4 x^3\right ) \log (3)+\left (1-2 x+x^2\right ) \log ^2(3)+\left (18-4 x^2+(2-2 x) \log (3)\right ) \log (5)+\log ^2(5)} \, dx=-\frac {e^{\left (2 \, x\right )}}{2 \, x^{2} + x \log \left (3\right ) - \log \left (5\right ) - \log \left (3\right ) - 9} \end {dmath*}
integrate((2*log(5)+(3-2*x)*log(3)-4*x^2+4*x+18)*exp(x)^2/(log(5)^2+((2-2* x)*log(3)-4*x^2+18)*log(5)+(x^2-2*x+1)*log(3)^2+(4*x^3-4*x^2-18*x+18)*log( 3)+4*x^4-36*x^2+81),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {e^{2 x} \left (18+4 x-4 x^2+(3-2 x) \log (3)+2 \log (5)\right )}{81-36 x^2+4 x^4+\left (18-18 x-4 x^2+4 x^3\right ) \log (3)+\left (1-2 x+x^2\right ) \log ^2(3)+\left (18-4 x^2+(2-2 x) \log (3)\right ) \log (5)+\log ^2(5)} \, dx=\int \frac {{\mathrm {e}}^{2\,x}\,\left (4\,x+2\,\ln \left (5\right )-\ln \left (3\right )\,\left (2\,x-3\right )-4\,x^2+18\right )}{{\ln \left (3\right )}^2\,\left (x^2-2\,x+1\right )-\ln \left (5\right )\,\left (\ln \left (3\right )\,\left (2\,x-2\right )+4\,x^2-18\right )-\ln \left (3\right )\,\left (-4\,x^3+4\,x^2+18\,x-18\right )+{\ln \left (5\right )}^2-36\,x^2+4\,x^4+81} \,d x \end {dmath*}
int((exp(2*x)*(4*x + 2*log(5) - log(3)*(2*x - 3) - 4*x^2 + 18))/(log(3)^2* (x^2 - 2*x + 1) - log(5)*(log(3)*(2*x - 2) + 4*x^2 - 18) - log(3)*(18*x + 4*x^2 - 4*x^3 - 18) + log(5)^2 - 36*x^2 + 4*x^4 + 81),x)