Integrand size = 91, antiderivative size = 24 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=x+\frac {e^x \log (3)}{x \left (x+\frac {48}{5} \left (3+x^3\right )\right )} \]
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=x+\frac {5 e^x \log (3)}{x \left (144+5 x+48 x^3\right )} \]
Integrate[(20736*x^2 + 1440*x^3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8 + E^x*(-720 + 670*x + 25*x^2 - 960*x^3 + 240*x^4)*Log[3])/(20736*x^2 + 144 0*x^3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8),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.
\(\displaystyle \int \frac {2304 x^8+480 x^6+13824 x^5+25 x^4+1440 x^3+20736 x^2+e^x \left (240 x^4-960 x^3+25 x^2+670 x-720\right ) \log (3)}{2304 x^8+480 x^6+13824 x^5+25 x^4+1440 x^3+20736 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2304 x^8+480 x^6+13824 x^5+25 x^4+1440 x^3+20736 x^2+e^x \left (240 x^4-960 x^3+25 x^2+670 x-720\right ) \log (3)}{x^2 \left (2304 x^6+480 x^4+13824 x^3+25 x^2+1440 x+20736\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {2304 x^8+480 x^6+13824 x^5+25 x^4+1440 x^3+20736 x^2+e^x \left (240 x^4-960 x^3+25 x^2+670 x-720\right ) \log (3)}{x^2 \left (48 x^3+5 x+144\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {13824 x^3}{\left (48 x^3+5 x+144\right )^2}+\frac {1440 x}{\left (48 x^3+5 x+144\right )^2}+\frac {20736}{\left (48 x^3+5 x+144\right )^2}+\frac {2304 x^6}{\left (48 x^3+5 x+144\right )^2}+\frac {480 x^4}{\left (48 x^3+5 x+144\right )^2}+\frac {25 x^2}{\left (48 x^3+5 x+144\right )^2}+\frac {5 e^x \left (48 x^4-192 x^3+5 x^2+134 x-144\right ) \log (3)}{\left (48 x^3+5 x+144\right )^2 x^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {13824 x^3}{\left (48 x^3+5 x+144\right )^2}+\frac {1440 x}{\left (48 x^3+5 x+144\right )^2}+\frac {20736}{\left (48 x^3+5 x+144\right )^2}+\frac {2304 x^6}{\left (48 x^3+5 x+144\right )^2}+\frac {480 x^4}{\left (48 x^3+5 x+144\right )^2}+\frac {25 x^2}{\left (48 x^3+5 x+144\right )^2}+\frac {5 e^x \left (48 x^4-192 x^3+5 x^2+134 x-144\right ) \log (3)}{\left (48 x^3+5 x+144\right )^2 x^2}\right )dx\) |
Int[(20736*x^2 + 1440*x^3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8 + E^x* (-720 + 670*x + 25*x^2 - 960*x^3 + 240*x^4)*Log[3])/(20736*x^2 + 1440*x^3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8),x]
3.25.8.3.1 Defintions of rubi rules used
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])
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 = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x +\frac {5 \ln \left (3\right ) {\mathrm e}^{x}}{x \left (48 x^{3}+5 x +144\right )}\) | \(24\) |
parallelrisch | \(\frac {2304 x^{5}+240 x^{3}+240 \ln \left (3\right ) {\mathrm e}^{x}+6912 x^{2}}{48 x \left (48 x^{3}+5 x +144\right )}\) | \(40\) |
norman | \(\frac {-\frac {20736 x}{5}-\frac {6912 x^{4}}{5}+5 x^{3}+48 x^{5}+5 \ln \left (3\right ) {\mathrm e}^{x}}{x \left (48 x^{3}+5 x +144\right )}\) | \(42\) |
parts | \(x -\frac {25 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (360 \textit {\_R1}^{2}-15912 \textit {\_R1} +31129\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{6718589}+\frac {5 \ln \left (3\right ) {\mathrm e}^{x}}{x \left (48 x^{3}+5 x +144\right )}+\frac {5 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (9352563888 \textit {\_R1}^{2}-30971987712 \textit {\_R1} +1536585725\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{69658330752}+\frac {335 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (322665072 \textit {\_R1}^{2}-7637760 \textit {\_R1} +371027137\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{69658330752}-\frac {480 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (31104 \textit {\_R1}^{2}-31079 \textit {\_R1} +2110\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{6718589}+\frac {120 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (25 \textit {\_R1}^{2}-1105 \textit {\_R1} -91152\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{6718589}\) | \(259\) |
default | \(\text {Expression too large to display}\) | \(722\) |
int(((240*x^4-960*x^3+25*x^2+670*x-720)*ln(3)*exp(x)+2304*x^8+480*x^6+1382 4*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4+1440*x ^3+20736*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=\frac {48 \, x^{5} + 5 \, x^{3} + 144 \, x^{2} + 5 \, e^{x} \log \left (3\right )}{48 \, x^{4} + 5 \, x^{2} + 144 \, x} \]
integrate(((240*x^4-960*x^3+25*x^2+670*x-720)*log(3)*exp(x)+2304*x^8+480*x ^6+13824*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4 +1440*x^3+20736*x^2),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=x + \frac {5 e^{x} \log {\left (3 \right )}}{48 x^{4} + 5 x^{2} + 144 x} \]
integrate(((240*x**4-960*x**3+25*x**2+670*x-720)*ln(3)*exp(x)+2304*x**8+48 0*x**6+13824*x**5+25*x**4+1440*x**3+20736*x**2)/(2304*x**8+480*x**6+13824* x**5+25*x**4+1440*x**3+20736*x**2),x)
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=\frac {48 \, x^{5} + 5 \, x^{3} + 144 \, x^{2} + 5 \, e^{x} \log \left (3\right )}{48 \, x^{4} + 5 \, x^{2} + 144 \, x} \]
integrate(((240*x^4-960*x^3+25*x^2+670*x-720)*log(3)*exp(x)+2304*x^8+480*x ^6+13824*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4 +1440*x^3+20736*x^2),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=\frac {48 \, x^{5} + 5 \, x^{3} + 144 \, x^{2} + 5 \, e^{x} \log \left (3\right )}{48 \, x^{4} + 5 \, x^{2} + 144 \, x} \]
integrate(((240*x^4-960*x^3+25*x^2+670*x-720)*log(3)*exp(x)+2304*x^8+480*x ^6+13824*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4 +1440*x^3+20736*x^2),x, algorithm=\
Time = 11.69 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=x+\frac {5\,{\mathrm {e}}^x\,\ln \left (3\right )}{x\,\left (48\,x^3+5\,x+144\right )} \]