Optimal. Leaf size=30 \[ \left (-9-e^x+\frac {\left (-3-2 x+x^2\right )^2}{x^2}\right ) \log ^2\left (5+x^2\right ) \]
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Rubi [C] time = 5.81, antiderivative size = 620, normalized size of antiderivative = 20.67, number of steps used = 66, number of rules used = 43, integrand size = 108, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.398, Rules used = {1593, 6688, 14, 6742, 6725, 2178, 2194, 2554, 12, 1802, 635, 203, 260, 2528, 2448, 321, 2454, 2395, 36, 29, 31, 2455, 2389, 2295, 2288, 2392, 2391, 2462, 2416, 2390, 2301, 2394, 2393, 2450, 2476, 2470, 4920, 4854, 2402, 2315, 2397, 2457, 2296} \begin {gather*} -16 i \sqrt {5} \text {Li}_2\left (1-\frac {2 \sqrt {5}}{i x+\sqrt {5}}\right )-\frac {48 i \text {Li}_2\left (1-\frac {2 \sqrt {5}}{i x+\sqrt {5}}\right )}{\sqrt {5}}+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}-i x}{2 \sqrt {5}}\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {i x+\sqrt {5}}{2 \sqrt {5}}\right )-4 x \log ^2\left (x^2+5\right )+\frac {9 \left (x^2+5\right ) \log ^2\left (x^2+5\right )}{5 x^2}+\left (x^2+5\right ) \log ^2\left (x^2+5\right )+\frac {12 \log ^2\left (x^2+5\right )}{x}-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (-x+i \sqrt {5}\right ) \log \left (x^2+5\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (x+i \sqrt {5}\right ) \log \left (x^2+5\right )-\frac {e^x \log \left (x^2+5\right ) \left (x^2 \log \left (x^2+5\right )+5 \log \left (x^2+5\right )\right )}{x^2+5}-16 \sqrt {5} \log \left (x^2+5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-\frac {48 \log \left (x^2+5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5}}+\frac {1}{5} \left (89+32 i \sqrt {5}\right ) \log ^2\left (-x+i \sqrt {5}\right )+\frac {1}{5} \left (89-32 i \sqrt {5}\right ) \log ^2\left (x+i \sqrt {5}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (-x+i \sqrt {5}\right )}{2 \sqrt {5}}\right ) \log \left (x+i \sqrt {5}\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (-x+i \sqrt {5}\right ) \log \left (-\frac {i \left (x+i \sqrt {5}\right )}{2 \sqrt {5}}\right )-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-32 \sqrt {5} \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-\frac {96 \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 29
Rule 31
Rule 36
Rule 203
Rule 260
Rule 321
Rule 635
Rule 1593
Rule 1802
Rule 2178
Rule 2194
Rule 2288
Rule 2295
Rule 2296
Rule 2301
Rule 2315
Rule 2389
Rule 2390
Rule 2391
Rule 2392
Rule 2393
Rule 2394
Rule 2395
Rule 2397
Rule 2402
Rule 2416
Rule 2448
Rule 2450
Rule 2454
Rule 2455
Rule 2457
Rule 2462
Rule 2470
Rule 2476
Rule 2528
Rule 2554
Rule 4854
Rule 4920
Rule 6688
Rule 6725
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (36 x^2+48 x^3-44 x^4-4 e^x x^4-16 x^5+4 x^6\right ) \log \left (5+x^2\right )+\left (-90-60 x-18 x^2-32 x^3+10 x^4-4 x^5+2 x^6+e^x \left (-5 x^3-x^5\right )\right ) \log ^2\left (5+x^2\right )}{x^3 \left (5+x^2\right )} \, dx\\ &=\int \frac {\log \left (5+x^2\right ) \left (\frac {4 x^2 \left (9+12 x-\left (11+e^x\right ) x^2-4 x^3+x^4\right )}{5+x^2}+\left (-18-12 x-\left (4+e^x\right ) x^3+2 x^4\right ) \log \left (5+x^2\right )\right )}{x^3} \, dx\\ &=\int \left (-\frac {e^x \log \left (5+x^2\right ) \left (4 x+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+\frac {2 \log \left (5+x^2\right ) \left (18 x^2+24 x^3-22 x^4-8 x^5+2 x^6-45 \log \left (5+x^2\right )-30 x \log \left (5+x^2\right )-9 x^2 \log \left (5+x^2\right )-16 x^3 \log \left (5+x^2\right )+5 x^4 \log \left (5+x^2\right )-2 x^5 \log \left (5+x^2\right )+x^6 \log \left (5+x^2\right )\right )}{x^3 \left (5+x^2\right )}\right ) \, dx\\ &=2 \int \frac {\log \left (5+x^2\right ) \left (18 x^2+24 x^3-22 x^4-8 x^5+2 x^6-45 \log \left (5+x^2\right )-30 x \log \left (5+x^2\right )-9 x^2 \log \left (5+x^2\right )-16 x^3 \log \left (5+x^2\right )+5 x^4 \log \left (5+x^2\right )-2 x^5 \log \left (5+x^2\right )+x^6 \log \left (5+x^2\right )\right )}{x^3 \left (5+x^2\right )} \, dx-\int \frac {e^x \log \left (5+x^2\right ) \left (4 x+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2} \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \int \frac {\log \left (5+x^2\right ) \left (2 x^2 \left (9+12 x-11 x^2-4 x^3+x^4\right )+\left (-45-30 x-9 x^2-16 x^3+5 x^4-2 x^5+x^6\right ) \log \left (5+x^2\right )\right )}{x^3 \left (5+x^2\right )} \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \int \left (\frac {2 \left (-3-5 x+x^2\right ) \left (-3+x+x^2\right ) \log \left (5+x^2\right )}{x \left (5+x^2\right )}+\frac {(-3+x) (1+x) \left (3+x^2\right ) \log ^2\left (5+x^2\right )}{x^3}\right ) \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \int \frac {(-3+x) (1+x) \left (3+x^2\right ) \log ^2\left (5+x^2\right )}{x^3} \, dx+4 \int \frac {\left (-3-5 x+x^2\right ) \left (-3+x+x^2\right ) \log \left (5+x^2\right )}{x \left (5+x^2\right )} \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \int \left (-2 \log ^2\left (5+x^2\right )-\frac {9 \log ^2\left (5+x^2\right )}{x^3}-\frac {6 \log ^2\left (5+x^2\right )}{x^2}+x \log ^2\left (5+x^2\right )\right ) \, dx+4 \int \left (-4 \log \left (5+x^2\right )+\frac {9 \log \left (5+x^2\right )}{5 x}+x \log \left (5+x^2\right )+\frac {(160-89 x) \log \left (5+x^2\right )}{5 \left (5+x^2\right )}\right ) \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+\frac {4}{5} \int \frac {(160-89 x) \log \left (5+x^2\right )}{5+x^2} \, dx+2 \int x \log ^2\left (5+x^2\right ) \, dx+4 \int x \log \left (5+x^2\right ) \, dx-4 \int \log ^2\left (5+x^2\right ) \, dx+\frac {36}{5} \int \frac {\log \left (5+x^2\right )}{x} \, dx-12 \int \frac {\log ^2\left (5+x^2\right )}{x^2} \, dx-16 \int \log \left (5+x^2\right ) \, dx-18 \int \frac {\log ^2\left (5+x^2\right )}{x^3} \, dx\\ &=-16 x \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+\frac {4}{5} \int \left (\frac {\left (445+160 i \sqrt {5}\right ) \log \left (5+x^2\right )}{10 \left (i \sqrt {5}-x\right )}+\frac {\left (-445+160 i \sqrt {5}\right ) \log \left (5+x^2\right )}{10 \left (i \sqrt {5}+x\right )}\right ) \, dx+2 \operatorname {Subst}\left (\int \log (5+x) \, dx,x,x^2\right )+\frac {18}{5} \operatorname {Subst}\left (\int \frac {\log (5+x)}{x} \, dx,x,x^2\right )-9 \operatorname {Subst}\left (\int \frac {\log ^2(5+x)}{x^2} \, dx,x,x^2\right )+16 \int \frac {x^2 \log \left (5+x^2\right )}{5+x^2} \, dx+32 \int \frac {x^2}{5+x^2} \, dx-48 \int \frac {\log \left (5+x^2\right )}{5+x^2} \, dx+\operatorname {Subst}\left (\int \log ^2(5+x) \, dx,x,x^2\right )\\ &=32 x+\frac {36}{5} \log (5) \log (x)-16 x \log \left (5+x^2\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \operatorname {Subst}\left (\int \log (x) \, dx,x,5+x^2\right )+\frac {18}{5} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx,x,x^2\right )-\frac {18}{5} \operatorname {Subst}\left (\int \frac {\log (5+x)}{x} \, dx,x,x^2\right )+16 \int \left (\log \left (5+x^2\right )-\frac {5 \log \left (5+x^2\right )}{5+x^2}\right ) \, dx+96 \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5} \left (5+x^2\right )} \, dx-160 \int \frac {1}{5+x^2} \, dx-\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {\log \left (5+x^2\right )}{i \sqrt {5}+x} \, dx+\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {\log \left (5+x^2\right )}{i \sqrt {5}-x} \, dx+\operatorname {Subst}\left (\int \log ^2(x) \, dx,x,5+x^2\right )\\ &=32 x-2 x^2-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-16 x \log \left (5+x^2\right )+2 \left (5+x^2\right ) \log \left (5+x^2\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {18}{5} \text {Li}_2\left (-\frac {x^2}{5}\right )-2 \operatorname {Subst}\left (\int \log (x) \, dx,x,5+x^2\right )-\frac {18}{5} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx,x,x^2\right )+16 \int \log \left (5+x^2\right ) \, dx-80 \int \frac {\log \left (5+x^2\right )}{5+x^2} \, dx+\frac {96 \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{5+x^2} \, dx}{\sqrt {5}}+\frac {1}{5} \left (4 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {x \log \left (i \sqrt {5}+x\right )}{5+x^2} \, dx+\frac {1}{5} \left (4 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {x \log \left (i \sqrt {5}-x\right )}{5+x^2} \, dx\\ &=32 x-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {96}{5} \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{i-\frac {x}{\sqrt {5}}} \, dx-32 \int \frac {x^2}{5+x^2} \, dx+160 \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5} \left (5+x^2\right )} \, dx+\frac {1}{5} \left (4 \left (89-32 i \sqrt {5}\right )\right ) \int \left (-\frac {\log \left (i \sqrt {5}+x\right )}{2 \left (i \sqrt {5}-x\right )}+\frac {\log \left (i \sqrt {5}+x\right )}{2 \left (i \sqrt {5}+x\right )}\right ) \, dx+\frac {1}{5} \left (4 \left (89+32 i \sqrt {5}\right )\right ) \int \left (-\frac {\log \left (i \sqrt {5}-x\right )}{2 \left (i \sqrt {5}-x\right )}+\frac {\log \left (i \sqrt {5}-x\right )}{2 \left (i \sqrt {5}+x\right )}\right ) \, dx\\ &=-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+\frac {96}{5} \int \frac {\log \left (\frac {2}{1+\frac {i x}{\sqrt {5}}}\right )}{1+\frac {x^2}{5}} \, dx+160 \int \frac {1}{5+x^2} \, dx+\left (32 \sqrt {5}\right ) \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{5+x^2} \, dx-\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {\log \left (i \sqrt {5}+x\right )}{i \sqrt {5}-x} \, dx+\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {\log \left (i \sqrt {5}+x\right )}{i \sqrt {5}+x} \, dx-\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {\log \left (i \sqrt {5}-x\right )}{i \sqrt {5}-x} \, dx+\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {\log \left (i \sqrt {5}-x\right )}{i \sqrt {5}+x} \, dx\\ &=-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right ) \log \left (i \sqrt {5}+x\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (-\frac {i \left (i \sqrt {5}+x\right )}{2 \sqrt {5}}\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-32 \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{i-\frac {x}{\sqrt {5}}} \, dx-\frac {(96 i) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i x}{\sqrt {5}}}\right )}{\sqrt {5}}-\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {\log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right )}{i \sqrt {5}+x} \, dx+\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,i \sqrt {5}+x\right )+\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {\log \left (\frac {i \left (-i \sqrt {5}-x\right )}{2 \sqrt {5}}\right )}{i \sqrt {5}-x} \, dx+\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,i \sqrt {5}-x\right )\\ &=-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2+\frac {1}{5} \left (89+32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}-x\right )-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right ) \log \left (i \sqrt {5}+x\right )+\frac {1}{5} \left (89-32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}+x\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (-\frac {i \left (i \sqrt {5}+x\right )}{2 \sqrt {5}}\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {48 i \text {Li}_2\left (1-\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}+32 \int \frac {\log \left (\frac {2}{1+\frac {i x}{\sqrt {5}}}\right )}{1+\frac {x^2}{5}} \, dx-\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i x}{2 \sqrt {5}}\right )}{x} \, dx,x,i \sqrt {5}+x\right )-\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i x}{2 \sqrt {5}}\right )}{x} \, dx,x,i \sqrt {5}-x\right )\\ &=-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2+\frac {1}{5} \left (89+32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}-x\right )-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right ) \log \left (i \sqrt {5}+x\right )+\frac {1}{5} \left (89-32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}+x\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (-\frac {i \left (i \sqrt {5}+x\right )}{2 \sqrt {5}}\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {48 i \text {Li}_2\left (1-\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}-i x}{2 \sqrt {5}}\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}+i x}{2 \sqrt {5}}\right )-\left (32 i \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i x}{\sqrt {5}}}\right )\\ &=-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2+\frac {1}{5} \left (89+32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}-x\right )-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right ) \log \left (i \sqrt {5}+x\right )+\frac {1}{5} \left (89-32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}+x\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (-\frac {i \left (i \sqrt {5}+x\right )}{2 \sqrt {5}}\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {48 i \text {Li}_2\left (1-\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-16 i \sqrt {5} \text {Li}_2\left (1-\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}-i x}{2 \sqrt {5}}\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}+i x}{2 \sqrt {5}}\right )\\ \end {aligned} \end {gather*}
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Mathematica [C] time = 1.06, size = 572, normalized size = 19.07 \begin {gather*} \frac {1}{5} \left (-128 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2+89 \log ^2\left (i \sqrt {5}-x\right )+32 i \sqrt {5} \log ^2\left (i \sqrt {5}-x\right )+89 \log ^2\left (i \sqrt {5}+x\right )-32 i \sqrt {5} \log ^2\left (i \sqrt {5}+x\right )-256 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {10 i}{5 i-\sqrt {5} x}\right )+178 \log \left (i \sqrt {5}-x\right ) \log \left (\frac {1}{10} \left (5-i \sqrt {5} x\right )\right )+64 i \sqrt {5} \log \left (i \sqrt {5}-x\right ) \log \left (\frac {1}{10} \left (5-i \sqrt {5} x\right )\right )+178 \log \left (i \sqrt {5}+x\right ) \log \left (\frac {1}{10} \left (5+i \sqrt {5} x\right )\right )-64 i \sqrt {5} \log \left (i \sqrt {5}+x\right ) \log \left (\frac {1}{10} \left (5+i \sqrt {5} x\right )\right )-128 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-178 \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-64 i \sqrt {5} \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-178 \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+64 i \sqrt {5} \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+34 \log ^2\left (5+x^2\right )-5 e^x \log ^2\left (5+x^2\right )+\frac {45 \log ^2\left (5+x^2\right )}{x^2}+\frac {60 \log ^2\left (5+x^2\right )}{x}-20 x \log ^2\left (5+x^2\right )+5 x^2 \log ^2\left (5+x^2\right )+2 \left (89-32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {1}{2}-\frac {i x}{2 \sqrt {5}}\right )+2 \left (89+32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {1}{2}+\frac {i x}{2 \sqrt {5}}\right )-128 i \sqrt {5} \text {Li}_2\left (\frac {5 i+\sqrt {5} x}{-5 i+\sqrt {5} x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 37, normalized size = 1.23 \begin {gather*} \frac {{\left (x^{4} - 4 \, x^{3} - x^{2} e^{x} - 11 \, x^{2} + 12 \, x + 9\right )} \log \left (x^{2} + 5\right )^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 79, normalized size = 2.63 \begin {gather*} \frac {x^{4} \log \left (x^{2} + 5\right )^{2} - 4 \, x^{3} \log \left (x^{2} + 5\right )^{2} - x^{2} e^{x} \log \left (x^{2} + 5\right )^{2} - 11 \, x^{2} \log \left (x^{2} + 5\right )^{2} + 12 \, x \log \left (x^{2} + 5\right )^{2} + 9 \, \log \left (x^{2} + 5\right )^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 38, normalized size = 1.27
method | result | size |
risch | \(\frac {\left (x^{4}-4 x^{3}-{\mathrm e}^{x} x^{2}-11 x^{2}+12 x +9\right ) \ln \left (x^{2}+5\right )^{2}}{x^{2}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 37, normalized size = 1.23 \begin {gather*} \frac {{\left (x^{4} - 4 \, x^{3} - x^{2} e^{x} - 11 \, x^{2} + 12 \, x + 9\right )} \log \left (x^{2} + 5\right )^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 49, normalized size = 1.63 \begin {gather*} -{\ln \left (x^2+5\right )}^2\,\left ({\mathrm {e}}^x+\frac {12\,x^3-6\,x^4}{x^2}-\frac {-5\,x^4+8\,x^3+12\,x+9}{x^2}+11\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.60, size = 41, normalized size = 1.37 \begin {gather*} - e^{x} \log {\left (x^{2} + 5 \right )}^{2} + \frac {\left (x^{4} - 4 x^{3} - 11 x^{2} + 12 x + 9\right ) \log {\left (x^{2} + 5 \right )}^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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