Integrand size = 60, antiderivative size = 24 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=x+e^{x+x \log (3)} \left (\frac {1}{2} \left (4+e^5\right )+x\right )^5 \]
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Time = 8.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88, number of steps used = 16, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6873, 6874, 2227, 2207, 2225} \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=\frac {1}{32} (3 e)^x \left (2 x+e^5+4\right )^5+x \]
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Rule 2207
Rule 2225
Rule 2227
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {4 \left (1+\frac {e^5}{4}\right )+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx \\ & = \int \left (1+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^4 \left (14+e^5 (1+\log (3))+x (2+\log (9))+\log (81)\right )\right ) \, dx \\ & = x+\frac {1}{32} \int (3 e)^x \left (4+e^5+2 x\right )^4 \left (14+e^5 (1+\log (3))+x (2+\log (9))+\log (81)\right ) \, dx \\ & = x+\frac {1}{32} \int \left (10 (3 e)^x \left (4+e^5+2 x\right )^4+(3 e)^x \left (4+e^5+2 x\right )^5 (1+\log (3))\right ) \, dx \\ & = x+\frac {5}{16} \int (3 e)^x \left (4+e^5+2 x\right )^4 \, dx+\frac {1}{32} (1+\log (3)) \int (3 e)^x \left (4+e^5+2 x\right )^5 \, dx \\ & = x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {5 (3 e)^x \left (4+e^5+2 x\right )^4}{16 (1+\log (3))}-\frac {5}{16} \int (3 e)^x \left (4+e^5+2 x\right )^4 \, dx-\frac {5 \int (3 e)^x \left (4+e^5+2 x\right )^3 \, dx}{2 (1+\log (3))} \\ & = x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5-\frac {5 (3 e)^x \left (4+e^5+2 x\right )^3}{2 (1+\log (3))^2}+\frac {15 \int (3 e)^x \left (4+e^5+2 x\right )^2 \, dx}{(1+\log (3))^2}+\frac {5 \int (3 e)^x \left (4+e^5+2 x\right )^3 \, dx}{2 (1+\log (3))} \\ & = x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {5\ 3^{1+x} e^x \left (4+e^5+2 x\right )^2}{(1+\log (3))^3}-\frac {60 \int (3 e)^x \left (4+e^5+2 x\right ) \, dx}{(1+\log (3))^3}-\frac {15 \int (3 e)^x \left (4+e^5+2 x\right )^2 \, dx}{(1+\log (3))^2} \\ & = x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5-\frac {20\ 3^{1+x} e^x \left (4+e^5+2 x\right )}{(1+\log (3))^4}+\frac {120 \int (3 e)^x \, dx}{(1+\log (3))^4}+\frac {60 \int (3 e)^x \left (4+e^5+2 x\right ) \, dx}{(1+\log (3))^3} \\ & = x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {40\ 3^{1+x} e^x}{(1+\log (3))^5}-\frac {120 \int (3 e)^x \, dx}{(1+\log (3))^4} \\ & = x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5 \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(909\) vs. \(2(24)=48\).
Time = 7.33 (sec) , antiderivative size = 909, normalized size of antiderivative = 37.88 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=\frac {3^x e^{25+x} (1+\log (3))^6+32\ 3^{1+x} e^x \left (5 x^4 (1+\log (3))^5+4 x^3 (1+\log (3))^4 (-5+(1+\log (3)) \log (9))-4 \left (40+20 \log ^4(3)-2 \log ^2(3) (-80+\log (9))+\log (3) (130-3 \log (81))-2 \log ^3(3) (-40+\log (81))+\log (9) (1+\log (81))\right )+4 x (1+\log (3)) \left (40+20 \log ^4(3)-2 \log ^2(3) (-80+\log (9))+\log (3) (130-3 \log (81))-2 \log ^3(3) (-40+\log (81))+\log (9) (1+\log (81))\right )+2 x^2 (1+\log (3))^2 \left (-20-\log (9)+\log ^2(3) (-40+\log (81))+2 \log ^3(3) \log (81)+\log (27) \log (81)-2 \log (3) (25+\log (81))\right )\right )+8\ 3^x e^{15+x} (1+\log (3))^3 \left (20+50 \log (3)+12 \log ^3(3)-\log ^2(9)+x^2 \left (5+13 \log (3)+3 \log ^3(3)+\log (9)+\log ^2(9)+\log ^2(3) (11+\log (9))\right )+\log (27) \log (81)+2 \log ^2(3) (26+\log (81))+x \left (20+56 \log (3)+12 \log ^3(3)+\log ^2(9)+\log (81)+\log (9) \log (81)+2 \log ^2(3) (24+\log (81))\right )+\log (59049)\right )+3^x e^{20+x} (1+\log (3))^4 \left (20+16 \log ^2(3)+\log (9)+\log (3) (38+\log (81))+x (1+\log (3)) (10+\log (59049))\right )+16 (1+\log (3))^3 \left ((3 e)^x x^5 (1+\log (3))^2 (2+\log (9))+16 (3 e)^x x^2 (1+\log (3))^2 (25+\log (81))+16 (3 e)^x \left (64-\log (3) (-28+\log (9))+\log (9)+\log ^2(3) (14+\log (81))\right )+4 (3 e)^x x^3 \left (50-8 \log (3) (-10+\log (9))-\log (9)+\log (27) \log (81)+2 \log ^2(3) (20+\log (81))\right )+2 x (1+\log (3)) \left (1+\log ^2(3)+\log (9)-8\ 3^{1+x} e^x \log (9)+8 (3 e)^x (-50+\log (3) \log (9)+\log (9) \log (81)+\log (6561))\right )+(3 e)^x x^4 (1+\log (3)) (-10+\log (3) (8 \log (9)+\log (81))+\log (59049))\right )+16\ 3^x e^{5+x} (1+\log (3)) \left (x^4 (1+\log (3))^4 (5+\log (243))+2 x^3 (1+\log (3))^3 \left (20+4 \log ^2(3)+\log (3) (34+6 \log (9)+\log (81))+\log (729)\right )+4 x (1+\log (3)) \left (40+8 \log ^4(3)+8 \log ^3(3) (14+\log (81))+12 \log ^2(3) (16+\log (81))+\log (3) (152+6 \log (9)+9 \log (81))+\log (6561)\right )+6 x^2 (1+\log (3))^2 \left (20+50 \log (3)+4 \log ^3(3)+\log ^2(9)+\log (27) \log (81)+2 \log ^2(3) (22+\log (6561))+\log (59049)\right )+4 \left (20+4 \log ^5(3)+4 \log ^4(3) (17+\log (81))+4 \log ^2(3) (46+3 \log (81))+\log ^3(3) (152-4 \log (9)+10 \log (81))+\log (4782969)+\log (3) (86+\log (43046721))\right )\right )+8\ 3^x e^{10+x} (1+\log (3))^2 \left (2 \left (40+16 \log ^4(3)+2 \log ^3(3) (62+3 \log (81))+\log (3) (154+9 \log (81))+\log (729)+6 \log ^2(3) (34+\log (729))\right )+x^3 (1+\log (3))^3 (10+\log (59049))+6 x (1+\log (3)) \left (20+50 \log (3)+8 \log ^3(3)+\log (27) \log (81)+3 \log ^2(3) (16+\log (81))+\log (59049)\right )+3 x^2 (1+\log (3))^2 (20+\log (3) \log (81)+(2+\log (9)) \log (6561)+\log (282429536481))\right )}{32 (1+\log (3))^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs. \(2(19)=38\).
Time = 0.73 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.75
method | result | size |
risch | \(x +\left (32+\frac {{\mathrm e}^{25}}{32}+\frac {5 x \,{\mathrm e}^{20}}{16}+\frac {5 x^{2} {\mathrm e}^{15}}{4}+\frac {5 x^{3} {\mathrm e}^{10}}{2}+\frac {5 x^{4} {\mathrm e}^{5}}{2}+\frac {5 \,{\mathrm e}^{20}}{8}+20 \,{\mathrm e}^{10}+30 x \,{\mathrm e}^{10}+5 x \,{\mathrm e}^{15}+15 x^{2} {\mathrm e}^{10}+20 x^{3} {\mathrm e}^{5}+5 \,{\mathrm e}^{15}+80 x +40 \,{\mathrm e}^{5}+60 x^{2} {\mathrm e}^{5}+80 x \,{\mathrm e}^{5}+x^{5}+10 x^{4}+40 x^{3}+80 x^{2}\right ) 3^{x} {\mathrm e}^{x}\) | \(114\) |
norman | \(x +\left (32+40 \,{\mathrm e}^{5}+\frac {{\mathrm e}^{25}}{32}+\frac {5 \,{\mathrm e}^{20}}{8}+5 \,{\mathrm e}^{15}+20 \,{\mathrm e}^{10}\right ) {\mathrm e}^{x \ln \left (3\right )+x}+{\mathrm e}^{x \ln \left (3\right )+x} x^{5}+\left (\frac {5 \,{\mathrm e}^{5}}{2}+10\right ) x^{4} {\mathrm e}^{x \ln \left (3\right )+x}+\left (\frac {5 \,{\mathrm e}^{10}}{2}+20 \,{\mathrm e}^{5}+40\right ) x^{3} {\mathrm e}^{x \ln \left (3\right )+x}+\left (\frac {5 \,{\mathrm e}^{15}}{4}+15 \,{\mathrm e}^{10}+60 \,{\mathrm e}^{5}+80\right ) x^{2} {\mathrm e}^{x \ln \left (3\right )+x}+\left (\frac {5 \,{\mathrm e}^{20}}{16}+5 \,{\mathrm e}^{15}+30 \,{\mathrm e}^{10}+80 \,{\mathrm e}^{5}+80\right ) x \,{\mathrm e}^{x \ln \left (3\right )+x}\) | \(154\) |
parallelrisch | \(x +\frac {{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{25}}{32}+\frac {5 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{20}}{8}+5 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{15}+20 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{10}+40 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{5}+32 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x}+\frac {5 \,{\mathrm e}^{15} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{2}}{4}+15 \,{\mathrm e}^{10} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{2}+60 \,{\mathrm e}^{5} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{2}+80 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{2}+\frac {5 \,{\mathrm e}^{10} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{3}}{2}+20 \,{\mathrm e}^{5} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{3}+40 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{3}+\frac {5 \,{\mathrm e}^{5} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{4}}{2}+10 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{4}+{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{5}+\frac {5 \,{\mathrm e}^{20} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x}{16}+5 \,{\mathrm e}^{15} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x +30 \,{\mathrm e}^{10} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x +80 \,{\mathrm e}^{5} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x +80 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x\) | \(276\) |
parts | \(\text {Expression too large to display}\) | \(1910\) |
derivativedivides | \(\text {Expression too large to display}\) | \(34958\) |
default | \(\text {Expression too large to display}\) | \(34958\) |
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.00 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=\frac {1}{32} \, {\left (32 \, x^{5} + 320 \, x^{4} + 1280 \, x^{3} + 2560 \, x^{2} + 10 \, {\left (x + 2\right )} e^{20} + 40 \, {\left (x^{2} + 4 \, x + 4\right )} e^{15} + 80 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} e^{10} + 80 \, {\left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 32 \, x + 16\right )} e^{5} + 2560 \, x + e^{25} + 1024\right )} e^{\left (x \log \left (3\right ) + x\right )} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (19) = 38\).
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 5.75 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=x + \frac {\left (32 x^{5} + 320 x^{4} + 80 x^{4} e^{5} + 1280 x^{3} + 640 x^{3} e^{5} + 80 x^{3} e^{10} + 2560 x^{2} + 1920 x^{2} e^{5} + 480 x^{2} e^{10} + 40 x^{2} e^{15} + 2560 x + 2560 x e^{5} + 960 x e^{10} + 160 x e^{15} + 10 x e^{20} + 1024 + 1280 e^{5} + 640 e^{10} + 160 e^{15} + 20 e^{20} + e^{25}\right ) e^{x + x \log {\left (3 \right )}}}{32} \]
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\[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=\int { \frac {{\left ({\left (2 \, x + e^{5} + 4\right )} \log \left (3\right ) + 2 \, x + e^{5} + 14\right )} {\left (2 \, x + e^{5} + 4\right )}^{5} e^{\left (x \log \left (3\right ) + x\right )} + 64 \, x + 32 \, e^{5} + 128}{32 \, {\left (2 \, x + e^{5} + 4\right )}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (20) = 40\).
Time = 0.37 (sec) , antiderivative size = 240, normalized size of antiderivative = 10.00 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=x^{5} e^{\left (x \log \left (3\right ) + x\right )} + \frac {5}{2} \, x^{4} e^{\left (x \log \left (3\right ) + x + 5\right )} + 10 \, x^{4} e^{\left (x \log \left (3\right ) + x\right )} + \frac {5}{2} \, x^{3} e^{\left (x \log \left (3\right ) + x + 10\right )} + 20 \, x^{3} e^{\left (x \log \left (3\right ) + x + 5\right )} + 40 \, x^{3} e^{\left (x \log \left (3\right ) + x\right )} + \frac {5}{4} \, x^{2} e^{\left (x \log \left (3\right ) + x + 15\right )} + 15 \, x^{2} e^{\left (x \log \left (3\right ) + x + 10\right )} + 60 \, x^{2} e^{\left (x \log \left (3\right ) + x + 5\right )} + 80 \, x^{2} e^{\left (x \log \left (3\right ) + x\right )} + \frac {5}{16} \, x e^{\left (x \log \left (3\right ) + x + 20\right )} + 5 \, x e^{\left (x \log \left (3\right ) + x + 15\right )} + 30 \, x e^{\left (x \log \left (3\right ) + x + 10\right )} + 80 \, x e^{\left (x \log \left (3\right ) + x + 5\right )} + 80 \, x e^{\left (x \log \left (3\right ) + x\right )} + x + \frac {1}{32} \, e^{\left (x \log \left (3\right ) + x + 25\right )} + \frac {5}{8} \, e^{\left (x \log \left (3\right ) + x + 20\right )} + 5 \, e^{\left (x \log \left (3\right ) + x + 15\right )} + 20 \, e^{\left (x \log \left (3\right ) + x + 10\right )} + 40 \, e^{\left (x \log \left (3\right ) + x + 5\right )} + 32 \, e^{\left (x \log \left (3\right ) + x\right )} \]
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Time = 12.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.17 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=x+3^x\,x^5\,{\mathrm {e}}^x+\frac {3^x\,{\mathrm {e}}^x\,\left (1280\,{\mathrm {e}}^5+640\,{\mathrm {e}}^{10}+160\,{\mathrm {e}}^{15}+20\,{\mathrm {e}}^{20}+{\mathrm {e}}^{25}+1024\right )}{32}+\frac {3^x\,x^4\,{\mathrm {e}}^x\,\left (80\,{\mathrm {e}}^5+320\right )}{32}+\frac {5\,3^x\,x^2\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^3}{4}+\frac {5\,3^x\,x^3\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^2}{2}+\frac {5\,3^x\,x\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^4}{16} \]
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