Integrand size = 127, antiderivative size = 24 \begin {dmath*} \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {x (i \pi +\log (3))}{e+\left (3-x+x^4\right )^2} \end {dmath*}
Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {x (i \pi +\log (3))}{e+\left (3-x+x^4\right )^2} \end {dmath*}
Integrate[((9 + E - x^2 - 18*x^4 + 8*x^5 - 7*x^8)*(I*Pi + Log[3]))/(81 + E ^2 - 108*x + 54*x^2 - 12*x^3 + 109*x^4 - 108*x^5 + 36*x^6 - 4*x^7 + 54*x^8 - 36*x^9 + 6*x^10 + 12*x^12 - 4*x^13 + x^16 + E*(18 - 12*x + 2*x^2 + 12*x ^4 - 4*x^5 + 2*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.
\(\displaystyle \int \frac {\left (-7 x^8+8 x^5-18 x^4-x^2+e+9\right ) (\log (3)+i \pi )}{x^{16}-4 x^{13}+12 x^{12}+6 x^{10}-36 x^9+54 x^8-4 x^7+36 x^6-108 x^5+109 x^4-12 x^3+54 x^2+e \left (2 x^8-4 x^5+12 x^4+2 x^2-12 x+18\right )-108 x+e^2+81} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle (\log (3)+i \pi ) \int \frac {-7 x^8+8 x^5-18 x^4-x^2+e+9}{x^{16}-4 x^{13}+12 x^{12}+6 x^{10}-36 x^9+54 x^8-4 x^7+36 x^6-108 x^5+109 x^4-12 x^3+54 x^2-108 x+2 e \left (x^8-2 x^5+6 x^4+x^2-6 x+9\right )+e^2+81}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle (\log (3)+i \pi ) \int \left (\frac {2 \left (-3 x^5+12 x^4+3 x^2-21 x+36 \left (1+\frac {e}{9}\right )\right )}{\left (x^8-2 x^5+6 x^4+x^2-6 x+9 \left (1+\frac {e}{9}\right )\right )^2}+\frac {7}{-x^8+2 x^5-6 x^4-x^2+6 x-9 \left (1+\frac {e}{9}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle (\log (3)+i \pi ) \left (-\frac {2 (9+e) \int \frac {1}{\left (-x^4+x+i \sqrt {e}-3\right )^2}dx}{e}+\frac {2 (9+e) \int \frac {1}{-i x^4+i x+\sqrt {e}-3 i}dx}{e^{3/2}}-\frac {7 \int \frac {1}{-i x^4+i x+\sqrt {e}-3 i}dx}{2 \sqrt {e}}+\frac {2 (9+e) \int \frac {1}{i x^4-i x+\sqrt {e}+3 i}dx}{e^{3/2}}-\frac {7 \int \frac {1}{i x^4-i x+\sqrt {e}+3 i}dx}{2 \sqrt {e}}-\frac {2 (9+e) \int \frac {1}{\left (x^4-x+i \sqrt {e}+3\right )^2}dx}{e}-42 \int \frac {x}{\left (\left (x^4-x+3\right )^2+e\right )^2}dx+24 \int \frac {x^4}{\left (\left (x^4-x+3\right )^2+e\right )^2}dx-6 \int \frac {x^5}{\left (\left (x^4-x+3\right )^2+e\right )^2}dx+6 \int \frac {x^2}{\left (\left (x^4-x+3\right )^2+e\right )^2}dx\right )\) |
Int[((9 + E - x^2 - 18*x^4 + 8*x^5 - 7*x^8)*(I*Pi + Log[3]))/(81 + E^2 - 1 08*x + 54*x^2 - 12*x^3 + 109*x^4 - 108*x^5 + 36*x^6 - 4*x^7 + 54*x^8 - 36* x^9 + 6*x^10 + 12*x^12 - 4*x^13 + x^16 + E*(18 - 12*x + 2*x^2 + 12*x^4 - 4 *x^5 + 2*x^8)),x]
3.8.4.3.1 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[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 2.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
method | result | size |
gosper | \(\frac {\left (\ln \left (3\right )+i \pi \right ) x}{x^{8}-2 x^{5}+6 x^{4}+x^{2}+{\mathrm e}-6 x +9}\) | \(35\) |
norman | \(\frac {\left (\ln \left (3\right )+i \pi \right ) x}{x^{8}-2 x^{5}+6 x^{4}+x^{2}+{\mathrm e}-6 x +9}\) | \(35\) |
risch | \(\frac {\left (\ln \left (3\right )+i \pi \right ) x}{x^{8}-2 x^{5}+6 x^{4}+x^{2}+{\mathrm e}-6 x +9}\) | \(35\) |
parallelrisch | \(\frac {\left (\ln \left (3\right )+i \pi \right ) x}{x^{8}-2 x^{5}+6 x^{4}+x^{2}+{\mathrm e}-6 x +9}\) | \(35\) |
int((exp(1)-7*x^8+8*x^5-18*x^4-x^2+9)*(ln(3)+I*Pi)/(exp(1)^2+(2*x^8-4*x^5+ 12*x^4+2*x^2-12*x+18)*exp(1)+x^16-4*x^13+12*x^12+6*x^10-36*x^9+54*x^8-4*x^ 7+36*x^6-108*x^5+109*x^4-12*x^3+54*x^2-108*x+81),x,method=_RETURNVERBOSE)
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \begin {dmath*} \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {i \, \pi x + x \log \left (3\right )}{x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + e + 9} \end {dmath*}
integrate((exp(1)-7*x^8+8*x^5-18*x^4-x^2+9)*(log(3)+I*pi)/(exp(1)^2+(2*x^8 -4*x^5+12*x^4+2*x^2-12*x+18)*exp(1)+x^16-4*x^13+12*x^12+6*x^10-36*x^9+54*x ^8-4*x^7+36*x^6-108*x^5+109*x^4-12*x^3+54*x^2-108*x+81),x, algorithm=\
Time = 152.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \begin {dmath*} \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=- \frac {x \left (- \log {\left (3 \right )} - i \pi \right )}{x^{8} - 2 x^{5} + 6 x^{4} + x^{2} - 6 x + e + 9} \end {dmath*}
integrate((exp(1)-7*x**8+8*x**5-18*x**4-x**2+9)*(ln(3)+I*pi)/(exp(1)**2+(2 *x**8-4*x**5+12*x**4+2*x**2-12*x+18)*exp(1)+x**16-4*x**13+12*x**12+6*x**10 -36*x**9+54*x**8-4*x**7+36*x**6-108*x**5+109*x**4-12*x**3+54*x**2-108*x+81 ),x)
Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \begin {dmath*} \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {{\left (i \, \pi + \log \left (3\right )\right )} x}{x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + e + 9} \end {dmath*}
integrate((exp(1)-7*x^8+8*x^5-18*x^4-x^2+9)*(log(3)+I*pi)/(exp(1)^2+(2*x^8 -4*x^5+12*x^4+2*x^2-12*x+18)*exp(1)+x^16-4*x^13+12*x^12+6*x^10-36*x^9+54*x ^8-4*x^7+36*x^6-108*x^5+109*x^4-12*x^3+54*x^2-108*x+81),x, algorithm=\
Time = 0.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \begin {dmath*} \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=-\frac {{\left (-i \, \pi - \log \left (3\right )\right )} x}{x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + e + 9} \end {dmath*}
integrate((exp(1)-7*x^8+8*x^5-18*x^4-x^2+9)*(log(3)+I*pi)/(exp(1)^2+(2*x^8 -4*x^5+12*x^4+2*x^2-12*x+18)*exp(1)+x^16-4*x^13+12*x^12+6*x^10-36*x^9+54*x ^8-4*x^7+36*x^6-108*x^5+109*x^4-12*x^3+54*x^2-108*x+81),x, algorithm=\
Time = 27.89 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \begin {dmath*} \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {x\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}{x^8-2\,x^5+6\,x^4+x^2-6\,x+\mathrm {e}+9} \end {dmath*}