Integrand size = 230, antiderivative size = 22 \[ \int \frac {2+e \left (432+14256 x+20880 x^2+230416 x^3+185600 x^4+1216512 x^5+458752 x^6+2097152 x^7\right )+e \left (-432-14112 x-13872 x^2-150528 x^3-61440 x^4-393216 x^5\right ) \log (4)+e \left (144+4656 x+2304 x^2+24576 x^3\right ) \log ^2(4)+e (-16-512 x) \log ^3(4)}{x+e \left (162+216 x+3564 x^2+3480 x^3+28802 x^4+18560 x^5+101376 x^6+32768 x^7+131072 x^8\right )+e \left (-216-216 x-3528 x^2-2312 x^3-18816 x^4-6144 x^5-32768 x^6\right ) \log (4)+e \left (108+72 x+1164 x^2+384 x^3+3072 x^4\right ) \log ^2(4)+e \left (-24-8 x-128 x^2\right ) \log ^3(4)+2 e \log ^4(4)} \, dx=\log \left (\left (x+2 e \left (3+x+16 x^2-\log (4)\right )^4\right )^2\right ) \] Output:
ln((x+2*exp(1)*(3+16*x^2-2*ln(2)+x)^4)^2)
Leaf count is larger than twice the leaf count of optimal. \(182\) vs. \(2(22)=44\).
Time = 0.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 8.27 \[ \int \frac {2+e \left (432+14256 x+20880 x^2+230416 x^3+185600 x^4+1216512 x^5+458752 x^6+2097152 x^7\right )+e \left (-432-14112 x-13872 x^2-150528 x^3-61440 x^4-393216 x^5\right ) \log (4)+e \left (144+4656 x+2304 x^2+24576 x^3\right ) \log ^2(4)+e (-16-512 x) \log ^3(4)}{x+e \left (162+216 x+3564 x^2+3480 x^3+28802 x^4+18560 x^5+101376 x^6+32768 x^7+131072 x^8\right )+e \left (-216-216 x-3528 x^2-2312 x^3-18816 x^4-6144 x^5-32768 x^6\right ) \log (4)+e \left (108+72 x+1164 x^2+384 x^3+3072 x^4\right ) \log ^2(4)+e \left (-24-8 x-128 x^2\right ) \log ^3(4)+2 e \log ^4(4)} \, dx=2 \log \left (162 e+x+216 e x+3564 e x^2+3480 e x^3+28802 e x^4+18560 e x^5+101376 e x^6+32768 e x^7+131072 e x^8-216 e \log (4)-216 e x \log (4)-3528 e x^2 \log (4)-2312 e x^3 \log (4)-18816 e x^4 \log (4)-6144 e x^5 \log (4)-32768 e x^6 \log (4)+108 e \log ^2(4)+72 e x \log ^2(4)+1164 e x^2 \log ^2(4)+384 e x^3 \log ^2(4)+3072 e x^4 \log ^2(4)-24 e \log ^3(4)-8 e x \log ^3(4)-128 e x^2 \log ^3(4)+2 e \log ^4(4)\right ) \] Input:
Integrate[(2 + E*(432 + 14256*x + 20880*x^2 + 230416*x^3 + 185600*x^4 + 12 16512*x^5 + 458752*x^6 + 2097152*x^7) + E*(-432 - 14112*x - 13872*x^2 - 15 0528*x^3 - 61440*x^4 - 393216*x^5)*Log[4] + E*(144 + 4656*x + 2304*x^2 + 2 4576*x^3)*Log[4]^2 + E*(-16 - 512*x)*Log[4]^3)/(x + E*(162 + 216*x + 3564* x^2 + 3480*x^3 + 28802*x^4 + 18560*x^5 + 101376*x^6 + 32768*x^7 + 131072*x ^8) + E*(-216 - 216*x - 3528*x^2 - 2312*x^3 - 18816*x^4 - 6144*x^5 - 32768 *x^6)*Log[4] + E*(108 + 72*x + 1164*x^2 + 384*x^3 + 3072*x^4)*Log[4]^2 + E *(-24 - 8*x - 128*x^2)*Log[4]^3 + 2*E*Log[4]^4),x]
Output:
2*Log[162*E + x + 216*E*x + 3564*E*x^2 + 3480*E*x^3 + 28802*E*x^4 + 18560* E*x^5 + 101376*E*x^6 + 32768*E*x^7 + 131072*E*x^8 - 216*E*Log[4] - 216*E*x *Log[4] - 3528*E*x^2*Log[4] - 2312*E*x^3*Log[4] - 18816*E*x^4*Log[4] - 614 4*E*x^5*Log[4] - 32768*E*x^6*Log[4] + 108*E*Log[4]^2 + 72*E*x*Log[4]^2 + 1 164*E*x^2*Log[4]^2 + 384*E*x^3*Log[4]^2 + 3072*E*x^4*Log[4]^2 - 24*E*Log[4 ]^3 - 8*E*x*Log[4]^3 - 128*E*x^2*Log[4]^3 + 2*E*Log[4]^4]
Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(22)=44\).
Time = 0.39 (sec) , antiderivative size = 132, normalized size of antiderivative = 6.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.004, Rules used = {2020}
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.
| \(\displaystyle \int \frac {e \left (24576 x^3+2304 x^2+4656 x+144\right ) \log ^2(4)+e \left (-393216 x^5-61440 x^4-150528 x^3-13872 x^2-14112 x-432\right ) \log (4)+e \left (2097152 x^7+458752 x^6+1216512 x^5+185600 x^4+230416 x^3+20880 x^2+14256 x+432\right )+e (-512 x-16) \log ^3(4)+2}{e \left (-128 x^2-8 x-24\right ) \log ^3(4)+e \left (3072 x^4+384 x^3+1164 x^2+72 x+108\right ) \log ^2(4)+e \left (-32768 x^6-6144 x^5-18816 x^4-2312 x^3-3528 x^2-216 x-216\right ) \log (4)+e \left (131072 x^8+32768 x^7+101376 x^6+18560 x^5+28802 x^4+3480 x^3+3564 x^2+216 x+162\right )+x+2 e \log ^4(4)} \, dx\) |
| \(\Big \downarrow \) 2020 |
| \(\displaystyle 2 \log \left (-8 e \left (16 x^2+x+3\right ) \log ^3(4)+12 e \left (256 x^4+32 x^3+97 x^2+6 x+9\right ) \log ^2(4)-8 e \left (4096 x^6+768 x^5+2352 x^4+289 x^3+441 x^2+27 x+27\right ) \log (4)+2 e \left (65536 x^8+16384 x^7+50688 x^6+9280 x^5+14401 x^4+1740 x^3+1782 x^2+108 x+81\right )+x+2 e \log ^4(4)\right )\) |
Input:
Int[(2 + E*(432 + 14256*x + 20880*x^2 + 230416*x^3 + 185600*x^4 + 1216512* x^5 + 458752*x^6 + 2097152*x^7) + E*(-432 - 14112*x - 13872*x^2 - 150528*x ^3 - 61440*x^4 - 393216*x^5)*Log[4] + E*(144 + 4656*x + 2304*x^2 + 24576*x ^3)*Log[4]^2 + E*(-16 - 512*x)*Log[4]^3)/(x + E*(162 + 216*x + 3564*x^2 + 3480*x^3 + 28802*x^4 + 18560*x^5 + 101376*x^6 + 32768*x^7 + 131072*x^8) + E*(-216 - 216*x - 3528*x^2 - 2312*x^3 - 18816*x^4 - 6144*x^5 - 32768*x^6)* Log[4] + E*(108 + 72*x + 1164*x^2 + 384*x^3 + 3072*x^4)*Log[4]^2 + E*(-24 - 8*x - 128*x^2)*Log[4]^3 + 2*E*Log[4]^4),x]
Output:
2*Log[x + 2*E*(81 + 108*x + 1782*x^2 + 1740*x^3 + 14401*x^4 + 9280*x^5 + 5 0688*x^6 + 16384*x^7 + 65536*x^8) - 8*E*(27 + 27*x + 441*x^2 + 289*x^3 + 2 352*x^4 + 768*x^5 + 4096*x^6)*Log[4] + 12*E*(9 + 6*x + 97*x^2 + 32*x^3 + 2 56*x^4)*Log[4]^2 - 8*E*(3 + x + 16*x^2)*Log[4]^3 + 2*E*Log[4]^4]
Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Si mp[Coeff[Pp, x, p]*(Log[RemoveContent[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; E qQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))*D[Qq , x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]
Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(23)=46\).
Time = 2.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 8.64
| method | result | size |
| risch | \(2 \ln \left (131072 x^{8} {\mathrm e}+32768 \,{\mathrm e} x^{7}+\left (-65536 \ln \left (2\right ) {\mathrm e}+101376 \,{\mathrm e}\right ) x^{6}+\left (-12288 \ln \left (2\right ) {\mathrm e}+18560 \,{\mathrm e}\right ) x^{5}+\left (12288 \ln \left (2\right )^{2} {\mathrm e}-37632 \ln \left (2\right ) {\mathrm e}+28802 \,{\mathrm e}\right ) x^{4}+\left (1536 \ln \left (2\right )^{2} {\mathrm e}-4624 \ln \left (2\right ) {\mathrm e}+3480 \,{\mathrm e}\right ) x^{3}+\left (-1024 \ln \left (2\right )^{3} {\mathrm e}+4656 \ln \left (2\right )^{2} {\mathrm e}-7056 \ln \left (2\right ) {\mathrm e}+3564 \,{\mathrm e}\right ) x^{2}+\left (-64 \ln \left (2\right )^{3} {\mathrm e}+288 \ln \left (2\right )^{2} {\mathrm e}-432 \ln \left (2\right ) {\mathrm e}+216 \,{\mathrm e}+1\right ) x +32 \,{\mathrm e} \ln \left (2\right )^{4}-192 \ln \left (2\right )^{3} {\mathrm e}+432 \ln \left (2\right )^{2} {\mathrm e}-432 \ln \left (2\right ) {\mathrm e}+162 \,{\mathrm e}\right )\) | \(190\) |
| default | \(2 \ln \left (x +162 \,{\mathrm e}+32 \,{\mathrm e} \ln \left (2\right )^{4}+101376 x^{6} {\mathrm e}+18560 x^{5} {\mathrm e}+131072 x^{8} {\mathrm e}+3480 x^{3} {\mathrm e}+28802 x^{4} {\mathrm e}+3564 x^{2} {\mathrm e}+216 x \,{\mathrm e}+288 \ln \left (2\right )^{2} {\mathrm e} x -7056 \ln \left (2\right ) {\mathrm e} x^{2}-432 \ln \left (2\right ) {\mathrm e} x -65536 \ln \left (2\right ) {\mathrm e} x^{6}+12288 \ln \left (2\right )^{2} {\mathrm e} x^{4}-12288 \ln \left (2\right ) {\mathrm e} x^{5}-1024 \ln \left (2\right )^{3} {\mathrm e} x^{2}+1536 \ln \left (2\right )^{2} {\mathrm e} x^{3}-37632 \ln \left (2\right ) {\mathrm e} x^{4}-64 \ln \left (2\right )^{3} {\mathrm e} x +4656 \ln \left (2\right )^{2} {\mathrm e} x^{2}-4624 \ln \left (2\right ) {\mathrm e} x^{3}+32768 \,{\mathrm e} x^{7}-192 \ln \left (2\right )^{3} {\mathrm e}+432 \ln \left (2\right )^{2} {\mathrm e}-432 \ln \left (2\right ) {\mathrm e}\right )\) | \(208\) |
| norman | \(2 \ln \left (x +162 \,{\mathrm e}+32 \,{\mathrm e} \ln \left (2\right )^{4}+101376 x^{6} {\mathrm e}+18560 x^{5} {\mathrm e}+131072 x^{8} {\mathrm e}+3480 x^{3} {\mathrm e}+28802 x^{4} {\mathrm e}+3564 x^{2} {\mathrm e}+216 x \,{\mathrm e}+288 \ln \left (2\right )^{2} {\mathrm e} x -7056 \ln \left (2\right ) {\mathrm e} x^{2}-432 \ln \left (2\right ) {\mathrm e} x -65536 \ln \left (2\right ) {\mathrm e} x^{6}+12288 \ln \left (2\right )^{2} {\mathrm e} x^{4}-12288 \ln \left (2\right ) {\mathrm e} x^{5}-1024 \ln \left (2\right )^{3} {\mathrm e} x^{2}+1536 \ln \left (2\right )^{2} {\mathrm e} x^{3}-37632 \ln \left (2\right ) {\mathrm e} x^{4}-64 \ln \left (2\right )^{3} {\mathrm e} x +4656 \ln \left (2\right )^{2} {\mathrm e} x^{2}-4624 \ln \left (2\right ) {\mathrm e} x^{3}+32768 \,{\mathrm e} x^{7}-192 \ln \left (2\right )^{3} {\mathrm e}+432 \ln \left (2\right )^{2} {\mathrm e}-432 \ln \left (2\right ) {\mathrm e}\right )\) | \(208\) |
| parallelrisch | \(2 \ln \left (\frac {\left (x +162 \,{\mathrm e}+32 \,{\mathrm e} \ln \left (2\right )^{4}+101376 x^{6} {\mathrm e}+18560 x^{5} {\mathrm e}+131072 x^{8} {\mathrm e}+3480 x^{3} {\mathrm e}+28802 x^{4} {\mathrm e}+3564 x^{2} {\mathrm e}+216 x \,{\mathrm e}+288 \ln \left (2\right )^{2} {\mathrm e} x -7056 \ln \left (2\right ) {\mathrm e} x^{2}-432 \ln \left (2\right ) {\mathrm e} x -65536 \ln \left (2\right ) {\mathrm e} x^{6}+12288 \ln \left (2\right )^{2} {\mathrm e} x^{4}-12288 \ln \left (2\right ) {\mathrm e} x^{5}-1024 \ln \left (2\right )^{3} {\mathrm e} x^{2}+1536 \ln \left (2\right )^{2} {\mathrm e} x^{3}-37632 \ln \left (2\right ) {\mathrm e} x^{4}-64 \ln \left (2\right )^{3} {\mathrm e} x +4656 \ln \left (2\right )^{2} {\mathrm e} x^{2}-4624 \ln \left (2\right ) {\mathrm e} x^{3}+32768 \,{\mathrm e} x^{7}-192 \ln \left (2\right )^{3} {\mathrm e}+432 \ln \left (2\right )^{2} {\mathrm e}-432 \ln \left (2\right ) {\mathrm e}\right ) {\mathrm e}^{-1}}{131072}\right )\) | \(214\) |
Input:
int((8*(-512*x-16)*exp(1)*ln(2)^3+4*(24576*x^3+2304*x^2+4656*x+144)*exp(1) *ln(2)^2+2*(-393216*x^5-61440*x^4-150528*x^3-13872*x^2-14112*x-432)*exp(1) *ln(2)+(2097152*x^7+458752*x^6+1216512*x^5+185600*x^4+230416*x^3+20880*x^2 +14256*x+432)*exp(1)+2)/(32*exp(1)*ln(2)^4+8*(-128*x^2-8*x-24)*exp(1)*ln(2 )^3+4*(3072*x^4+384*x^3+1164*x^2+72*x+108)*exp(1)*ln(2)^2+2*(-32768*x^6-61 44*x^5-18816*x^4-2312*x^3-3528*x^2-216*x-216)*exp(1)*ln(2)+(131072*x^8+327 68*x^7+101376*x^6+18560*x^5+28802*x^4+3480*x^3+3564*x^2+216*x+162)*exp(1)+ x),x,method=_RETURNVERBOSE)
Output:
2*ln(131072*x^8*exp(1)+32768*exp(1)*x^7+(-65536*ln(2)*exp(1)+101376*exp(1) )*x^6+(-12288*ln(2)*exp(1)+18560*exp(1))*x^5+(12288*ln(2)^2*exp(1)-37632*l n(2)*exp(1)+28802*exp(1))*x^4+(1536*ln(2)^2*exp(1)-4624*ln(2)*exp(1)+3480* exp(1))*x^3+(-1024*ln(2)^3*exp(1)+4656*ln(2)^2*exp(1)-7056*ln(2)*exp(1)+35 64*exp(1))*x^2+(-64*ln(2)^3*exp(1)+288*ln(2)^2*exp(1)-432*ln(2)*exp(1)+216 *exp(1)+1)*x+32*exp(1)*ln(2)^4-192*ln(2)^3*exp(1)+432*ln(2)^2*exp(1)-432*l n(2)*exp(1)+162*exp(1))
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 6.23 \[ \int \frac {2+e \left (432+14256 x+20880 x^2+230416 x^3+185600 x^4+1216512 x^5+458752 x^6+2097152 x^7\right )+e \left (-432-14112 x-13872 x^2-150528 x^3-61440 x^4-393216 x^5\right ) \log (4)+e \left (144+4656 x+2304 x^2+24576 x^3\right ) \log ^2(4)+e (-16-512 x) \log ^3(4)}{x+e \left (162+216 x+3564 x^2+3480 x^3+28802 x^4+18560 x^5+101376 x^6+32768 x^7+131072 x^8\right )+e \left (-216-216 x-3528 x^2-2312 x^3-18816 x^4-6144 x^5-32768 x^6\right ) \log (4)+e \left (108+72 x+1164 x^2+384 x^3+3072 x^4\right ) \log ^2(4)+e \left (-24-8 x-128 x^2\right ) \log ^3(4)+2 e \log ^4(4)} \, dx=2 \, \log \left (-64 \, {\left (16 \, x^{2} + x + 3\right )} e \log \left (2\right )^{3} + 32 \, e \log \left (2\right )^{4} + 48 \, {\left (256 \, x^{4} + 32 \, x^{3} + 97 \, x^{2} + 6 \, x + 9\right )} e \log \left (2\right )^{2} - 16 \, {\left (4096 \, x^{6} + 768 \, x^{5} + 2352 \, x^{4} + 289 \, x^{3} + 441 \, x^{2} + 27 \, x + 27\right )} e \log \left (2\right ) + 2 \, {\left (65536 \, x^{8} + 16384 \, x^{7} + 50688 \, x^{6} + 9280 \, x^{5} + 14401 \, x^{4} + 1740 \, x^{3} + 1782 \, x^{2} + 108 \, x + 81\right )} e + x\right ) \] Input:
integrate((8*(-512*x-16)*exp(1)*log(2)^3+4*(24576*x^3+2304*x^2+4656*x+144) *exp(1)*log(2)^2+2*(-393216*x^5-61440*x^4-150528*x^3-13872*x^2-14112*x-432 )*exp(1)*log(2)+(2097152*x^7+458752*x^6+1216512*x^5+185600*x^4+230416*x^3+ 20880*x^2+14256*x+432)*exp(1)+2)/(32*exp(1)*log(2)^4+8*(-128*x^2-8*x-24)*e xp(1)*log(2)^3+4*(3072*x^4+384*x^3+1164*x^2+72*x+108)*exp(1)*log(2)^2+2*(- 32768*x^6-6144*x^5-18816*x^4-2312*x^3-3528*x^2-216*x-216)*exp(1)*log(2)+(1 31072*x^8+32768*x^7+101376*x^6+18560*x^5+28802*x^4+3480*x^3+3564*x^2+216*x +162)*exp(1)+x),x, algorithm="fricas")
Output:
2*log(-64*(16*x^2 + x + 3)*e*log(2)^3 + 32*e*log(2)^4 + 48*(256*x^4 + 32*x ^3 + 97*x^2 + 6*x + 9)*e*log(2)^2 - 16*(4096*x^6 + 768*x^5 + 2352*x^4 + 28 9*x^3 + 441*x^2 + 27*x + 27)*e*log(2) + 2*(65536*x^8 + 16384*x^7 + 50688*x ^6 + 9280*x^5 + 14401*x^4 + 1740*x^3 + 1782*x^2 + 108*x + 81)*e + x)
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (24) = 48\).
Time = 2.56 (sec) , antiderivative size = 226, normalized size of antiderivative = 10.27 \[ \int \frac {2+e \left (432+14256 x+20880 x^2+230416 x^3+185600 x^4+1216512 x^5+458752 x^6+2097152 x^7\right )+e \left (-432-14112 x-13872 x^2-150528 x^3-61440 x^4-393216 x^5\right ) \log (4)+e \left (144+4656 x+2304 x^2+24576 x^3\right ) \log ^2(4)+e (-16-512 x) \log ^3(4)}{x+e \left (162+216 x+3564 x^2+3480 x^3+28802 x^4+18560 x^5+101376 x^6+32768 x^7+131072 x^8\right )+e \left (-216-216 x-3528 x^2-2312 x^3-18816 x^4-6144 x^5-32768 x^6\right ) \log (4)+e \left (108+72 x+1164 x^2+384 x^3+3072 x^4\right ) \log ^2(4)+e \left (-24-8 x-128 x^2\right ) \log ^3(4)+2 e \log ^4(4)} \, dx=2 \log {\left (131072 e x^{8} + 32768 e x^{7} + x^{6} \left (- 65536 e \log {\left (2 \right )} + 101376 e\right ) + x^{5} \left (- 12288 e \log {\left (2 \right )} + 18560 e\right ) + x^{4} \left (- 37632 e \log {\left (2 \right )} + 12288 e \log {\left (2 \right )}^{2} + 28802 e\right ) + x^{3} \left (- 4624 e \log {\left (2 \right )} + 1536 e \log {\left (2 \right )}^{2} + 3480 e\right ) + x^{2} \left (- 7056 e \log {\left (2 \right )} - 1024 e \log {\left (2 \right )}^{3} + 4656 e \log {\left (2 \right )}^{2} + 3564 e\right ) + x \left (- 432 e \log {\left (2 \right )} - 64 e \log {\left (2 \right )}^{3} + 1 + 288 e \log {\left (2 \right )}^{2} + 216 e\right ) - 432 e \log {\left (2 \right )} - 192 e \log {\left (2 \right )}^{3} + 32 e \log {\left (2 \right )}^{4} + 162 e + 432 e \log {\left (2 \right )}^{2} \right )} \] Input:
integrate((8*(-512*x-16)*exp(1)*ln(2)**3+4*(24576*x**3+2304*x**2+4656*x+14 4)*exp(1)*ln(2)**2+2*(-393216*x**5-61440*x**4-150528*x**3-13872*x**2-14112 *x-432)*exp(1)*ln(2)+(2097152*x**7+458752*x**6+1216512*x**5+185600*x**4+23 0416*x**3+20880*x**2+14256*x+432)*exp(1)+2)/(32*exp(1)*ln(2)**4+8*(-128*x* *2-8*x-24)*exp(1)*ln(2)**3+4*(3072*x**4+384*x**3+1164*x**2+72*x+108)*exp(1 )*ln(2)**2+2*(-32768*x**6-6144*x**5-18816*x**4-2312*x**3-3528*x**2-216*x-2 16)*exp(1)*ln(2)+(131072*x**8+32768*x**7+101376*x**6+18560*x**5+28802*x**4 +3480*x**3+3564*x**2+216*x+162)*exp(1)+x),x)
Output:
2*log(131072*E*x**8 + 32768*E*x**7 + x**6*(-65536*E*log(2) + 101376*E) + x **5*(-12288*E*log(2) + 18560*E) + x**4*(-37632*E*log(2) + 12288*E*log(2)** 2 + 28802*E) + x**3*(-4624*E*log(2) + 1536*E*log(2)**2 + 3480*E) + x**2*(- 7056*E*log(2) - 1024*E*log(2)**3 + 4656*E*log(2)**2 + 3564*E) + x*(-432*E* log(2) - 64*E*log(2)**3 + 1 + 288*E*log(2)**2 + 216*E) - 432*E*log(2) - 19 2*E*log(2)**3 + 32*E*log(2)**4 + 162*E + 432*E*log(2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (23) = 46\).
Time = 0.04 (sec) , antiderivative size = 195, normalized size of antiderivative = 8.86 \[ \int \frac {2+e \left (432+14256 x+20880 x^2+230416 x^3+185600 x^4+1216512 x^5+458752 x^6+2097152 x^7\right )+e \left (-432-14112 x-13872 x^2-150528 x^3-61440 x^4-393216 x^5\right ) \log (4)+e \left (144+4656 x+2304 x^2+24576 x^3\right ) \log ^2(4)+e (-16-512 x) \log ^3(4)}{x+e \left (162+216 x+3564 x^2+3480 x^3+28802 x^4+18560 x^5+101376 x^6+32768 x^7+131072 x^8\right )+e \left (-216-216 x-3528 x^2-2312 x^3-18816 x^4-6144 x^5-32768 x^6\right ) \log (4)+e \left (108+72 x+1164 x^2+384 x^3+3072 x^4\right ) \log ^2(4)+e \left (-24-8 x-128 x^2\right ) \log ^3(4)+2 e \log ^4(4)} \, dx=2 \, \log \left (131072 \, x^{8} e + 32768 \, x^{7} e - 1024 \, {\left (64 \, e \log \left (2\right ) - 99 \, e\right )} x^{6} - 128 \, {\left (96 \, e \log \left (2\right ) - 145 \, e\right )} x^{5} + 2 \, {\left (6144 \, e \log \left (2\right )^{2} - 18816 \, e \log \left (2\right ) + 14401 \, e\right )} x^{4} + 32 \, e \log \left (2\right )^{4} + 8 \, {\left (192 \, e \log \left (2\right )^{2} - 578 \, e \log \left (2\right ) + 435 \, e\right )} x^{3} - 192 \, e \log \left (2\right )^{3} - 4 \, {\left (256 \, e \log \left (2\right )^{3} - 1164 \, e \log \left (2\right )^{2} + 1764 \, e \log \left (2\right ) - 891 \, e\right )} x^{2} + 432 \, e \log \left (2\right )^{2} - {\left (64 \, e \log \left (2\right )^{3} - 288 \, e \log \left (2\right )^{2} + 432 \, e \log \left (2\right ) - 216 \, e - 1\right )} x - 432 \, e \log \left (2\right ) + 162 \, e\right ) \] Input:
integrate((8*(-512*x-16)*exp(1)*log(2)^3+4*(24576*x^3+2304*x^2+4656*x+144) *exp(1)*log(2)^2+2*(-393216*x^5-61440*x^4-150528*x^3-13872*x^2-14112*x-432 )*exp(1)*log(2)+(2097152*x^7+458752*x^6+1216512*x^5+185600*x^4+230416*x^3+ 20880*x^2+14256*x+432)*exp(1)+2)/(32*exp(1)*log(2)^4+8*(-128*x^2-8*x-24)*e xp(1)*log(2)^3+4*(3072*x^4+384*x^3+1164*x^2+72*x+108)*exp(1)*log(2)^2+2*(- 32768*x^6-6144*x^5-18816*x^4-2312*x^3-3528*x^2-216*x-216)*exp(1)*log(2)+(1 31072*x^8+32768*x^7+101376*x^6+18560*x^5+28802*x^4+3480*x^3+3564*x^2+216*x +162)*exp(1)+x),x, algorithm="maxima")
Output:
2*log(131072*x^8*e + 32768*x^7*e - 1024*(64*e*log(2) - 99*e)*x^6 - 128*(96 *e*log(2) - 145*e)*x^5 + 2*(6144*e*log(2)^2 - 18816*e*log(2) + 14401*e)*x^ 4 + 32*e*log(2)^4 + 8*(192*e*log(2)^2 - 578*e*log(2) + 435*e)*x^3 - 192*e* log(2)^3 - 4*(256*e*log(2)^3 - 1164*e*log(2)^2 + 1764*e*log(2) - 891*e)*x^ 2 + 432*e*log(2)^2 - (64*e*log(2)^3 - 288*e*log(2)^2 + 432*e*log(2) - 216* e - 1)*x - 432*e*log(2) + 162*e)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (23) = 46\).
Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 7.23 \[ \int \frac {2+e \left (432+14256 x+20880 x^2+230416 x^3+185600 x^4+1216512 x^5+458752 x^6+2097152 x^7\right )+e \left (-432-14112 x-13872 x^2-150528 x^3-61440 x^4-393216 x^5\right ) \log (4)+e \left (144+4656 x+2304 x^2+24576 x^3\right ) \log ^2(4)+e (-16-512 x) \log ^3(4)}{x+e \left (162+216 x+3564 x^2+3480 x^3+28802 x^4+18560 x^5+101376 x^6+32768 x^7+131072 x^8\right )+e \left (-216-216 x-3528 x^2-2312 x^3-18816 x^4-6144 x^5-32768 x^6\right ) \log (4)+e \left (108+72 x+1164 x^2+384 x^3+3072 x^4\right ) \log ^2(4)+e \left (-24-8 x-128 x^2\right ) \log ^3(4)+2 e \log ^4(4)} \, dx=2 \, \log \left (-64 \, {\left (16 \, x^{2} + x\right )} e \log \left (2\right )^{3} + 32 \, e \log \left (2\right )^{4} + 48 \, {\left (256 \, x^{4} + 32 \, x^{3} + 97 \, x^{2} + 6 \, x\right )} e \log \left (2\right )^{2} - 192 \, e \log \left (2\right )^{3} - 16 \, {\left (4096 \, x^{6} + 768 \, x^{5} + 2352 \, x^{4} + 289 \, x^{3} + 441 \, x^{2} + 27 \, x\right )} e \log \left (2\right ) + 432 \, e \log \left (2\right )^{2} + 2 \, {\left (65536 \, x^{8} + 16384 \, x^{7} + 50688 \, x^{6} + 9280 \, x^{5} + 14401 \, x^{4} + 1740 \, x^{3} + 1782 \, x^{2} + 108 \, x\right )} e - 432 \, e \log \left (2\right ) + x + 162 \, e\right ) \] Input:
integrate((8*(-512*x-16)*exp(1)*log(2)^3+4*(24576*x^3+2304*x^2+4656*x+144) *exp(1)*log(2)^2+2*(-393216*x^5-61440*x^4-150528*x^3-13872*x^2-14112*x-432 )*exp(1)*log(2)+(2097152*x^7+458752*x^6+1216512*x^5+185600*x^4+230416*x^3+ 20880*x^2+14256*x+432)*exp(1)+2)/(32*exp(1)*log(2)^4+8*(-128*x^2-8*x-24)*e xp(1)*log(2)^3+4*(3072*x^4+384*x^3+1164*x^2+72*x+108)*exp(1)*log(2)^2+2*(- 32768*x^6-6144*x^5-18816*x^4-2312*x^3-3528*x^2-216*x-216)*exp(1)*log(2)+(1 31072*x^8+32768*x^7+101376*x^6+18560*x^5+28802*x^4+3480*x^3+3564*x^2+216*x +162)*exp(1)+x),x, algorithm="giac")
Output:
2*log(-64*(16*x^2 + x)*e*log(2)^3 + 32*e*log(2)^4 + 48*(256*x^4 + 32*x^3 + 97*x^2 + 6*x)*e*log(2)^2 - 192*e*log(2)^3 - 16*(4096*x^6 + 768*x^5 + 2352 *x^4 + 289*x^3 + 441*x^2 + 27*x)*e*log(2) + 432*e*log(2)^2 + 2*(65536*x^8 + 16384*x^7 + 50688*x^6 + 9280*x^5 + 14401*x^4 + 1740*x^3 + 1782*x^2 + 108 *x)*e - 432*e*log(2) + x + 162*e)
Timed out. \[ \int \frac {2+e \left (432+14256 x+20880 x^2+230416 x^3+185600 x^4+1216512 x^5+458752 x^6+2097152 x^7\right )+e \left (-432-14112 x-13872 x^2-150528 x^3-61440 x^4-393216 x^5\right ) \log (4)+e \left (144+4656 x+2304 x^2+24576 x^3\right ) \log ^2(4)+e (-16-512 x) \log ^3(4)}{x+e \left (162+216 x+3564 x^2+3480 x^3+28802 x^4+18560 x^5+101376 x^6+32768 x^7+131072 x^8\right )+e \left (-216-216 x-3528 x^2-2312 x^3-18816 x^4-6144 x^5-32768 x^6\right ) \log (4)+e \left (108+72 x+1164 x^2+384 x^3+3072 x^4\right ) \log ^2(4)+e \left (-24-8 x-128 x^2\right ) \log ^3(4)+2 e \log ^4(4)} \, dx=\text {Hanged} \] Input:
int((exp(1)*(14256*x + 20880*x^2 + 230416*x^3 + 185600*x^4 + 1216512*x^5 + 458752*x^6 + 2097152*x^7 + 432) + 4*exp(1)*log(2)^2*(4656*x + 2304*x^2 + 24576*x^3 + 144) - 2*exp(1)*log(2)*(14112*x + 13872*x^2 + 150528*x^3 + 614 40*x^4 + 393216*x^5 + 432) - 8*exp(1)*log(2)^3*(512*x + 16) + 2)/(x + exp( 1)*(216*x + 3564*x^2 + 3480*x^3 + 28802*x^4 + 18560*x^5 + 101376*x^6 + 327 68*x^7 + 131072*x^8 + 162) + 32*exp(1)*log(2)^4 + 4*exp(1)*log(2)^2*(72*x + 1164*x^2 + 384*x^3 + 3072*x^4 + 108) - 2*exp(1)*log(2)*(216*x + 3528*x^2 + 2312*x^3 + 18816*x^4 + 6144*x^5 + 32768*x^6 + 216) - 8*exp(1)*log(2)^3* (8*x + 128*x^2 + 24)),x)
Output:
\text{Hanged}
Time = 0.18 (sec) , antiderivative size = 182, normalized size of antiderivative = 8.27 \[ \int \frac {2+e \left (432+14256 x+20880 x^2+230416 x^3+185600 x^4+1216512 x^5+458752 x^6+2097152 x^7\right )+e \left (-432-14112 x-13872 x^2-150528 x^3-61440 x^4-393216 x^5\right ) \log (4)+e \left (144+4656 x+2304 x^2+24576 x^3\right ) \log ^2(4)+e (-16-512 x) \log ^3(4)}{x+e \left (162+216 x+3564 x^2+3480 x^3+28802 x^4+18560 x^5+101376 x^6+32768 x^7+131072 x^8\right )+e \left (-216-216 x-3528 x^2-2312 x^3-18816 x^4-6144 x^5-32768 x^6\right ) \log (4)+e \left (108+72 x+1164 x^2+384 x^3+3072 x^4\right ) \log ^2(4)+e \left (-24-8 x-128 x^2\right ) \log ^3(4)+2 e \log ^4(4)} \, dx=2 \,\mathrm {log}\left (x +162 e +32 \mathrm {log}\left (2\right )^{4} e -192 \mathrm {log}\left (2\right )^{3} e +432 \mathrm {log}\left (2\right )^{2} e -432 \,\mathrm {log}\left (2\right ) e +131072 e \,x^{8}+32768 e \,x^{7}+101376 e \,x^{6}+18560 e \,x^{5}+3480 e \,x^{3}-1024 \mathrm {log}\left (2\right )^{3} e \,x^{2}-64 \mathrm {log}\left (2\right )^{3} e x +12288 \mathrm {log}\left (2\right )^{2} e \,x^{4}+1536 \mathrm {log}\left (2\right )^{2} e \,x^{3}+4656 \mathrm {log}\left (2\right )^{2} e \,x^{2}+288 \mathrm {log}\left (2\right )^{2} e x -65536 \,\mathrm {log}\left (2\right ) e \,x^{6}-12288 \,\mathrm {log}\left (2\right ) e \,x^{5}-37632 \,\mathrm {log}\left (2\right ) e \,x^{4}-4624 \,\mathrm {log}\left (2\right ) e \,x^{3}-7056 \,\mathrm {log}\left (2\right ) e \,x^{2}-432 \,\mathrm {log}\left (2\right ) e x +3564 e \,x^{2}+216 e x +28802 e \,x^{4}\right ) \] Input:
int((8*(-512*x-16)*exp(1)*log(2)^3+4*(24576*x^3+2304*x^2+4656*x+144)*exp(1 )*log(2)^2+2*(-393216*x^5-61440*x^4-150528*x^3-13872*x^2-14112*x-432)*exp( 1)*log(2)+(2097152*x^7+458752*x^6+1216512*x^5+185600*x^4+230416*x^3+20880* x^2+14256*x+432)*exp(1)+2)/(32*exp(1)*log(2)^4+8*(-128*x^2-8*x-24)*exp(1)* log(2)^3+4*(3072*x^4+384*x^3+1164*x^2+72*x+108)*exp(1)*log(2)^2+2*(-32768* x^6-6144*x^5-18816*x^4-2312*x^3-3528*x^2-216*x-216)*exp(1)*log(2)+(131072* x^8+32768*x^7+101376*x^6+18560*x^5+28802*x^4+3480*x^3+3564*x^2+216*x+162)* exp(1)+x),x)
Output:
2*log(32*log(2)**4*e - 1024*log(2)**3*e*x**2 - 64*log(2)**3*e*x - 192*log( 2)**3*e + 12288*log(2)**2*e*x**4 + 1536*log(2)**2*e*x**3 + 4656*log(2)**2* e*x**2 + 288*log(2)**2*e*x + 432*log(2)**2*e - 65536*log(2)*e*x**6 - 12288 *log(2)*e*x**5 - 37632*log(2)*e*x**4 - 4624*log(2)*e*x**3 - 7056*log(2)*e* x**2 - 432*log(2)*e*x - 432*log(2)*e + 131072*e*x**8 + 32768*e*x**7 + 1013 76*e*x**6 + 18560*e*x**5 + 28802*e*x**4 + 3480*e*x**3 + 3564*e*x**2 + 216* e*x + 162*e + x)