Integrand size = 98, antiderivative size = 27 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\left (1+\left (-x+\frac {x^2}{256}\right )^2\right ) \left (4+\log \left (1+e^3-x\right )\right ) \] Output:
(ln(exp(3)-x+1)+4)*(1+(1/256*x^2-x)^2)
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(27)=54\).
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {262144 x^2-2048 x^3+4 x^4+65536 \log \left (1+e^3-x\right )+65536 x^2 \log \left (1+e^3-x\right )-512 x^3 \log \left (1+e^3-x\right )+x^4 \log \left (1+e^3-x\right )}{65536} \] Input:
Integrate[(-65536 + 524288*x - 595968*x^2 + 6672*x^3 - 17*x^4 + E^3*(52428 8*x - 6144*x^2 + 16*x^3) + (131072*x - 132608*x^2 + 1540*x^3 - 4*x^4 + E^3 *(131072*x - 1536*x^2 + 4*x^3))*Log[1 + E^3 - x])/(65536 + 65536*E^3 - 655 36*x),x]
Output:
(262144*x^2 - 2048*x^3 + 4*x^4 + 65536*Log[1 + E^3 - x] + 65536*x^2*Log[1 + E^3 - x] - 512*x^3*Log[1 + E^3 - x] + x^4*Log[1 + E^3 - x])/65536
Leaf count is larger than twice the leaf count of optimal. \(303\) vs. \(2(27)=54\).
Time = 1.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 11.22, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {-17 x^4+6672 x^3-595968 x^2+e^3 \left (16 x^3-6144 x^2+524288 x\right )+\left (-4 x^4+1540 x^3-132608 x^2+e^3 \left (4 x^3-1536 x^2+131072 x\right )+131072 x\right ) \log \left (-x+e^3+1\right )+524288 x-65536}{-65536 x+65536 e^3+65536} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {17 x^4}{65536 \left (-x+e^3+1\right )}+\frac {417 x^3}{4096 \left (-x+e^3+1\right )}-\frac {291 x^2}{32 \left (-x+e^3+1\right )}+\frac {e^3 (x-256) (x-128) x}{4096 \left (-x+e^3+1\right )}+\frac {8 x}{-x+e^3+1}+\frac {1}{x-e^3-1}+\frac {(x-256) (x-128) x \log \left (-x+e^3+1\right )}{16384}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^4}{16384}+\frac {x^4 \log \left (-x+e^3+1\right )}{65536}+\frac {\left (1+e^3\right ) x^3}{12288}-\frac {e^3 x^3}{12288}-\frac {385 x^3}{12288}-\frac {1}{128} x^3 \log \left (-x+e^3+1\right )+\frac {\left (1+e^3\right )^2 x^2}{8192}-\frac {385 \left (1+e^3\right ) x^2}{8192}+\frac {e^3 \left (383-e^3\right ) x^2}{8192}+\frac {259 x^2}{64}+x^2 \log \left (-x+e^3+1\right )-\frac {e^3 \left (32385-382 e^3+e^6\right ) x}{4096}+\frac {\left (1+e^3\right )^3 x}{4096}-\frac {385 \left (1+e^3\right )^2 x}{4096}+\frac {259}{32} \left (1+e^3\right ) x-8 x-\frac {e^3 \left (32385+32003 e^3-381 e^6+e^9\right ) \log \left (-x+e^3+1\right )}{4096}+\frac {\left (1+e^3\right )^4 \log \left (-x+e^3+1\right )}{4096}-\frac {385 \left (1+e^3\right )^3 \log \left (-x+e^3+1\right )}{4096}+\frac {259}{32} \left (1+e^3\right )^2 \log \left (-x+e^3+1\right )-8 \left (1+e^3\right ) \log \left (-x+e^3+1\right )+\log \left (-x+e^3+1\right )\) |
Input:
Int[(-65536 + 524288*x - 595968*x^2 + 6672*x^3 - 17*x^4 + E^3*(524288*x - 6144*x^2 + 16*x^3) + (131072*x - 132608*x^2 + 1540*x^3 - 4*x^4 + E^3*(1310 72*x - 1536*x^2 + 4*x^3))*Log[1 + E^3 - x])/(65536 + 65536*E^3 - 65536*x), x]
Output:
-8*x + (259*(1 + E^3)*x)/32 - (385*(1 + E^3)^2*x)/4096 + ((1 + E^3)^3*x)/4 096 - (E^3*(32385 - 382*E^3 + E^6)*x)/4096 + (259*x^2)/64 + (E^3*(383 - E^ 3)*x^2)/8192 - (385*(1 + E^3)*x^2)/8192 + ((1 + E^3)^2*x^2)/8192 - (385*x^ 3)/12288 - (E^3*x^3)/12288 + ((1 + E^3)*x^3)/12288 + x^4/16384 + Log[1 + E ^3 - x] - 8*(1 + E^3)*Log[1 + E^3 - x] + (259*(1 + E^3)^2*Log[1 + E^3 - x] )/32 - (385*(1 + E^3)^3*Log[1 + E^3 - x])/4096 + ((1 + E^3)^4*Log[1 + E^3 - x])/4096 - (E^3*(32385 + 32003*E^3 - 381*E^6 + E^9)*Log[1 + E^3 - x])/40 96 + x^2*Log[1 + E^3 - x] - (x^3*Log[1 + E^3 - x])/128 + (x^4*Log[1 + E^3 - x])/65536
Time = 1.57 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78
| method | result | size |
| risch | \(\left (\frac {1}{65536} x^{4}-\frac {1}{128} x^{3}+x^{2}\right ) \ln \left ({\mathrm e}^{3}-x +1\right )+\frac {x^{4}}{16384}-\frac {x^{3}}{32}+4 x^{2}+\ln \left (x -{\mathrm e}^{3}-1\right )\) | \(48\) |
| norman | \(\ln \left ({\mathrm e}^{3}-x +1\right )+\ln \left ({\mathrm e}^{3}-x +1\right ) x^{2}+4 x^{2}-\frac {x^{3}}{32}+\frac {x^{4}}{16384}-\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) x^{3}}{128}+\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) x^{4}}{65536}\) | \(63\) |
| parallelrisch | \(\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) x^{4}}{65536}+\frac {x^{4}}{16384}-\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) x^{3}}{128}-\frac {x^{3}}{32}+\ln \left ({\mathrm e}^{3}-x +1\right ) x^{2}-4 \,{\mathrm e}^{6}-4+4 x^{2}-8 \,{\mathrm e}^{3}+\ln \left ({\mathrm e}^{3}-x +1\right )\) | \(74\) |
| parts | \(-\frac {64515 x}{65536}+\frac {x^{3} {\mathrm e}^{3}}{196608}+\frac {32385}{16384}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{4}}{262144}-\frac {127 \left ({\mathrm e}^{3}-x +1\right )^{3}}{49152}+\frac {\left ({\mathrm e}^{12}-508 \,{\mathrm e}^{9}+64006 \,{\mathrm e}^{6}+129540 \,{\mathrm e}^{3}+130561\right ) \ln \left (x -{\mathrm e}^{3}-1\right )}{65536}+\frac {x^{2} {\mathrm e}^{6}}{131072}+\frac {64515 x \,{\mathrm e}^{3}}{65536}-\frac {255 x^{2} {\mathrm e}^{3}}{65536}+\frac {x \,{\mathrm e}^{9}}{65536}-\frac {509 x \,{\mathrm e}^{6}}{65536}-\frac {32003 \left ({\mathrm e}^{3}-x +1\right )^{2}}{65536}+\frac {32385 \,{\mathrm e}^{3}}{16384}+\frac {589313 x^{2}}{131072}-\frac {6655 x^{3}}{196608}+\frac {17 x^{4}}{262144}-\frac {32003 \,{\mathrm e}^{3} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}-\frac {32385 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )}{16384}-\frac {3 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{3}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{3}}{9}\right )}{16384}-\frac {381 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{8192}+\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{4}}{65536}+\frac {127 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{16384}+\frac {32003 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}+\frac {3 \,{\mathrm e}^{6} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{16384}-\frac {{\mathrm e}^{9} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {381 \,{\mathrm e}^{6} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}\) | \(392\) |
| derivativedivides | \(-\frac {127 \,{\mathrm e}^{9} \ln \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {{\mathrm e}^{12} \ln \left ({\mathrm e}^{3}-x +1\right )}{65536}-\frac {5 \,{\mathrm e}^{9} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {27 \,{\mathrm e}^{6} \left ({\mathrm e}^{3}-x +1\right )^{2}}{65536}+\frac {1905 \,{\mathrm e}^{6} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32385 x}{4096}-\frac {32385}{4096}+\frac {\left ({\mathrm e}^{3}-x +1\right )^{4}}{16384}+\frac {127 \left ({\mathrm e}^{3}-x +1\right )^{3}}{4096}+\frac {130561 \ln \left ({\mathrm e}^{3}-x +1\right )}{65536}+\frac {32385 \,{\mathrm e}^{3} \ln \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32003 \left ({\mathrm e}^{3}-x +1\right )^{2}}{8192}-\frac {32385 \,{\mathrm e}^{3}}{4096}-\frac {32003 \,{\mathrm e}^{3} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}-\frac {32385 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )}{16384}-\frac {3 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{3}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{3}}{9}\right )}{16384}-\frac {381 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{8192}+\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{4}}{65536}-\frac {13 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )^{3}}{49152}+\frac {127 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{16384}-\frac {3429 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}+\frac {32003 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}-\frac {160015 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {3 \,{\mathrm e}^{6} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{16384}-\frac {{\mathrm e}^{9} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {381 \,{\mathrm e}^{6} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {32003 \,{\mathrm e}^{6} \ln \left ({\mathrm e}^{3}-x +1\right )}{32768}\) | \(451\) |
| default | \(-\frac {127 \,{\mathrm e}^{9} \ln \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {{\mathrm e}^{12} \ln \left ({\mathrm e}^{3}-x +1\right )}{65536}-\frac {5 \,{\mathrm e}^{9} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {27 \,{\mathrm e}^{6} \left ({\mathrm e}^{3}-x +1\right )^{2}}{65536}+\frac {1905 \,{\mathrm e}^{6} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32385 x}{4096}-\frac {32385}{4096}+\frac {\left ({\mathrm e}^{3}-x +1\right )^{4}}{16384}+\frac {127 \left ({\mathrm e}^{3}-x +1\right )^{3}}{4096}+\frac {130561 \ln \left ({\mathrm e}^{3}-x +1\right )}{65536}+\frac {32385 \,{\mathrm e}^{3} \ln \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32003 \left ({\mathrm e}^{3}-x +1\right )^{2}}{8192}-\frac {32385 \,{\mathrm e}^{3}}{4096}-\frac {32003 \,{\mathrm e}^{3} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}-\frac {32385 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )}{16384}-\frac {3 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{3}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{3}}{9}\right )}{16384}-\frac {381 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{8192}+\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{4}}{65536}-\frac {13 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )^{3}}{49152}+\frac {127 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{16384}-\frac {3429 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}+\frac {32003 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}-\frac {160015 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {3 \,{\mathrm e}^{6} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{16384}-\frac {{\mathrm e}^{9} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {381 \,{\mathrm e}^{6} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {32003 \,{\mathrm e}^{6} \ln \left ({\mathrm e}^{3}-x +1\right )}{32768}\) | \(451\) |
Input:
int((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131072*x) *ln(exp(3)-x+1)+(16*x^3-6144*x^2+524288*x)*exp(3)-17*x^4+6672*x^3-595968*x ^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x,method=_RETURNVERBOSE)
Output:
(1/65536*x^4-1/128*x^3+x^2)*ln(exp(3)-x+1)+1/16384*x^4-1/32*x^3+4*x^2+ln(x -exp(3)-1)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {1}{16384} \, x^{4} - \frac {1}{32} \, x^{3} + 4 \, x^{2} + \frac {1}{65536} \, {\left (x^{4} - 512 \, x^{3} + 65536 \, x^{2} + 65536\right )} \log \left (-x + e^{3} + 1\right ) \] Input:
integrate((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131 072*x)*log(exp(3)-x+1)+(16*x^3-6144*x^2+524288*x)*exp(3)-17*x^4+6672*x^3-5 95968*x^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x, algorithm="frica s")
Output:
1/16384*x^4 - 1/32*x^3 + 4*x^2 + 1/65536*(x^4 - 512*x^3 + 65536*x^2 + 6553 6)*log(-x + e^3 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {x^{4}}{16384} - \frac {x^{3}}{32} + 4 x^{2} + \left (\frac {x^{4}}{65536} - \frac {x^{3}}{128} + x^{2}\right ) \log {\left (- x + 1 + e^{3} \right )} + \log {\left (x - e^{3} - 1 \right )} \] Input:
integrate((((4*x**3-1536*x**2+131072*x)*exp(3)-4*x**4+1540*x**3-132608*x** 2+131072*x)*ln(exp(3)-x+1)+(16*x**3-6144*x**2+524288*x)*exp(3)-17*x**4+667 2*x**3-595968*x**2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x)
Output:
x**4/16384 - x**3/32 + 4*x**2 + (x**4/65536 - x**3/128 + x**2)*log(-x + 1 + exp(3)) + log(x - exp(3) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (23) = 46\).
Time = 0.05 (sec) , antiderivative size = 856, normalized size of antiderivative = 31.70 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\text {Too large to display} \] Input:
integrate((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131 072*x)*log(exp(3)-x+1)+(16*x^3-6144*x^2+524288*x)*exp(3)-17*x^4+6672*x^3-5 95968*x^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x, algorithm="maxim a")
Output:
1/16384*x^4 + 11/147456*x^3*(e^3 + 1) - 4619/147456*x^3 + 19/196608*x^2*(e ^6 + 2*e^3 + 1) - 8083/196608*x^2*(e^3 + 1) - 1/32768*(e^12 + 4*e^9 + 6*e^ 6 + 4*e^3 + 1)*log(x - e^3 - 1)^2 + 385/32768*(e^9 + 3*e^6 + 3*e^3 + 1)*lo g(x - e^3 - 1)^2 - 259/256*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1)^2 + (e^3 + 1 )*log(x - e^3 - 1)^2 - 1/98304*(2*x^3 + 3*x^2*(e^3 + 1) + 6*x*(e^6 + 2*e^3 + 1) + 6*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1))*e^3*log(-x + e^3 + 1 ) + 3/256*(x^2 + 2*x*(e^3 + 1) + 2*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1))*e^3 *log(-x + e^3 + 1) - 2*((e^3 + 1)*log(x - e^3 - 1) + x)*e^3*log(-x + e^3 + 1) + 2069/512*x^2 + 13/98304*x*(e^9 + 3*e^6 + 3*e^3 + 1) - 5773/98304*x*( e^6 + 2*e^3 + 1) + 1551/256*x*(e^3 + 1) + 1/589824*(4*x^3 + 15*x^2*(e^3 + 1) + 18*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1)^2 + 66*x*(e^6 + 2*e^3 + 1) + 66*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1))*e^3 - 1/24576*(2*x^3 + 3*x^2*(e^3 + 1) + 6*x*(e^6 + 2*e^3 + 1) + 6*(e^9 + 3*e^6 + 3*e^3 + 1)*lo g(x - e^3 - 1))*e^3 - 3/512*(2*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1)^2 + x^2 + 6*x*(e^3 + 1) + 6*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1))*e^3 + ((e^3 + 1)*l og(x - e^3 - 1)^2 + 2*(e^3 + 1)*log(x - e^3 - 1) + 2*x)*e^3 + 3/64*(x^2 + 2*x*(e^3 + 1) + 2*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1))*e^3 - 8*((e^3 + 1)*l og(x - e^3 - 1) + x)*e^3 + 13/98304*(e^12 + 4*e^9 + 6*e^6 + 4*e^3 + 1)*log (x - e^3 - 1) - 5773/98304*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1) + 15 51/256*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1) - 6*(e^3 + 1)*log(x - e^3 - 1...
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 366, normalized size of antiderivative = 13.56 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {1}{65536} \, {\left (x - e^{3} - 1\right )}^{4} \log \left (-x + e^{3} + 1\right ) + \frac {1}{16384} \, {\left (x - e^{3} - 1\right )}^{3} e^{3} \log \left (-x + e^{3} + 1\right ) + \frac {1}{16384} \, {\left (x - e^{3} - 1\right )}^{4} + \frac {1}{4096} \, {\left (x - e^{3} - 1\right )}^{3} e^{3} - \frac {127}{16384} \, {\left (x - e^{3} - 1\right )}^{3} \log \left (-x + e^{3} + 1\right ) + \frac {3}{32768} \, {\left (x - e^{3} - 1\right )}^{2} e^{6} \log \left (-x + e^{3} + 1\right ) - \frac {381}{16384} \, {\left (x - e^{3} - 1\right )}^{2} e^{3} \log \left (-x + e^{3} + 1\right ) - \frac {127}{4096} \, {\left (x - e^{3} - 1\right )}^{3} + \frac {3}{8192} \, {\left (x - e^{3} - 1\right )}^{2} e^{6} - \frac {381}{4096} \, {\left (x - e^{3} - 1\right )}^{2} e^{3} + \frac {32003}{32768} \, {\left (x - e^{3} - 1\right )}^{2} \log \left (-x + e^{3} + 1\right ) + \frac {1}{16384} \, {\left (x - e^{3} - 1\right )} e^{9} \log \left (-x + e^{3} + 1\right ) - \frac {381}{16384} \, {\left (x - e^{3} - 1\right )} e^{6} \log \left (-x + e^{3} + 1\right ) + \frac {32003}{16384} \, {\left (x - e^{3} - 1\right )} e^{3} \log \left (-x + e^{3} + 1\right ) + \frac {32003}{8192} \, {\left (x - e^{3} - 1\right )}^{2} + \frac {1}{4096} \, {\left (x - e^{3} - 1\right )} e^{9} - \frac {381}{4096} \, {\left (x - e^{3} - 1\right )} e^{6} + \frac {32003}{4096} \, {\left (x - e^{3} - 1\right )} e^{3} + \frac {32385}{16384} \, {\left (x - e^{3} - 1\right )} \log \left (-x + e^{3} + 1\right ) + \frac {1}{65536} \, e^{12} \log \left (-x + e^{3} + 1\right ) - \frac {127}{16384} \, e^{9} \log \left (-x + e^{3} + 1\right ) + \frac {32003}{32768} \, e^{6} \log \left (-x + e^{3} + 1\right ) + \frac {32385}{16384} \, e^{3} \log \left (-x + e^{3} + 1\right ) + \frac {32385}{4096} \, x - \frac {32385}{4096} \, e^{3} + \frac {130561}{65536} \, \log \left (-x + e^{3} + 1\right ) - \frac {32385}{4096} \] Input:
integrate((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131 072*x)*log(exp(3)-x+1)+(16*x^3-6144*x^2+524288*x)*exp(3)-17*x^4+6672*x^3-5 95968*x^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x, algorithm="giac" )
Output:
1/65536*(x - e^3 - 1)^4*log(-x + e^3 + 1) + 1/16384*(x - e^3 - 1)^3*e^3*lo g(-x + e^3 + 1) + 1/16384*(x - e^3 - 1)^4 + 1/4096*(x - e^3 - 1)^3*e^3 - 1 27/16384*(x - e^3 - 1)^3*log(-x + e^3 + 1) + 3/32768*(x - e^3 - 1)^2*e^6*l og(-x + e^3 + 1) - 381/16384*(x - e^3 - 1)^2*e^3*log(-x + e^3 + 1) - 127/4 096*(x - e^3 - 1)^3 + 3/8192*(x - e^3 - 1)^2*e^6 - 381/4096*(x - e^3 - 1)^ 2*e^3 + 32003/32768*(x - e^3 - 1)^2*log(-x + e^3 + 1) + 1/16384*(x - e^3 - 1)*e^9*log(-x + e^3 + 1) - 381/16384*(x - e^3 - 1)*e^6*log(-x + e^3 + 1) + 32003/16384*(x - e^3 - 1)*e^3*log(-x + e^3 + 1) + 32003/8192*(x - e^3 - 1)^2 + 1/4096*(x - e^3 - 1)*e^9 - 381/4096*(x - e^3 - 1)*e^6 + 32003/4096* (x - e^3 - 1)*e^3 + 32385/16384*(x - e^3 - 1)*log(-x + e^3 + 1) + 1/65536* e^12*log(-x + e^3 + 1) - 127/16384*e^9*log(-x + e^3 + 1) + 32003/32768*e^6 *log(-x + e^3 + 1) + 32385/16384*e^3*log(-x + e^3 + 1) + 32385/4096*x - 32 385/4096*e^3 + 130561/65536*log(-x + e^3 + 1) - 32385/4096
Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\ln \left (x-{\mathrm {e}}^3-1\right )+\ln \left ({\mathrm {e}}^3-x+1\right )\,\left (\frac {x^4}{65536}-\frac {x^3}{128}+x^2\right )+4\,x^2-\frac {x^3}{32}+\frac {x^4}{16384} \] Input:
int((524288*x + exp(3)*(524288*x - 6144*x^2 + 16*x^3) + log(exp(3) - x + 1 )*(131072*x + exp(3)*(131072*x - 1536*x^2 + 4*x^3) - 132608*x^2 + 1540*x^3 - 4*x^4) - 595968*x^2 + 6672*x^3 - 17*x^4 - 65536)/(65536*exp(3) - 65536* x + 65536),x)
Output:
log(x - exp(3) - 1) + log(exp(3) - x + 1)*(x^2 - x^3/128 + x^4/65536) + 4* x^2 - x^3/32 + x^4/16384
Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.44 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {\mathrm {log}\left (e^{3}-x +1\right ) x^{4}}{65536}-\frac {\mathrm {log}\left (e^{3}-x +1\right ) x^{3}}{128}+\mathrm {log}\left (e^{3}-x +1\right ) x^{2}+\mathrm {log}\left (e^{3}-x +1\right )+\frac {x^{4}}{16384}-\frac {x^{3}}{32}+4 x^{2} \] Input:
int((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131072*x) *log(exp(3)-x+1)+(16*x^3-6144*x^2+524288*x)*exp(3)-17*x^4+6672*x^3-595968* x^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x)
Output:
(log(e**3 - x + 1)*x**4 - 512*log(e**3 - x + 1)*x**3 + 65536*log(e**3 - x + 1)*x**2 + 65536*log(e**3 - x + 1) + 4*x**4 - 2048*x**3 + 262144*x**2)/65 536