Integrand size = 127, antiderivative size = 24 \[ \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {x (i \pi +\log (3))}{e+\left (3-x+x^4\right )^2} \]
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\[ \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = (i \pi +\log (3)) \int \frac {9+e-x^2-18 x^4+8 x^5-7 x^8}{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 \\ & = (i \pi +\log (3)) \int \left (\frac {7}{-9 \left (1+\frac {e}{9}\right )+6 x-x^2-6 x^4+2 x^5-x^8}+\frac {2 \left (36 \left (1+\frac {e}{9}\right )-21 x+3 x^2+12 x^4-3 x^5\right )}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}\right ) \, dx \\ & = (2 (i \pi +\log (3))) \int \frac {36 \left (1+\frac {e}{9}\right )-21 x+3 x^2+12 x^4-3 x^5}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx+(7 (i \pi +\log (3))) \int \frac {1}{-9 \left (1+\frac {e}{9}\right )+6 x-x^2-6 x^4+2 x^5-x^8} \, dx \\ & = (2 (i \pi +\log (3))) \int \frac {4 e+3 \left (12-7 x+x^2+4 x^4-x^5\right )}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(7 (i \pi +\log (3))) \int \frac {1}{-e-\left (3-x+x^4\right )^2} \, dx \\ & = (2 (i \pi +\log (3))) \int \left (\frac {36 \left (1+\frac {e}{9}\right )}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}-\frac {21 x}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}+\frac {3 x^2}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}+\frac {12 x^4}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}-\frac {3 x^5}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}\right ) \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {\sqrt {e}}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 e}-\frac {(7 (i \pi +\log (3))) \int \frac {\sqrt {e}}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 e} \\ & = (6 (i \pi +\log (3))) \int \frac {x^2}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+(8 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx \\ & = (6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+(8 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx \\ & = (6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+(8 (9+e) (i \pi +\log (3))) \int \left (-\frac {1}{4 e \left (-3+i \sqrt {e}+x-x^4\right )^2}-\frac {1}{4 e \left (3+i \sqrt {e}-x+x^4\right )^2}-\frac {1}{2 e \left (-e-\left (3-x+x^4\right )^2\right )}\right ) \, dx \\ & = (6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (-3+i \sqrt {e}+x-x^4\right )^2} \, dx}{e}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (3+i \sqrt {e}-x+x^4\right )^2} \, dx}{e}-\frac {(4 (9+e) (i \pi +\log (3))) \int \frac {1}{-e-\left (3-x+x^4\right )^2} \, dx}{e} \\ & = (6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {\sqrt {e}}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{e^2}+\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {\sqrt {e}}{3 i+\sqrt {e}-i x+i x^4} \, dx}{e^2}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (-3+i \sqrt {e}+x-x^4\right )^2} \, dx}{e}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (3+i \sqrt {e}-x+x^4\right )^2} \, dx}{e} \\ & = (6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{e^{3/2}}+\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{e^{3/2}}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (-3+i \sqrt {e}+x-x^4\right )^2} \, dx}{e}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (3+i \sqrt {e}-x+x^4\right )^2} \, dx}{e} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {x (i \pi +\log (3))}{e+\left (3-x+x^4\right )^2} \]
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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 | \(\frac {\left (\ln \left (3\right )+i \pi \right ) x}{x^{8}-2 x^{5}+6 x^{4}+x^{2}+{\mathrm e}-6 x +9}\) | \(35\) |
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {i \, \pi x + x \log \left (3\right )}{x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + e + 9} \]
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Time = 152.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=- \frac {x \left (- \log {\left (3 \right )} - i \pi \right )}{x^{8} - 2 x^{5} + 6 x^{4} + x^{2} - 6 x + e + 9} \]
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Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {{\left (i \, \pi + \log \left (3\right )\right )} x}{x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + e + 9} \]
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Time = 0.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=-\frac {{\left (-i \, \pi - \log \left (3\right )\right )} x}{x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + e + 9} \]
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Time = 27.89 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx=\frac {x\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}{x^8-2\,x^5+6\,x^4+x^2-6\,x+\mathrm {e}+9} \]
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