Integrand size = 504, antiderivative size = 26 \[ \int \frac {-20 x^6+5 x^7+\left (120 x^6-30 x^7\right ) \log (5)+\left (-300 x^6+75 x^7\right ) \log ^2(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (-300 x^6+75 x^7\right ) \log ^4(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (-20 x^6+5 x^7\right ) \log ^6(5)+\left (-25 x^4 \log ^2(5)+100 x^4 \log ^3(5)-150 x^4 \log ^4(5)+100 x^4 \log ^5(5)-25 x^4 \log ^6(5)\right ) \log (\log (3))}{64 x^6+48 x^7+12 x^8+x^9+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log (5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^2(5)+\left (-1280 x^6-960 x^7-240 x^8-20 x^9\right ) \log ^3(5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^4(5)+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log ^5(5)+\left (64 x^6+48 x^7+12 x^8+x^9\right ) \log ^6(5)+\left (\left (48 x^4+24 x^5+3 x^6\right ) \log ^2(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^3(5)+\left (288 x^4+144 x^5+18 x^6\right ) \log ^4(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^5(5)+\left (48 x^4+24 x^5+3 x^6\right ) \log ^6(5)\right ) \log (\log (3))+\left (\left (12 x^2+3 x^3\right ) \log ^4(5)+\left (-24 x^2-6 x^3\right ) \log ^5(5)+\left (12 x^2+3 x^3\right ) \log ^6(5)\right ) \log ^2(\log (3))+\log ^6(5) \log ^3(\log (3))} \, dx=2-\frac {5 x}{\left (4+x+\frac {\log (\log (3))}{\left (-x+\frac {x}{\log (5)}\right )^2}\right )^2} \] Output:
2-5/(ln(ln(3))/(x/ln(5)-x)^2+x+4)^2*x
Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-20 x^6+5 x^7+\left (120 x^6-30 x^7\right ) \log (5)+\left (-300 x^6+75 x^7\right ) \log ^2(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (-300 x^6+75 x^7\right ) \log ^4(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (-20 x^6+5 x^7\right ) \log ^6(5)+\left (-25 x^4 \log ^2(5)+100 x^4 \log ^3(5)-150 x^4 \log ^4(5)+100 x^4 \log ^5(5)-25 x^4 \log ^6(5)\right ) \log (\log (3))}{64 x^6+48 x^7+12 x^8+x^9+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log (5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^2(5)+\left (-1280 x^6-960 x^7-240 x^8-20 x^9\right ) \log ^3(5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^4(5)+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log ^5(5)+\left (64 x^6+48 x^7+12 x^8+x^9\right ) \log ^6(5)+\left (\left (48 x^4+24 x^5+3 x^6\right ) \log ^2(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^3(5)+\left (288 x^4+144 x^5+18 x^6\right ) \log ^4(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^5(5)+\left (48 x^4+24 x^5+3 x^6\right ) \log ^6(5)\right ) \log (\log (3))+\left (\left (12 x^2+3 x^3\right ) \log ^4(5)+\left (-24 x^2-6 x^3\right ) \log ^5(5)+\left (12 x^2+3 x^3\right ) \log ^6(5)\right ) \log ^2(\log (3))+\log ^6(5) \log ^3(\log (3))} \, dx=-\frac {5 x^5 (-1+\log (5))^4}{\left (4 x^2 (-1+\log (5))^2+x^3 (-1+\log (5))^2+\log ^2(5) \log (\log (3))\right )^2} \] Input:
Integrate[(-20*x^6 + 5*x^7 + (120*x^6 - 30*x^7)*Log[5] + (-300*x^6 + 75*x^ 7)*Log[5]^2 + (400*x^6 - 100*x^7)*Log[5]^3 + (-300*x^6 + 75*x^7)*Log[5]^4 + (120*x^6 - 30*x^7)*Log[5]^5 + (-20*x^6 + 5*x^7)*Log[5]^6 + (-25*x^4*Log[ 5]^2 + 100*x^4*Log[5]^3 - 150*x^4*Log[5]^4 + 100*x^4*Log[5]^5 - 25*x^4*Log [5]^6)*Log[Log[3]])/(64*x^6 + 48*x^7 + 12*x^8 + x^9 + (-384*x^6 - 288*x^7 - 72*x^8 - 6*x^9)*Log[5] + (960*x^6 + 720*x^7 + 180*x^8 + 15*x^9)*Log[5]^2 + (-1280*x^6 - 960*x^7 - 240*x^8 - 20*x^9)*Log[5]^3 + (960*x^6 + 720*x^7 + 180*x^8 + 15*x^9)*Log[5]^4 + (-384*x^6 - 288*x^7 - 72*x^8 - 6*x^9)*Log[5 ]^5 + (64*x^6 + 48*x^7 + 12*x^8 + x^9)*Log[5]^6 + ((48*x^4 + 24*x^5 + 3*x^ 6)*Log[5]^2 + (-192*x^4 - 96*x^5 - 12*x^6)*Log[5]^3 + (288*x^4 + 144*x^5 + 18*x^6)*Log[5]^4 + (-192*x^4 - 96*x^5 - 12*x^6)*Log[5]^5 + (48*x^4 + 24*x ^5 + 3*x^6)*Log[5]^6)*Log[Log[3]] + ((12*x^2 + 3*x^3)*Log[5]^4 + (-24*x^2 - 6*x^3)*Log[5]^5 + (12*x^2 + 3*x^3)*Log[5]^6)*Log[Log[3]]^2 + Log[5]^6*Lo g[Log[3]]^3),x]
Output:
(-5*x^5*(-1 + Log[5])^4)/(4*x^2*(-1 + Log[5])^2 + x^3*(-1 + Log[5])^2 + Lo g[5]^2*Log[Log[3]])^2
Time = 11.64 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6, 6, 6, 6, 2462, 7239, 27, 25, 2021}
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.
| \(\displaystyle \int \frac {5 x^7-20 x^6+\log (\log (3)) \left (-25 x^4 \log ^6(5)+100 x^4 \log ^5(5)-150 x^4 \log ^4(5)+100 x^4 \log ^3(5)-25 x^4 \log ^2(5)\right )+\left (5 x^7-20 x^6\right ) \log ^6(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (75 x^7-300 x^6\right ) \log ^4(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (75 x^7-300 x^6\right ) \log ^2(5)+\left (120 x^6-30 x^7\right ) \log (5)}{x^9+12 x^8+48 x^7+64 x^6+\log ^2(\log (3)) \left (\left (3 x^3+12 x^2\right ) \log ^6(5)+\left (-6 x^3-24 x^2\right ) \log ^5(5)+\left (3 x^3+12 x^2\right ) \log ^4(5)\right )+\log (\log (3)) \left (\left (3 x^6+24 x^5+48 x^4\right ) \log ^6(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^5(5)+\left (18 x^6+144 x^5+288 x^4\right ) \log ^4(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^3(5)+\left (3 x^6+24 x^5+48 x^4\right ) \log ^2(5)\right )+\left (x^9+12 x^8+48 x^7+64 x^6\right ) \log ^6(5)+\left (-6 x^9-72 x^8-288 x^7-384 x^6\right ) \log ^5(5)+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \log ^4(5)+\left (-20 x^9-240 x^8-960 x^7-1280 x^6\right ) \log ^3(5)+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \log ^2(5)+\left (-6 x^9-72 x^8-288 x^7-384 x^6\right ) \log (5)+\log ^6(5) \log ^3(\log (3))} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {5 x^7-20 x^6+\log (\log (3)) \left (-25 x^4 \log ^6(5)+100 x^4 \log ^5(5)-150 x^4 \log ^4(5)+100 x^4 \log ^3(5)-25 x^4 \log ^2(5)\right )+\left (5 x^7-20 x^6\right ) \log ^6(5)+\left (120 x^6-30 x^7\right ) \left (\log ^5(5)+\log (5)\right )+\left (75 x^7-300 x^6\right ) \log ^4(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (75 x^7-300 x^6\right ) \log ^2(5)}{x^9+12 x^8+48 x^7+64 x^6+\log ^2(\log (3)) \left (\left (3 x^3+12 x^2\right ) \log ^6(5)+\left (-6 x^3-24 x^2\right ) \log ^5(5)+\left (3 x^3+12 x^2\right ) \log ^4(5)\right )+\log (\log (3)) \left (\left (3 x^6+24 x^5+48 x^4\right ) \log ^6(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^5(5)+\left (18 x^6+144 x^5+288 x^4\right ) \log ^4(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^3(5)+\left (3 x^6+24 x^5+48 x^4\right ) \log ^2(5)\right )+\left (x^9+12 x^8+48 x^7+64 x^6\right ) \log ^6(5)+\left (-6 x^9-72 x^8-288 x^7-384 x^6\right ) \log ^5(5)+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \log ^4(5)+\left (-20 x^9-240 x^8-960 x^7-1280 x^6\right ) \log ^3(5)+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \log ^2(5)+\left (-6 x^9-72 x^8-288 x^7-384 x^6\right ) \log (5)+\log ^6(5) \log ^3(\log (3))}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {5 x^7-20 x^6+\log (\log (3)) \left (-25 x^4 \log ^6(5)+100 x^4 \log ^5(5)-150 x^4 \log ^4(5)+100 x^4 \log ^3(5)-25 x^4 \log ^2(5)\right )+\left (5 x^7-20 x^6\right ) \log ^6(5)+\left (120 x^6-30 x^7\right ) \left (\log ^5(5)+\log (5)\right )+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (75 x^7-300 x^6\right ) \left (\log ^4(5)+\log ^2(5)\right )}{x^9+12 x^8+48 x^7+64 x^6+\log ^2(\log (3)) \left (\left (3 x^3+12 x^2\right ) \log ^6(5)+\left (-6 x^3-24 x^2\right ) \log ^5(5)+\left (3 x^3+12 x^2\right ) \log ^4(5)\right )+\log (\log (3)) \left (\left (3 x^6+24 x^5+48 x^4\right ) \log ^6(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^5(5)+\left (18 x^6+144 x^5+288 x^4\right ) \log ^4(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^3(5)+\left (3 x^6+24 x^5+48 x^4\right ) \log ^2(5)\right )+\left (x^9+12 x^8+48 x^7+64 x^6\right ) \log ^6(5)+\left (-6 x^9-72 x^8-288 x^7-384 x^6\right ) \log ^5(5)+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \log ^4(5)+\left (-20 x^9-240 x^8-960 x^7-1280 x^6\right ) \log ^3(5)+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \log ^2(5)+\left (-6 x^9-72 x^8-288 x^7-384 x^6\right ) \log (5)+\log ^6(5) \log ^3(\log (3))}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {5 x^7-20 x^6+\log (\log (3)) \left (-25 x^4 \log ^6(5)+100 x^4 \log ^5(5)-150 x^4 \log ^4(5)+100 x^4 \log ^3(5)-25 x^4 \log ^2(5)\right )+\left (5 x^7-20 x^6\right ) \log ^6(5)+\left (120 x^6-30 x^7\right ) \left (\log ^5(5)+\log (5)\right )+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (75 x^7-300 x^6\right ) \left (\log ^4(5)+\log ^2(5)\right )}{x^9+12 x^8+48 x^7+64 x^6+\log ^2(\log (3)) \left (\left (3 x^3+12 x^2\right ) \log ^6(5)+\left (-6 x^3-24 x^2\right ) \log ^5(5)+\left (3 x^3+12 x^2\right ) \log ^4(5)\right )+\log (\log (3)) \left (\left (3 x^6+24 x^5+48 x^4\right ) \log ^6(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^5(5)+\left (18 x^6+144 x^5+288 x^4\right ) \log ^4(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^3(5)+\left (3 x^6+24 x^5+48 x^4\right ) \log ^2(5)\right )+\left (x^9+12 x^8+48 x^7+64 x^6\right ) \log ^6(5)+\left (-6 x^9-72 x^8-288 x^7-384 x^6\right ) \left (\log ^5(5)+\log (5)\right )+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \log ^4(5)+\left (-20 x^9-240 x^8-960 x^7-1280 x^6\right ) \log ^3(5)+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \log ^2(5)+\log ^6(5) \log ^3(\log (3))}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {5 x^7-20 x^6+\log (\log (3)) \left (-25 x^4 \log ^6(5)+100 x^4 \log ^5(5)-150 x^4 \log ^4(5)+100 x^4 \log ^3(5)-25 x^4 \log ^2(5)\right )+\left (5 x^7-20 x^6\right ) \log ^6(5)+\left (120 x^6-30 x^7\right ) \left (\log ^5(5)+\log (5)\right )+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (75 x^7-300 x^6\right ) \left (\log ^4(5)+\log ^2(5)\right )}{x^9+12 x^8+48 x^7+64 x^6+\log ^2(\log (3)) \left (\left (3 x^3+12 x^2\right ) \log ^6(5)+\left (-6 x^3-24 x^2\right ) \log ^5(5)+\left (3 x^3+12 x^2\right ) \log ^4(5)\right )+\log (\log (3)) \left (\left (3 x^6+24 x^5+48 x^4\right ) \log ^6(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^5(5)+\left (18 x^6+144 x^5+288 x^4\right ) \log ^4(5)+\left (-12 x^6-96 x^5-192 x^4\right ) \log ^3(5)+\left (3 x^6+24 x^5+48 x^4\right ) \log ^2(5)\right )+\left (x^9+12 x^8+48 x^7+64 x^6\right ) \log ^6(5)+\left (-6 x^9-72 x^8-288 x^7-384 x^6\right ) \left (\log ^5(5)+\log (5)\right )+\left (-20 x^9-240 x^8-960 x^7-1280 x^6\right ) \log ^3(5)+\left (15 x^9+180 x^8+720 x^7+960 x^6\right ) \left (\log ^4(5)+\log ^2(5)\right )+\log ^6(5) \log ^3(\log (3))}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {5 (x-12) (1-\log (5))^2}{x^3 (1-\log (5))^2+4 x^2 (1-\log (5))^2+\log ^2(5) \log (\log (3))}+\frac {10 (1-\log (5))^2 \left (-16 x^2 (1-\log (5))^2 \left (64+\log ^2(5) (64+5 \log (\log (3)))-128 \log (5)\right )+x \log ^2(5) \log (\log (3)) \left (64+\log ^2(5) (64+3 \log (\log (3)))-128 \log (5)\right )-16 \log ^2(5) \log (\log (3)) \left (16+\log ^2(5) (16+\log (\log (3)))-32 \log (5)\right )\right )}{\left (x^3 (1-\log (5))^2+4 x^2 (1-\log (5))^2+\log ^2(5) \log (\log (3))\right )^3}+\frac {5 (1-\log (5))^2 \left (80 x^2 (1-\log (5))^2-x \left (128+\log ^2(5) (128+7 \log (\log (3)))-256 \log (5)\right )+4 \left (128+\frac {1}{4} \log ^2(5) (512+44 \log (\log (3)))-256 \log (5)\right )\right )}{\left (x^3 (1-\log (5))^2+4 x^2 (1-\log (5))^2+\log ^2(5) \log (\log (3))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 x^4 (1-\log (5))^4 \left (x^3 (\log (5)-1)^2-4 x^2 (\log (5)-1)^2-5 \log ^2(5) \log (\log (3))\right )}{\left (x^3 (1-\log (5))^2+4 x^2 (1-\log (5))^2+\log ^2(5) \log (\log (3))\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 (1-\log (5))^4 \int -\frac {x^4 \left (-(1-\log (5))^2 x^3+4 (1-\log (5))^2 x^2+5 \log ^2(5) \log (\log (3))\right )}{\left ((1-\log (5))^2 x^3+4 (1-\log (5))^2 x^2+\log ^2(5) \log (\log (3))\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 (1-\log (5))^4 \int \frac {x^4 \left (-(1-\log (5))^2 x^3+4 (1-\log (5))^2 x^2+5 \log ^2(5) \log (\log (3))\right )}{\left ((1-\log (5))^2 x^3+4 (1-\log (5))^2 x^2+\log ^2(5) \log (\log (3))\right )^3}dx\) |
| \(\Big \downarrow \) 2021 |
| \(\displaystyle -\frac {5 x^5 (1-\log (5))^4}{\left (x^3 (1-\log (5))^2+4 x^2 (1-\log (5))^2+\log ^2(5) \log (\log (3))\right )^2}\) |
Input:
Int[(-20*x^6 + 5*x^7 + (120*x^6 - 30*x^7)*Log[5] + (-300*x^6 + 75*x^7)*Log [5]^2 + (400*x^6 - 100*x^7)*Log[5]^3 + (-300*x^6 + 75*x^7)*Log[5]^4 + (120 *x^6 - 30*x^7)*Log[5]^5 + (-20*x^6 + 5*x^7)*Log[5]^6 + (-25*x^4*Log[5]^2 + 100*x^4*Log[5]^3 - 150*x^4*Log[5]^4 + 100*x^4*Log[5]^5 - 25*x^4*Log[5]^6) *Log[Log[3]])/(64*x^6 + 48*x^7 + 12*x^8 + x^9 + (-384*x^6 - 288*x^7 - 72*x ^8 - 6*x^9)*Log[5] + (960*x^6 + 720*x^7 + 180*x^8 + 15*x^9)*Log[5]^2 + (-1 280*x^6 - 960*x^7 - 240*x^8 - 20*x^9)*Log[5]^3 + (960*x^6 + 720*x^7 + 180* x^8 + 15*x^9)*Log[5]^4 + (-384*x^6 - 288*x^7 - 72*x^8 - 6*x^9)*Log[5]^5 + (64*x^6 + 48*x^7 + 12*x^8 + x^9)*Log[5]^6 + ((48*x^4 + 24*x^5 + 3*x^6)*Log [5]^2 + (-192*x^4 - 96*x^5 - 12*x^6)*Log[5]^3 + (288*x^4 + 144*x^5 + 18*x^ 6)*Log[5]^4 + (-192*x^4 - 96*x^5 - 12*x^6)*Log[5]^5 + (48*x^4 + 24*x^5 + 3 *x^6)*Log[5]^6)*Log[Log[3]] + ((12*x^2 + 3*x^3)*Log[5]^4 + (-24*x^2 - 6*x^ 3)*Log[5]^5 + (12*x^2 + 3*x^3)*Log[5]^6)*Log[Log[3]]^2 + Log[5]^6*Log[Log[ 3]]^3),x]
Output:
(-5*x^5*(1 - Log[5])^4)/(4*x^2*(1 - Log[5])^2 + x^3*(1 - Log[5])^2 + Log[5 ]^2*Log[Log[3]])^2
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x ]}, Simp[Coeff[Pp, x, p]*x^(p - q + 1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp , Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; Free Q[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(26)=52\).
Time = 0.95 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.04
| method | result | size |
| norman | \(\frac {\left (-5 \ln \left (5\right )^{4}+20 \ln \left (5\right )^{3}-30 \ln \left (5\right )^{2}+20 \ln \left (5\right )-5\right ) x^{5}}{\left (x^{3} \ln \left (5\right )^{2}+4 x^{2} \ln \left (5\right )^{2}-2 x^{3} \ln \left (5\right )+\ln \left (5\right )^{2} \ln \left (\ln \left (3\right )\right )-8 x^{2} \ln \left (5\right )+x^{3}+4 x^{2}\right )^{2}}\) | \(79\) |
| default | \(-\frac {\left (5 \ln \left (5\right )^{4}-20 \ln \left (5\right )^{3}+30 \ln \left (5\right )^{2}-20 \ln \left (5\right )+5\right ) x^{5}}{\left (x^{3} \ln \left (5\right )^{2}+4 x^{2} \ln \left (5\right )^{2}-2 x^{3} \ln \left (5\right )+\ln \left (5\right )^{2} \ln \left (\ln \left (3\right )\right )-8 x^{2} \ln \left (5\right )+x^{3}+4 x^{2}\right )^{2}}\) | \(80\) |
| gosper | \(-\frac {5 x^{5} \left (\ln \left (5\right )^{4}-4 \ln \left (5\right )^{3}+6 \ln \left (5\right )^{2}-4 \ln \left (5\right )+1\right )}{\ln \left (5\right )^{4} x^{6}+8 \ln \left (5\right )^{4} x^{5}-4 \ln \left (5\right )^{3} x^{6}+2 \ln \left (5\right )^{4} x^{3} \ln \left (\ln \left (3\right )\right )+16 x^{4} \ln \left (5\right )^{4}-32 x^{5} \ln \left (5\right )^{3}+6 x^{6} \ln \left (5\right )^{2}+8 x^{2} \ln \left (5\right )^{4} \ln \left (\ln \left (3\right )\right )-4 \ln \left (5\right )^{3} x^{3} \ln \left (\ln \left (3\right )\right )-64 x^{4} \ln \left (5\right )^{3}+48 x^{5} \ln \left (5\right )^{2}-4 x^{6} \ln \left (5\right )+\ln \left (5\right )^{4} \ln \left (\ln \left (3\right )\right )^{2}-16 \ln \left (5\right )^{3} x^{2} \ln \left (\ln \left (3\right )\right )+2 x^{3} \ln \left (5\right )^{2} \ln \left (\ln \left (3\right )\right )+96 x^{4} \ln \left (5\right )^{2}-32 x^{5} \ln \left (5\right )+x^{6}+8 \ln \left (5\right )^{2} x^{2} \ln \left (\ln \left (3\right )\right )-64 x^{4} \ln \left (5\right )+8 x^{5}+16 x^{4}}\) | \(227\) |
| risch | \(\frac {\left (-5 \ln \left (5\right )^{4}+20 \ln \left (5\right )^{3}-30 \ln \left (5\right )^{2}+20 \ln \left (5\right )-5\right ) x^{5}}{\ln \left (5\right )^{4} x^{6}+8 \ln \left (5\right )^{4} x^{5}-4 \ln \left (5\right )^{3} x^{6}+2 \ln \left (5\right )^{4} x^{3} \ln \left (\ln \left (3\right )\right )+16 x^{4} \ln \left (5\right )^{4}-32 x^{5} \ln \left (5\right )^{3}+6 x^{6} \ln \left (5\right )^{2}+8 x^{2} \ln \left (5\right )^{4} \ln \left (\ln \left (3\right )\right )-4 \ln \left (5\right )^{3} x^{3} \ln \left (\ln \left (3\right )\right )-64 x^{4} \ln \left (5\right )^{3}+48 x^{5} \ln \left (5\right )^{2}-4 x^{6} \ln \left (5\right )+\ln \left (5\right )^{4} \ln \left (\ln \left (3\right )\right )^{2}-16 \ln \left (5\right )^{3} x^{2} \ln \left (\ln \left (3\right )\right )+2 x^{3} \ln \left (5\right )^{2} \ln \left (\ln \left (3\right )\right )+96 x^{4} \ln \left (5\right )^{2}-32 x^{5} \ln \left (5\right )+x^{6}+8 \ln \left (5\right )^{2} x^{2} \ln \left (\ln \left (3\right )\right )-64 x^{4} \ln \left (5\right )+8 x^{5}+16 x^{4}}\) | \(228\) |
| parallelrisch | \(\frac {5 \ln \left (5\right )^{8} x^{6}+140 \ln \left (5\right )^{6} x^{6}+2240 x^{4} \ln \left (5\right )^{6}-4480 x^{4} \ln \left (5\right )^{5}-40 x^{6} \ln \left (5\right )+140 x^{6} \ln \left (5\right )^{2}-4480 x^{4} \ln \left (5\right )^{3}+5600 x^{4} \ln \left (5\right )^{4}+2240 x^{4} \ln \left (5\right )^{2}-640 x^{4} \ln \left (5\right )+30 \ln \left (5\right )^{6} \ln \left (\ln \left (3\right )\right )^{2}-20 \ln \left (5\right )^{5} \ln \left (\ln \left (3\right )\right )^{2}+5 \ln \left (5\right )^{8} \ln \left (\ln \left (3\right )\right )^{2}-20 \ln \left (5\right )^{7} \ln \left (\ln \left (3\right )\right )^{2}+5 \ln \left (5\right )^{4} \ln \left (\ln \left (3\right )\right )^{2}+5 x^{6}+80 x^{4}+80 \ln \left (5\right )^{8} x^{4}+350 \ln \left (5\right )^{4} x^{6}+150 \ln \left (5\right )^{4} x^{3} \ln \left (\ln \left (3\right )\right )-200 \ln \left (5\right )^{5} \ln \left (\ln \left (3\right )\right ) x^{3}+150 \ln \left (5\right )^{6} \ln \left (\ln \left (3\right )\right ) x^{3}-280 \ln \left (5\right )^{5} x^{6}+10 x^{3} \ln \left (5\right )^{2} \ln \left (\ln \left (3\right )\right )-60 \ln \left (5\right )^{3} x^{3} \ln \left (\ln \left (3\right )\right )+600 x^{2} \ln \left (5\right )^{4} \ln \left (\ln \left (3\right )\right )+40 \ln \left (5\right )^{2} x^{2} \ln \left (\ln \left (3\right )\right )-240 \ln \left (5\right )^{3} x^{2} \ln \left (\ln \left (3\right )\right )+600 \ln \left (5\right )^{6} \ln \left (\ln \left (3\right )\right ) x^{2}-800 \ln \left (5\right )^{5} \ln \left (\ln \left (3\right )\right ) x^{2}+10 \ln \left (5\right )^{8} \ln \left (\ln \left (3\right )\right ) x^{3}+40 \ln \left (5\right )^{8} \ln \left (\ln \left (3\right )\right ) x^{2}-60 \ln \left (5\right )^{7} \ln \left (\ln \left (3\right )\right ) x^{3}-240 \ln \left (5\right )^{7} \ln \left (\ln \left (3\right )\right ) x^{2}-40 \ln \left (5\right )^{7} x^{6}-640 \ln \left (5\right )^{7} x^{4}-280 \ln \left (5\right )^{3} x^{6}}{8 \left (\ln \left (5\right )^{2}-2 \ln \left (5\right )+1\right )^{2} \left (\ln \left (5\right )^{4} x^{6}+8 \ln \left (5\right )^{4} x^{5}-4 \ln \left (5\right )^{3} x^{6}+2 \ln \left (5\right )^{4} x^{3} \ln \left (\ln \left (3\right )\right )+16 x^{4} \ln \left (5\right )^{4}-32 x^{5} \ln \left (5\right )^{3}+6 x^{6} \ln \left (5\right )^{2}+8 x^{2} \ln \left (5\right )^{4} \ln \left (\ln \left (3\right )\right )-4 \ln \left (5\right )^{3} x^{3} \ln \left (\ln \left (3\right )\right )-64 x^{4} \ln \left (5\right )^{3}+48 x^{5} \ln \left (5\right )^{2}-4 x^{6} \ln \left (5\right )+\ln \left (5\right )^{4} \ln \left (\ln \left (3\right )\right )^{2}-16 \ln \left (5\right )^{3} x^{2} \ln \left (\ln \left (3\right )\right )+2 x^{3} \ln \left (5\right )^{2} \ln \left (\ln \left (3\right )\right )+96 x^{4} \ln \left (5\right )^{2}-32 x^{5} \ln \left (5\right )+x^{6}+8 \ln \left (5\right )^{2} x^{2} \ln \left (\ln \left (3\right )\right )-64 x^{4} \ln \left (5\right )+8 x^{5}+16 x^{4}\right )}\) | \(588\) |
Input:
int(((-25*x^4*ln(5)^6+100*x^4*ln(5)^5-150*x^4*ln(5)^4+100*x^4*ln(5)^3-25*x ^4*ln(5)^2)*ln(ln(3))+(5*x^7-20*x^6)*ln(5)^6+(-30*x^7+120*x^6)*ln(5)^5+(75 *x^7-300*x^6)*ln(5)^4+(-100*x^7+400*x^6)*ln(5)^3+(75*x^7-300*x^6)*ln(5)^2+ (-30*x^7+120*x^6)*ln(5)+5*x^7-20*x^6)/(ln(5)^6*ln(ln(3))^3+((3*x^3+12*x^2) *ln(5)^6+(-6*x^3-24*x^2)*ln(5)^5+(3*x^3+12*x^2)*ln(5)^4)*ln(ln(3))^2+((3*x ^6+24*x^5+48*x^4)*ln(5)^6+(-12*x^6-96*x^5-192*x^4)*ln(5)^5+(18*x^6+144*x^5 +288*x^4)*ln(5)^4+(-12*x^6-96*x^5-192*x^4)*ln(5)^3+(3*x^6+24*x^5+48*x^4)*l n(5)^2)*ln(ln(3))+(x^9+12*x^8+48*x^7+64*x^6)*ln(5)^6+(-6*x^9-72*x^8-288*x^ 7-384*x^6)*ln(5)^5+(15*x^9+180*x^8+720*x^7+960*x^6)*ln(5)^4+(-20*x^9-240*x ^8-960*x^7-1280*x^6)*ln(5)^3+(15*x^9+180*x^8+720*x^7+960*x^6)*ln(5)^2+(-6* x^9-72*x^8-288*x^7-384*x^6)*ln(5)+x^9+12*x^8+48*x^7+64*x^6),x,method=_RETU RNVERBOSE)
Output:
(-5*ln(5)^4+20*ln(5)^3-30*ln(5)^2+20*ln(5)-5)*x^5/(x^3*ln(5)^2+4*x^2*ln(5) ^2-2*x^3*ln(5)+ln(5)^2*ln(ln(3))-8*x^2*ln(5)+x^3+4*x^2)^2
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 7.35 \[ \int \frac {-20 x^6+5 x^7+\left (120 x^6-30 x^7\right ) \log (5)+\left (-300 x^6+75 x^7\right ) \log ^2(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (-300 x^6+75 x^7\right ) \log ^4(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (-20 x^6+5 x^7\right ) \log ^6(5)+\left (-25 x^4 \log ^2(5)+100 x^4 \log ^3(5)-150 x^4 \log ^4(5)+100 x^4 \log ^5(5)-25 x^4 \log ^6(5)\right ) \log (\log (3))}{64 x^6+48 x^7+12 x^8+x^9+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log (5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^2(5)+\left (-1280 x^6-960 x^7-240 x^8-20 x^9\right ) \log ^3(5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^4(5)+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log ^5(5)+\left (64 x^6+48 x^7+12 x^8+x^9\right ) \log ^6(5)+\left (\left (48 x^4+24 x^5+3 x^6\right ) \log ^2(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^3(5)+\left (288 x^4+144 x^5+18 x^6\right ) \log ^4(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^5(5)+\left (48 x^4+24 x^5+3 x^6\right ) \log ^6(5)\right ) \log (\log (3))+\left (\left (12 x^2+3 x^3\right ) \log ^4(5)+\left (-24 x^2-6 x^3\right ) \log ^5(5)+\left (12 x^2+3 x^3\right ) \log ^6(5)\right ) \log ^2(\log (3))+\log ^6(5) \log ^3(\log (3))} \, dx=-\frac {5 \, {\left (x^{5} \log \left (5\right )^{4} - 4 \, x^{5} \log \left (5\right )^{3} + 6 \, x^{5} \log \left (5\right )^{2} - 4 \, x^{5} \log \left (5\right ) + x^{5}\right )}}{x^{6} + \log \left (5\right )^{4} \log \left (\log \left (3\right )\right )^{2} + 8 \, x^{5} + {\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} \log \left (5\right )^{4} + 16 \, x^{4} - 4 \, {\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} \log \left (5\right )^{3} + 6 \, {\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} \log \left (5\right )^{2} - 4 \, {\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} \log \left (5\right ) + 2 \, {\left ({\left (x^{3} + 4 \, x^{2}\right )} \log \left (5\right )^{4} - 2 \, {\left (x^{3} + 4 \, x^{2}\right )} \log \left (5\right )^{3} + {\left (x^{3} + 4 \, x^{2}\right )} \log \left (5\right )^{2}\right )} \log \left (\log \left (3\right )\right )} \] Input:
integrate(((-25*x^4*log(5)^6+100*x^4*log(5)^5-150*x^4*log(5)^4+100*x^4*log (5)^3-25*x^4*log(5)^2)*log(log(3))+(5*x^7-20*x^6)*log(5)^6+(-30*x^7+120*x^ 6)*log(5)^5+(75*x^7-300*x^6)*log(5)^4+(-100*x^7+400*x^6)*log(5)^3+(75*x^7- 300*x^6)*log(5)^2+(-30*x^7+120*x^6)*log(5)+5*x^7-20*x^6)/(log(5)^6*log(log (3))^3+((3*x^3+12*x^2)*log(5)^6+(-6*x^3-24*x^2)*log(5)^5+(3*x^3+12*x^2)*lo g(5)^4)*log(log(3))^2+((3*x^6+24*x^5+48*x^4)*log(5)^6+(-12*x^6-96*x^5-192* x^4)*log(5)^5+(18*x^6+144*x^5+288*x^4)*log(5)^4+(-12*x^6-96*x^5-192*x^4)*l og(5)^3+(3*x^6+24*x^5+48*x^4)*log(5)^2)*log(log(3))+(x^9+12*x^8+48*x^7+64* x^6)*log(5)^6+(-6*x^9-72*x^8-288*x^7-384*x^6)*log(5)^5+(15*x^9+180*x^8+720 *x^7+960*x^6)*log(5)^4+(-20*x^9-240*x^8-960*x^7-1280*x^6)*log(5)^3+(15*x^9 +180*x^8+720*x^7+960*x^6)*log(5)^2+(-6*x^9-72*x^8-288*x^7-384*x^6)*log(5)+ x^9+12*x^8+48*x^7+64*x^6),x, algorithm="fricas")
Output:
-5*(x^5*log(5)^4 - 4*x^5*log(5)^3 + 6*x^5*log(5)^2 - 4*x^5*log(5) + x^5)/( x^6 + log(5)^4*log(log(3))^2 + 8*x^5 + (x^6 + 8*x^5 + 16*x^4)*log(5)^4 + 1 6*x^4 - 4*(x^6 + 8*x^5 + 16*x^4)*log(5)^3 + 6*(x^6 + 8*x^5 + 16*x^4)*log(5 )^2 - 4*(x^6 + 8*x^5 + 16*x^4)*log(5) + 2*((x^3 + 4*x^2)*log(5)^4 - 2*(x^3 + 4*x^2)*log(5)^3 + (x^3 + 4*x^2)*log(5)^2)*log(log(3)))
Timed out. \[ \int \frac {-20 x^6+5 x^7+\left (120 x^6-30 x^7\right ) \log (5)+\left (-300 x^6+75 x^7\right ) \log ^2(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (-300 x^6+75 x^7\right ) \log ^4(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (-20 x^6+5 x^7\right ) \log ^6(5)+\left (-25 x^4 \log ^2(5)+100 x^4 \log ^3(5)-150 x^4 \log ^4(5)+100 x^4 \log ^5(5)-25 x^4 \log ^6(5)\right ) \log (\log (3))}{64 x^6+48 x^7+12 x^8+x^9+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log (5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^2(5)+\left (-1280 x^6-960 x^7-240 x^8-20 x^9\right ) \log ^3(5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^4(5)+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log ^5(5)+\left (64 x^6+48 x^7+12 x^8+x^9\right ) \log ^6(5)+\left (\left (48 x^4+24 x^5+3 x^6\right ) \log ^2(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^3(5)+\left (288 x^4+144 x^5+18 x^6\right ) \log ^4(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^5(5)+\left (48 x^4+24 x^5+3 x^6\right ) \log ^6(5)\right ) \log (\log (3))+\left (\left (12 x^2+3 x^3\right ) \log ^4(5)+\left (-24 x^2-6 x^3\right ) \log ^5(5)+\left (12 x^2+3 x^3\right ) \log ^6(5)\right ) \log ^2(\log (3))+\log ^6(5) \log ^3(\log (3))} \, dx=\text {Timed out} \] Input:
integrate(((-25*x**4*ln(5)**6+100*x**4*ln(5)**5-150*x**4*ln(5)**4+100*x**4 *ln(5)**3-25*x**4*ln(5)**2)*ln(ln(3))+(5*x**7-20*x**6)*ln(5)**6+(-30*x**7+ 120*x**6)*ln(5)**5+(75*x**7-300*x**6)*ln(5)**4+(-100*x**7+400*x**6)*ln(5)* *3+(75*x**7-300*x**6)*ln(5)**2+(-30*x**7+120*x**6)*ln(5)+5*x**7-20*x**6)/( ln(5)**6*ln(ln(3))**3+((3*x**3+12*x**2)*ln(5)**6+(-6*x**3-24*x**2)*ln(5)** 5+(3*x**3+12*x**2)*ln(5)**4)*ln(ln(3))**2+((3*x**6+24*x**5+48*x**4)*ln(5)* *6+(-12*x**6-96*x**5-192*x**4)*ln(5)**5+(18*x**6+144*x**5+288*x**4)*ln(5)* *4+(-12*x**6-96*x**5-192*x**4)*ln(5)**3+(3*x**6+24*x**5+48*x**4)*ln(5)**2) *ln(ln(3))+(x**9+12*x**8+48*x**7+64*x**6)*ln(5)**6+(-6*x**9-72*x**8-288*x* *7-384*x**6)*ln(5)**5+(15*x**9+180*x**8+720*x**7+960*x**6)*ln(5)**4+(-20*x **9-240*x**8-960*x**7-1280*x**6)*ln(5)**3+(15*x**9+180*x**8+720*x**7+960*x **6)*ln(5)**2+(-6*x**9-72*x**8-288*x**7-384*x**6)*ln(5)+x**9+12*x**8+48*x* *7+64*x**6),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (25) = 50\).
Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 6.38 \[ \int \frac {-20 x^6+5 x^7+\left (120 x^6-30 x^7\right ) \log (5)+\left (-300 x^6+75 x^7\right ) \log ^2(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (-300 x^6+75 x^7\right ) \log ^4(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (-20 x^6+5 x^7\right ) \log ^6(5)+\left (-25 x^4 \log ^2(5)+100 x^4 \log ^3(5)-150 x^4 \log ^4(5)+100 x^4 \log ^5(5)-25 x^4 \log ^6(5)\right ) \log (\log (3))}{64 x^6+48 x^7+12 x^8+x^9+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log (5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^2(5)+\left (-1280 x^6-960 x^7-240 x^8-20 x^9\right ) \log ^3(5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^4(5)+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log ^5(5)+\left (64 x^6+48 x^7+12 x^8+x^9\right ) \log ^6(5)+\left (\left (48 x^4+24 x^5+3 x^6\right ) \log ^2(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^3(5)+\left (288 x^4+144 x^5+18 x^6\right ) \log ^4(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^5(5)+\left (48 x^4+24 x^5+3 x^6\right ) \log ^6(5)\right ) \log (\log (3))+\left (\left (12 x^2+3 x^3\right ) \log ^4(5)+\left (-24 x^2-6 x^3\right ) \log ^5(5)+\left (12 x^2+3 x^3\right ) \log ^6(5)\right ) \log ^2(\log (3))+\log ^6(5) \log ^3(\log (3))} \, dx=-\frac {5 \, {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} + 6 \, \log \left (5\right )^{2} - 4 \, \log \left (5\right ) + 1\right )} x^{5}}{{\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} + 6 \, \log \left (5\right )^{2} - 4 \, \log \left (5\right ) + 1\right )} x^{6} + 8 \, {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} + 6 \, \log \left (5\right )^{2} - 4 \, \log \left (5\right ) + 1\right )} x^{5} + \log \left (5\right )^{4} \log \left (\log \left (3\right )\right )^{2} + 16 \, {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} + 6 \, \log \left (5\right )^{2} - 4 \, \log \left (5\right ) + 1\right )} x^{4} + 2 \, {\left (\log \left (5\right )^{4} - 2 \, \log \left (5\right )^{3} + \log \left (5\right )^{2}\right )} x^{3} \log \left (\log \left (3\right )\right ) + 8 \, {\left (\log \left (5\right )^{4} - 2 \, \log \left (5\right )^{3} + \log \left (5\right )^{2}\right )} x^{2} \log \left (\log \left (3\right )\right )} \] Input:
integrate(((-25*x^4*log(5)^6+100*x^4*log(5)^5-150*x^4*log(5)^4+100*x^4*log (5)^3-25*x^4*log(5)^2)*log(log(3))+(5*x^7-20*x^6)*log(5)^6+(-30*x^7+120*x^ 6)*log(5)^5+(75*x^7-300*x^6)*log(5)^4+(-100*x^7+400*x^6)*log(5)^3+(75*x^7- 300*x^6)*log(5)^2+(-30*x^7+120*x^6)*log(5)+5*x^7-20*x^6)/(log(5)^6*log(log (3))^3+((3*x^3+12*x^2)*log(5)^6+(-6*x^3-24*x^2)*log(5)^5+(3*x^3+12*x^2)*lo g(5)^4)*log(log(3))^2+((3*x^6+24*x^5+48*x^4)*log(5)^6+(-12*x^6-96*x^5-192* x^4)*log(5)^5+(18*x^6+144*x^5+288*x^4)*log(5)^4+(-12*x^6-96*x^5-192*x^4)*l og(5)^3+(3*x^6+24*x^5+48*x^4)*log(5)^2)*log(log(3))+(x^9+12*x^8+48*x^7+64* x^6)*log(5)^6+(-6*x^9-72*x^8-288*x^7-384*x^6)*log(5)^5+(15*x^9+180*x^8+720 *x^7+960*x^6)*log(5)^4+(-20*x^9-240*x^8-960*x^7-1280*x^6)*log(5)^3+(15*x^9 +180*x^8+720*x^7+960*x^6)*log(5)^2+(-6*x^9-72*x^8-288*x^7-384*x^6)*log(5)+ x^9+12*x^8+48*x^7+64*x^6),x, algorithm="maxima")
Output:
-5*(log(5)^4 - 4*log(5)^3 + 6*log(5)^2 - 4*log(5) + 1)*x^5/((log(5)^4 - 4* log(5)^3 + 6*log(5)^2 - 4*log(5) + 1)*x^6 + 8*(log(5)^4 - 4*log(5)^3 + 6*l og(5)^2 - 4*log(5) + 1)*x^5 + log(5)^4*log(log(3))^2 + 16*(log(5)^4 - 4*lo g(5)^3 + 6*log(5)^2 - 4*log(5) + 1)*x^4 + 2*(log(5)^4 - 2*log(5)^3 + log(5 )^2)*x^3*log(log(3)) + 8*(log(5)^4 - 2*log(5)^3 + log(5)^2)*x^2*log(log(3) ))
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (25) = 50\).
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.42 \[ \int \frac {-20 x^6+5 x^7+\left (120 x^6-30 x^7\right ) \log (5)+\left (-300 x^6+75 x^7\right ) \log ^2(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (-300 x^6+75 x^7\right ) \log ^4(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (-20 x^6+5 x^7\right ) \log ^6(5)+\left (-25 x^4 \log ^2(5)+100 x^4 \log ^3(5)-150 x^4 \log ^4(5)+100 x^4 \log ^5(5)-25 x^4 \log ^6(5)\right ) \log (\log (3))}{64 x^6+48 x^7+12 x^8+x^9+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log (5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^2(5)+\left (-1280 x^6-960 x^7-240 x^8-20 x^9\right ) \log ^3(5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^4(5)+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log ^5(5)+\left (64 x^6+48 x^7+12 x^8+x^9\right ) \log ^6(5)+\left (\left (48 x^4+24 x^5+3 x^6\right ) \log ^2(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^3(5)+\left (288 x^4+144 x^5+18 x^6\right ) \log ^4(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^5(5)+\left (48 x^4+24 x^5+3 x^6\right ) \log ^6(5)\right ) \log (\log (3))+\left (\left (12 x^2+3 x^3\right ) \log ^4(5)+\left (-24 x^2-6 x^3\right ) \log ^5(5)+\left (12 x^2+3 x^3\right ) \log ^6(5)\right ) \log ^2(\log (3))+\log ^6(5) \log ^3(\log (3))} \, dx=-\frac {5 \, {\left (x^{5} \log \left (5\right )^{4} - 4 \, x^{5} \log \left (5\right )^{3} + 6 \, x^{5} \log \left (5\right )^{2} - 4 \, x^{5} \log \left (5\right ) + x^{5}\right )}}{{\left (x^{3} \log \left (5\right )^{2} - 2 \, x^{3} \log \left (5\right ) + 4 \, x^{2} \log \left (5\right )^{2} + x^{3} - 8 \, x^{2} \log \left (5\right ) + \log \left (5\right )^{2} \log \left (\log \left (3\right )\right ) + 4 \, x^{2}\right )}^{2}} \] Input:
integrate(((-25*x^4*log(5)^6+100*x^4*log(5)^5-150*x^4*log(5)^4+100*x^4*log (5)^3-25*x^4*log(5)^2)*log(log(3))+(5*x^7-20*x^6)*log(5)^6+(-30*x^7+120*x^ 6)*log(5)^5+(75*x^7-300*x^6)*log(5)^4+(-100*x^7+400*x^6)*log(5)^3+(75*x^7- 300*x^6)*log(5)^2+(-30*x^7+120*x^6)*log(5)+5*x^7-20*x^6)/(log(5)^6*log(log (3))^3+((3*x^3+12*x^2)*log(5)^6+(-6*x^3-24*x^2)*log(5)^5+(3*x^3+12*x^2)*lo g(5)^4)*log(log(3))^2+((3*x^6+24*x^5+48*x^4)*log(5)^6+(-12*x^6-96*x^5-192* x^4)*log(5)^5+(18*x^6+144*x^5+288*x^4)*log(5)^4+(-12*x^6-96*x^5-192*x^4)*l og(5)^3+(3*x^6+24*x^5+48*x^4)*log(5)^2)*log(log(3))+(x^9+12*x^8+48*x^7+64* x^6)*log(5)^6+(-6*x^9-72*x^8-288*x^7-384*x^6)*log(5)^5+(15*x^9+180*x^8+720 *x^7+960*x^6)*log(5)^4+(-20*x^9-240*x^8-960*x^7-1280*x^6)*log(5)^3+(15*x^9 +180*x^8+720*x^7+960*x^6)*log(5)^2+(-6*x^9-72*x^8-288*x^7-384*x^6)*log(5)+ x^9+12*x^8+48*x^7+64*x^6),x, algorithm="giac")
Output:
-5*(x^5*log(5)^4 - 4*x^5*log(5)^3 + 6*x^5*log(5)^2 - 4*x^5*log(5) + x^5)/( x^3*log(5)^2 - 2*x^3*log(5) + 4*x^2*log(5)^2 + x^3 - 8*x^2*log(5) + log(5) ^2*log(log(3)) + 4*x^2)^2
Time = 8.95 (sec) , antiderivative size = 33761, normalized size of antiderivative = 1298.50 \[ \int \frac {-20 x^6+5 x^7+\left (120 x^6-30 x^7\right ) \log (5)+\left (-300 x^6+75 x^7\right ) \log ^2(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (-300 x^6+75 x^7\right ) \log ^4(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (-20 x^6+5 x^7\right ) \log ^6(5)+\left (-25 x^4 \log ^2(5)+100 x^4 \log ^3(5)-150 x^4 \log ^4(5)+100 x^4 \log ^5(5)-25 x^4 \log ^6(5)\right ) \log (\log (3))}{64 x^6+48 x^7+12 x^8+x^9+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log (5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^2(5)+\left (-1280 x^6-960 x^7-240 x^8-20 x^9\right ) \log ^3(5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^4(5)+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log ^5(5)+\left (64 x^6+48 x^7+12 x^8+x^9\right ) \log ^6(5)+\left (\left (48 x^4+24 x^5+3 x^6\right ) \log ^2(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^3(5)+\left (288 x^4+144 x^5+18 x^6\right ) \log ^4(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^5(5)+\left (48 x^4+24 x^5+3 x^6\right ) \log ^6(5)\right ) \log (\log (3))+\left (\left (12 x^2+3 x^3\right ) \log ^4(5)+\left (-24 x^2-6 x^3\right ) \log ^5(5)+\left (12 x^2+3 x^3\right ) \log ^6(5)\right ) \log ^2(\log (3))+\log ^6(5) \log ^3(\log (3))} \, dx=\text {Too large to display} \] Input:
int(-(log(log(3))*(25*x^4*log(5)^2 - 100*x^4*log(5)^3 + 150*x^4*log(5)^4 - 100*x^4*log(5)^5 + 25*x^4*log(5)^6) - log(5)*(120*x^6 - 30*x^7) + 20*x^6 - 5*x^7 + log(5)^6*(20*x^6 - 5*x^7) - log(5)^5*(120*x^6 - 30*x^7) + log(5) ^2*(300*x^6 - 75*x^7) + log(5)^4*(300*x^6 - 75*x^7) - log(5)^3*(400*x^6 - 100*x^7))/(log(5)^6*(64*x^6 + 48*x^7 + 12*x^8 + x^9) - log(5)*(384*x^6 + 2 88*x^7 + 72*x^8 + 6*x^9) + log(5)^6*log(log(3))^3 - log(5)^5*(384*x^6 + 28 8*x^7 + 72*x^8 + 6*x^9) + log(5)^2*(960*x^6 + 720*x^7 + 180*x^8 + 15*x^9) + log(5)^4*(960*x^6 + 720*x^7 + 180*x^8 + 15*x^9) - log(5)^3*(1280*x^6 + 9 60*x^7 + 240*x^8 + 20*x^9) + log(log(3))*(log(5)^2*(48*x^4 + 24*x^5 + 3*x^ 6) + log(5)^6*(48*x^4 + 24*x^5 + 3*x^6) - log(5)^3*(192*x^4 + 96*x^5 + 12* x^6) - log(5)^5*(192*x^4 + 96*x^5 + 12*x^6) + log(5)^4*(288*x^4 + 144*x^5 + 18*x^6)) + 64*x^6 + 48*x^7 + 12*x^8 + x^9 + log(log(3))^2*(log(5)^4*(12* x^2 + 3*x^3) + log(5)^6*(12*x^2 + 3*x^3) - log(5)^5*(24*x^2 + 6*x^3))),x)
Output:
((32400*(2*log(5)^7*log(log(3))^3 - 4*log(5)^8*log(log(3))^3 + 2*log(5)^9* log(log(3))^3 - log(5)^6*log(25)*log(log(3))^3 + 2*log(5)^7*log(25)*log(lo g(3))^3 - log(5)^8*log(25)*log(log(3))^3))/(13824*log(5)^2*log(log(3)) - 2 62144*log(5) - 27648*log(5)^3*log(log(3)) + 13824*log(5)^4*log(log(3)) + 7 29*log(5)^4*log(log(3))^2 + 393216*log(5)^2 - 262144*log(5)^3 + 65536*log( 5)^4 + 65536) + (300*x^4*(2560*log(5)^3*log(log(3)) - 15360*log(5)^4*log(l og(3)) + 38400*log(5)^5*log(log(3)) - 51200*log(5)^6*log(log(3)) + 38400*l og(5)^7*log(log(3)) - 15360*log(5)^8*log(log(3)) + 2560*log(5)^9*log(log(3 )) - 54*log(5)^5*log(log(3))^2 + 216*log(5)^6*log(log(3))^2 - 324*log(5)^7 *log(log(3))^2 + 216*log(5)^8*log(log(3))^2 - 54*log(5)^9*log(log(3))^2 - 1280*log(5)^2*log(25)*log(log(3)) + 7680*log(5)^3*log(25)*log(log(3)) - 19 200*log(5)^4*log(25)*log(log(3)) + 25600*log(5)^5*log(25)*log(log(3)) - 19 200*log(5)^6*log(25)*log(log(3)) + 7680*log(5)^7*log(25)*log(log(3)) - 128 0*log(5)^8*log(25)*log(log(3)) + 27*log(5)^4*log(25)*log(log(3))^2 - 108*l og(5)^5*log(25)*log(log(3))^2 + 162*log(5)^6*log(25)*log(log(3))^2 - 108*l og(5)^7*log(25)*log(log(3))^2 + 27*log(5)^8*log(25)*log(log(3))^2))/(13824 *log(5)^2*log(log(3)) - 262144*log(5) - 27648*log(5)^3*log(log(3)) + 13824 *log(5)^4*log(log(3)) + 729*log(5)^4*log(log(3))^2 + 393216*log(5)^2 - 262 144*log(5)^3 + 65536*log(5)^4 + 65536) + (300*x*(256*log(5)^5*log(log(3))^ 2 - 1024*log(5)^6*log(log(3))^2 + 1536*log(5)^7*log(log(3))^2 - 54*log(...
Time = 0.17 (sec) , antiderivative size = 228, normalized size of antiderivative = 8.77 \[ \int \frac {-20 x^6+5 x^7+\left (120 x^6-30 x^7\right ) \log (5)+\left (-300 x^6+75 x^7\right ) \log ^2(5)+\left (400 x^6-100 x^7\right ) \log ^3(5)+\left (-300 x^6+75 x^7\right ) \log ^4(5)+\left (120 x^6-30 x^7\right ) \log ^5(5)+\left (-20 x^6+5 x^7\right ) \log ^6(5)+\left (-25 x^4 \log ^2(5)+100 x^4 \log ^3(5)-150 x^4 \log ^4(5)+100 x^4 \log ^5(5)-25 x^4 \log ^6(5)\right ) \log (\log (3))}{64 x^6+48 x^7+12 x^8+x^9+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log (5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^2(5)+\left (-1280 x^6-960 x^7-240 x^8-20 x^9\right ) \log ^3(5)+\left (960 x^6+720 x^7+180 x^8+15 x^9\right ) \log ^4(5)+\left (-384 x^6-288 x^7-72 x^8-6 x^9\right ) \log ^5(5)+\left (64 x^6+48 x^7+12 x^8+x^9\right ) \log ^6(5)+\left (\left (48 x^4+24 x^5+3 x^6\right ) \log ^2(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^3(5)+\left (288 x^4+144 x^5+18 x^6\right ) \log ^4(5)+\left (-192 x^4-96 x^5-12 x^6\right ) \log ^5(5)+\left (48 x^4+24 x^5+3 x^6\right ) \log ^6(5)\right ) \log (\log (3))+\left (\left (12 x^2+3 x^3\right ) \log ^4(5)+\left (-24 x^2-6 x^3\right ) \log ^5(5)+\left (12 x^2+3 x^3\right ) \log ^6(5)\right ) \log ^2(\log (3))+\log ^6(5) \log ^3(\log (3))} \, dx=\frac {5 x^{5} \left (-\mathrm {log}\left (5\right )^{4}+4 \mathrm {log}\left (5\right )^{3}-6 \mathrm {log}\left (5\right )^{2}+4 \,\mathrm {log}\left (5\right )-1\right )}{\mathrm {log}\left (\mathrm {log}\left (3\right )\right )^{2} \mathrm {log}\left (5\right )^{4}+2 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right ) \mathrm {log}\left (5\right )^{4} x^{3}+8 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right ) \mathrm {log}\left (5\right )^{4} x^{2}-4 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right ) \mathrm {log}\left (5\right )^{3} x^{3}-16 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right ) \mathrm {log}\left (5\right )^{3} x^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right ) \mathrm {log}\left (5\right )^{2} x^{3}+8 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right ) \mathrm {log}\left (5\right )^{2} x^{2}+\mathrm {log}\left (5\right )^{4} x^{6}+8 \mathrm {log}\left (5\right )^{4} x^{5}+16 \mathrm {log}\left (5\right )^{4} x^{4}-4 \mathrm {log}\left (5\right )^{3} x^{6}-32 x^{5} \mathrm {log}\left (5\right )^{3}-64 x^{4} \mathrm {log}\left (5\right )^{3}+6 x^{6} \mathrm {log}\left (5\right )^{2}+48 x^{5} \mathrm {log}\left (5\right )^{2}+96 x^{4} \mathrm {log}\left (5\right )^{2}-4 x^{6} \mathrm {log}\left (5\right )-32 \,\mathrm {log}\left (5\right ) x^{5}-64 \,\mathrm {log}\left (5\right ) x^{4}+x^{6}+8 x^{5}+16 x^{4}} \] Input:
int(((-25*x^4*log(5)^6+100*x^4*log(5)^5-150*x^4*log(5)^4+100*x^4*log(5)^3- 25*x^4*log(5)^2)*log(log(3))+(5*x^7-20*x^6)*log(5)^6+(-30*x^7+120*x^6)*log (5)^5+(75*x^7-300*x^6)*log(5)^4+(-100*x^7+400*x^6)*log(5)^3+(75*x^7-300*x^ 6)*log(5)^2+(-30*x^7+120*x^6)*log(5)+5*x^7-20*x^6)/(log(5)^6*log(log(3))^3 +((3*x^3+12*x^2)*log(5)^6+(-6*x^3-24*x^2)*log(5)^5+(3*x^3+12*x^2)*log(5)^4 )*log(log(3))^2+((3*x^6+24*x^5+48*x^4)*log(5)^6+(-12*x^6-96*x^5-192*x^4)*l og(5)^5+(18*x^6+144*x^5+288*x^4)*log(5)^4+(-12*x^6-96*x^5-192*x^4)*log(5)^ 3+(3*x^6+24*x^5+48*x^4)*log(5)^2)*log(log(3))+(x^9+12*x^8+48*x^7+64*x^6)*l og(5)^6+(-6*x^9-72*x^8-288*x^7-384*x^6)*log(5)^5+(15*x^9+180*x^8+720*x^7+9 60*x^6)*log(5)^4+(-20*x^9-240*x^8-960*x^7-1280*x^6)*log(5)^3+(15*x^9+180*x ^8+720*x^7+960*x^6)*log(5)^2+(-6*x^9-72*x^8-288*x^7-384*x^6)*log(5)+x^9+12 *x^8+48*x^7+64*x^6),x)
Output:
(5*x**5*( - log(5)**4 + 4*log(5)**3 - 6*log(5)**2 + 4*log(5) - 1))/(log(lo g(3))**2*log(5)**4 + 2*log(log(3))*log(5)**4*x**3 + 8*log(log(3))*log(5)** 4*x**2 - 4*log(log(3))*log(5)**3*x**3 - 16*log(log(3))*log(5)**3*x**2 + 2* log(log(3))*log(5)**2*x**3 + 8*log(log(3))*log(5)**2*x**2 + log(5)**4*x**6 + 8*log(5)**4*x**5 + 16*log(5)**4*x**4 - 4*log(5)**3*x**6 - 32*log(5)**3* x**5 - 64*log(5)**3*x**4 + 6*log(5)**2*x**6 + 48*log(5)**2*x**5 + 96*log(5 )**2*x**4 - 4*log(5)*x**6 - 32*log(5)*x**5 - 64*log(5)*x**4 + x**6 + 8*x** 5 + 16*x**4)