3.2.70 \(\int \frac {57460436295776367187500 x^{15}+148439460430755615234375 x^{15} \log (\frac {x^2}{4})+167592939196014404296875 x^{15} \log ^2(\frac {x^2}{4})+108004338592987060546875 x^{15} \log ^3(\frac {x^2}{4})+43450021273040771484375 x^{15} \log ^4(\frac {x^2}{4})+11172862613067626953125 x^{15} \log ^5(\frac {x^2}{4})+1793175481109619140625 x^{15} \log ^6(\frac {x^2}{4})+164210208892822265625 x^{15} \log ^7(\frac {x^2}{4})+6568408355712890625 x^{15} \log ^8(\frac {x^2}{4})}{295147905179352825856} \, dx\)

Optimal. Leaf size=19 \[ \frac {6568408355712890625 x^{16} \left (3+\log \left (\frac {x^2}{4}\right )\right )^8}{4722366482869645213696} \]

________________________________________________________________________________________

Rubi [B]  time = 0.57, antiderivative size = 142, normalized size of antiderivative = 7.47, number of steps used = 38, number of rules used = 3, integrand size = 128, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {12, 2304, 2305} \begin {gather*} \frac {43095327221832275390625 x^{16}}{4722366482869645213696}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {19705225067138671875 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {413809726409912109375 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {1241429179229736328125 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {18621437688446044921875 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{2361183241434822606848}+\frac {11172862613067626953125 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {33518587839202880859375 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {14365109073944091796875 x^{16} \log \left (\frac {x^2}{4}\right )}{590295810358705651712} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(57460436295776367187500*x^15 + 148439460430755615234375*x^15*Log[x^2/4] + 167592939196014404296875*x^15*L
og[x^2/4]^2 + 108004338592987060546875*x^15*Log[x^2/4]^3 + 43450021273040771484375*x^15*Log[x^2/4]^4 + 1117286
2613067626953125*x^15*Log[x^2/4]^5 + 1793175481109619140625*x^15*Log[x^2/4]^6 + 164210208892822265625*x^15*Log
[x^2/4]^7 + 6568408355712890625*x^15*Log[x^2/4]^8)/295147905179352825856,x]

[Out]

(43095327221832275390625*x^16)/4722366482869645213696 + (14365109073944091796875*x^16*Log[x^2/4])/590295810358
705651712 + (33518587839202880859375*x^16*Log[x^2/4]^2)/1180591620717411303424 + (11172862613067626953125*x^16
*Log[x^2/4]^3)/590295810358705651712 + (18621437688446044921875*x^16*Log[x^2/4]^4)/2361183241434822606848 + (1
241429179229736328125*x^16*Log[x^2/4]^5)/590295810358705651712 + (413809726409912109375*x^16*Log[x^2/4]^6)/118
0591620717411303424 + (19705225067138671875*x^16*Log[x^2/4]^7)/590295810358705651712 + (6568408355712890625*x^
16*Log[x^2/4]^8)/4722366482869645213696

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (57460436295776367187500 x^{15}+148439460430755615234375 x^{15} \log \left (\frac {x^2}{4}\right )+167592939196014404296875 x^{15} \log ^2\left (\frac {x^2}{4}\right )+108004338592987060546875 x^{15} \log ^3\left (\frac {x^2}{4}\right )+43450021273040771484375 x^{15} \log ^4\left (\frac {x^2}{4}\right )+11172862613067626953125 x^{15} \log ^5\left (\frac {x^2}{4}\right )+1793175481109619140625 x^{15} \log ^6\left (\frac {x^2}{4}\right )+164210208892822265625 x^{15} \log ^7\left (\frac {x^2}{4}\right )+6568408355712890625 x^{15} \log ^8\left (\frac {x^2}{4}\right )\right ) \, dx}{295147905179352825856}\\ &=\frac {14365109073944091796875 x^{16}}{1180591620717411303424}+\frac {6568408355712890625 \int x^{15} \log ^8\left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}+\frac {164210208892822265625 \int x^{15} \log ^7\left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}+\frac {1793175481109619140625 \int x^{15} \log ^6\left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}+\frac {11172862613067626953125 \int x^{15} \log ^5\left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}+\frac {43450021273040771484375 \int x^{15} \log ^4\left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}+\frac {108004338592987060546875 \int x^{15} \log ^3\left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}+\frac {148439460430755615234375 \int x^{15} \log \left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}+\frac {167592939196014404296875 \int x^{15} \log ^2\left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}\\ &=\frac {311244029935455322265625 x^{16}}{37778931862957161709568}+\frac {148439460430755615234375 x^{16} \log \left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {167592939196014404296875 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {108004338592987060546875 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {43450021273040771484375 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {11172862613067626953125 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {1793175481109619140625 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {164210208892822265625 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}-\frac {6568408355712890625 \int x^{15} \log ^7\left (\frac {x^2}{4}\right ) \, dx}{295147905179352825856}-\frac {1149471462249755859375 \int x^{15} \log ^6\left (\frac {x^2}{4}\right ) \, dx}{2361183241434822606848}-\frac {5379526443328857421875 \int x^{15} \log ^5\left (\frac {x^2}{4}\right ) \, dx}{1180591620717411303424}-\frac {55864313065338134765625 \int x^{15} \log ^4\left (\frac {x^2}{4}\right ) \, dx}{2361183241434822606848}-\frac {43450021273040771484375 \int x^{15} \log ^3\left (\frac {x^2}{4}\right ) \, dx}{590295810358705651712}-\frac {324013015778961181640625 \int x^{15} \log ^2\left (\frac {x^2}{4}\right ) \, dx}{2361183241434822606848}-\frac {167592939196014404296875 \int x^{15} \log \left (\frac {x^2}{4}\right ) \, dx}{1180591620717411303424}\\ &=\frac {1412569058937835693359375 x^{16}}{151115727451828646838272}+\frac {426164902527008056640625 x^{16} \log \left (\frac {x^2}{4}\right )}{18889465931478580854784}+\frac {1016730497789154052734375 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{37778931862957161709568}+\frac {172558655912933349609375 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{9444732965739290427392}+\frac {291735857118988037109375 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{37778931862957161709568}+\frac {39311924008941650390625 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{18889465931478580854784}+\frac {13195932386627197265625 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{37778931862957161709568}+\frac {19705225067138671875 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {45978858489990234375 \int x^{15} \log ^6\left (\frac {x^2}{4}\right ) \, dx}{2361183241434822606848}+\frac {3448414386749267578125 \int x^{15} \log ^5\left (\frac {x^2}{4}\right ) \, dx}{9444732965739290427392}+\frac {26897632216644287109375 \int x^{15} \log ^4\left (\frac {x^2}{4}\right ) \, dx}{9444732965739290427392}+\frac {55864313065338134765625 \int x^{15} \log ^3\left (\frac {x^2}{4}\right ) \, dx}{4722366482869645213696}+\frac {130350063819122314453125 \int x^{15} \log ^2\left (\frac {x^2}{4}\right ) \, dx}{4722366482869645213696}+\frac {324013015778961181640625 \int x^{15} \log \left (\frac {x^2}{4}\right ) \, dx}{9444732965739290427392}\\ &=\frac {10976539455723724365234375 x^{16}}{1208925819614629174706176}+\frac {3733332235995025634765625 x^{16} \log \left (\frac {x^2}{4}\right )}{151115727451828646838272}+\frac {2163811059397430419921875 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{75557863725914323419136}+\frac {1436333560368804931640625 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{75557863725914323419136}+\frac {1193841060692596435546875 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{151115727451828646838272}+\frac {317943806458282470703125 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{151115727451828646838272}+\frac {413809726409912109375 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {19705225067138671875 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}-\frac {137936575469970703125 \int x^{15} \log ^5\left (\frac {x^2}{4}\right ) \, dx}{9444732965739290427392}-\frac {17242071933746337890625 \int x^{15} \log ^4\left (\frac {x^2}{4}\right ) \, dx}{75557863725914323419136}-\frac {26897632216644287109375 \int x^{15} \log ^3\left (\frac {x^2}{4}\right ) \, dx}{18889465931478580854784}-\frac {167592939196014404296875 \int x^{15} \log ^2\left (\frac {x^2}{4}\right ) \, dx}{37778931862957161709568}-\frac {130350063819122314453125 \int x^{15} \log \left (\frac {x^2}{4}\right ) \, dx}{18889465931478580854784}\\ &=\frac {22083428975266571044921875 x^{16}}{2417851639229258349412352}+\frac {7336314408170928955078125 x^{16} \log \left (\frac {x^2}{4}\right )}{302231454903657293676544}+\frac {17142895535983428955078125 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{604462909807314587353088}+\frac {5718436609258575439453125 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{302231454903657293676544}+\frac {9533486413607025146484375 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{1208925819614629174706176}+\frac {1241429179229736328125 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {413809726409912109375 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {19705225067138671875 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {689682877349853515625 \int x^{15} \log ^4\left (\frac {x^2}{4}\right ) \, dx}{75557863725914323419136}+\frac {17242071933746337890625 \int x^{15} \log ^3\left (\frac {x^2}{4}\right ) \, dx}{151115727451828646838272}+\frac {80692896649932861328125 \int x^{15} \log ^2\left (\frac {x^2}{4}\right ) \, dx}{151115727451828646838272}+\frac {167592939196014404296875 \int x^{15} \log \left (\frac {x^2}{4}\right ) \, dx}{151115727451828646838272}\\ &=\frac {176499838862936553955078125 x^{16}}{19342813113834066795298816}+\frac {58858108204563446044921875 x^{16} \log \left (\frac {x^2}{4}\right )}{2417851639229258349412352}+\frac {68652275040583648681640625 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{2417851639229258349412352}+\frac {45764734946002349853515625 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{2417851639229258349412352}+\frac {18621437688446044921875 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{2361183241434822606848}+\frac {1241429179229736328125 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {413809726409912109375 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {19705225067138671875 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}-\frac {689682877349853515625 \int x^{15} \log ^3\left (\frac {x^2}{4}\right ) \, dx}{151115727451828646838272}-\frac {51726215801239013671875 \int x^{15} \log ^2\left (\frac {x^2}{4}\right ) \, dx}{1208925819614629174706176}-\frac {80692896649932861328125 \int x^{15} \log \left (\frac {x^2}{4}\right ) \, dx}{604462909807314587353088}\\ &=\frac {706080048348396148681640625 x^{16}}{77371252455336267181195264}+\frac {235351739921603851318359375 x^{16} \log \left (\frac {x^2}{4}\right )}{9671406556917033397649408}+\frac {549166474108867950439453125 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{19342813113834066795298816}+\frac {11172862613067626953125 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {18621437688446044921875 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{2361183241434822606848}+\frac {1241429179229736328125 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {413809726409912109375 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {19705225067138671875 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}+\frac {2069048632049560546875 \int x^{15} \log ^2\left (\frac {x^2}{4}\right ) \, dx}{1208925819614629174706176}+\frac {51726215801239013671875 \int x^{15} \log \left (\frac {x^2}{4}\right ) \, dx}{4835703278458516698824704}\\ &=\frac {5648588660571367950439453125 x^{16}}{618970019642690137449562112}+\frac {1882865645588632049560546875 x^{16} \log \left (\frac {x^2}{4}\right )}{77371252455336267181195264}+\frac {33518587839202880859375 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {11172862613067626953125 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {18621437688446044921875 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{2361183241434822606848}+\frac {1241429179229736328125 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {413809726409912109375 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {19705225067138671875 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}-\frac {2069048632049560546875 \int x^{15} \log \left (\frac {x^2}{4}\right ) \, dx}{4835703278458516698824704}\\ &=\frac {43095327221832275390625 x^{16}}{4722366482869645213696}+\frac {14365109073944091796875 x^{16} \log \left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {33518587839202880859375 x^{16} \log ^2\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {11172862613067626953125 x^{16} \log ^3\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {18621437688446044921875 x^{16} \log ^4\left (\frac {x^2}{4}\right )}{2361183241434822606848}+\frac {1241429179229736328125 x^{16} \log ^5\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {413809726409912109375 x^{16} \log ^6\left (\frac {x^2}{4}\right )}{1180591620717411303424}+\frac {19705225067138671875 x^{16} \log ^7\left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {6568408355712890625 x^{16} \log ^8\left (\frac {x^2}{4}\right )}{4722366482869645213696}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.14, size = 146, normalized size = 7.68 \begin {gather*} \frac {6568408355712890625 \left (\frac {6561 x^{16}}{16}+\frac {2187}{2} x^{16} \log \left (\frac {x^2}{4}\right )+\frac {5103}{4} x^{16} \log ^2\left (\frac {x^2}{4}\right )+\frac {1701}{2} x^{16} \log ^3\left (\frac {x^2}{4}\right )+\frac {2835}{8} x^{16} \log ^4\left (\frac {x^2}{4}\right )+\frac {189}{2} x^{16} \log ^5\left (\frac {x^2}{4}\right )+\frac {63}{4} x^{16} \log ^6\left (\frac {x^2}{4}\right )+\frac {3}{2} x^{16} \log ^7\left (\frac {x^2}{4}\right )+\frac {1}{16} x^{16} \log ^8\left (\frac {x^2}{4}\right )\right )}{295147905179352825856} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(57460436295776367187500*x^15 + 148439460430755615234375*x^15*Log[x^2/4] + 167592939196014404296875*
x^15*Log[x^2/4]^2 + 108004338592987060546875*x^15*Log[x^2/4]^3 + 43450021273040771484375*x^15*Log[x^2/4]^4 + 1
1172862613067626953125*x^15*Log[x^2/4]^5 + 1793175481109619140625*x^15*Log[x^2/4]^6 + 164210208892822265625*x^
15*Log[x^2/4]^7 + 6568408355712890625*x^15*Log[x^2/4]^8)/295147905179352825856,x]

[Out]

(6568408355712890625*((6561*x^16)/16 + (2187*x^16*Log[x^2/4])/2 + (5103*x^16*Log[x^2/4]^2)/4 + (1701*x^16*Log[
x^2/4]^3)/2 + (2835*x^16*Log[x^2/4]^4)/8 + (189*x^16*Log[x^2/4]^5)/2 + (63*x^16*Log[x^2/4]^6)/4 + (3*x^16*Log[
x^2/4]^7)/2 + (x^16*Log[x^2/4]^8)/16))/295147905179352825856

________________________________________________________________________________________

fricas [B]  time = 0.67, size = 108, normalized size = 5.68 \begin {gather*} \frac {6568408355712890625}{4722366482869645213696} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{8} + \frac {19705225067138671875}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{7} + \frac {413809726409912109375}{1180591620717411303424} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{6} + \frac {1241429179229736328125}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{5} + \frac {18621437688446044921875}{2361183241434822606848} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{4} + \frac {11172862613067626953125}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{3} + \frac {33518587839202880859375}{1180591620717411303424} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{2} + \frac {14365109073944091796875}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right ) + \frac {43095327221832275390625}{4722366482869645213696} \, x^{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6568408355712890625/295147905179352825856*x^15*log(1/4*x^2)^8+164210208892822265625/2951479051793528
25856*x^15*log(1/4*x^2)^7+1793175481109619140625/295147905179352825856*x^15*log(1/4*x^2)^6+1117286261306762695
3125/295147905179352825856*x^15*log(1/4*x^2)^5+43450021273040771484375/295147905179352825856*x^15*log(1/4*x^2)
^4+108004338592987060546875/295147905179352825856*x^15*log(1/4*x^2)^3+167592939196014404296875/295147905179352
825856*x^15*log(1/4*x^2)^2+148439460430755615234375/295147905179352825856*x^15*log(1/4*x^2)+143651090739440917
96875/73786976294838206464*x^15,x, algorithm="fricas")

[Out]

6568408355712890625/4722366482869645213696*x^16*log(1/4*x^2)^8 + 19705225067138671875/590295810358705651712*x^
16*log(1/4*x^2)^7 + 413809726409912109375/1180591620717411303424*x^16*log(1/4*x^2)^6 + 1241429179229736328125/
590295810358705651712*x^16*log(1/4*x^2)^5 + 18621437688446044921875/2361183241434822606848*x^16*log(1/4*x^2)^4
+ 11172862613067626953125/590295810358705651712*x^16*log(1/4*x^2)^3 + 33518587839202880859375/118059162071741
1303424*x^16*log(1/4*x^2)^2 + 14365109073944091796875/590295810358705651712*x^16*log(1/4*x^2) + 43095327221832
275390625/4722366482869645213696*x^16

________________________________________________________________________________________

giac [B]  time = 0.73, size = 108, normalized size = 5.68 \begin {gather*} \frac {6568408355712890625}{4722366482869645213696} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{8} + \frac {19705225067138671875}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{7} + \frac {413809726409912109375}{1180591620717411303424} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{6} + \frac {1241429179229736328125}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{5} + \frac {18621437688446044921875}{2361183241434822606848} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{4} + \frac {11172862613067626953125}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{3} + \frac {33518587839202880859375}{1180591620717411303424} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{2} + \frac {14365109073944091796875}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right ) + \frac {43095327221832275390625}{4722366482869645213696} \, x^{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6568408355712890625/295147905179352825856*x^15*log(1/4*x^2)^8+164210208892822265625/2951479051793528
25856*x^15*log(1/4*x^2)^7+1793175481109619140625/295147905179352825856*x^15*log(1/4*x^2)^6+1117286261306762695
3125/295147905179352825856*x^15*log(1/4*x^2)^5+43450021273040771484375/295147905179352825856*x^15*log(1/4*x^2)
^4+108004338592987060546875/295147905179352825856*x^15*log(1/4*x^2)^3+167592939196014404296875/295147905179352
825856*x^15*log(1/4*x^2)^2+148439460430755615234375/295147905179352825856*x^15*log(1/4*x^2)+143651090739440917
96875/73786976294838206464*x^15,x, algorithm="giac")

[Out]

6568408355712890625/4722366482869645213696*x^16*log(1/4*x^2)^8 + 19705225067138671875/590295810358705651712*x^
16*log(1/4*x^2)^7 + 413809726409912109375/1180591620717411303424*x^16*log(1/4*x^2)^6 + 1241429179229736328125/
590295810358705651712*x^16*log(1/4*x^2)^5 + 18621437688446044921875/2361183241434822606848*x^16*log(1/4*x^2)^4
+ 11172862613067626953125/590295810358705651712*x^16*log(1/4*x^2)^3 + 33518587839202880859375/118059162071741
1303424*x^16*log(1/4*x^2)^2 + 14365109073944091796875/590295810358705651712*x^16*log(1/4*x^2) + 43095327221832
275390625/4722366482869645213696*x^16

________________________________________________________________________________________

maple [B]  time = 0.05, size = 109, normalized size = 5.74

method	result	size
risch	\(\frac {6568408355712890625 x^{16} \ln \left (\frac {x^{2}}{4}\right )^{8}}{4722366482869645213696}+\frac {19705225067138671875 x^{16} \ln \left (\frac {x^{2}}{4}\right )^{7}}{590295810358705651712}+\frac {413809726409912109375 x^{16} \ln \left (\frac {x^{2}}{4}\right )^{6}}{1180591620717411303424}+\frac {1241429179229736328125 x^{16} \ln \left (\frac {x^{2}}{4}\right )^{5}}{590295810358705651712}+\frac {18621437688446044921875 x^{16} \ln \left (\frac {x^{2}}{4}\right )^{4}}{2361183241434822606848}+\frac {11172862613067626953125 x^{16} \ln \left (\frac {x^{2}}{4}\right )^{3}}{590295810358705651712}+\frac {33518587839202880859375 x^{16} \ln \left (\frac {x^{2}}{4}\right )^{2}}{1180591620717411303424}+\frac {14365109073944091796875 x^{16} \ln \left (\frac {x^{2}}{4}\right )}{590295810358705651712}+\frac {43095327221832275390625 x^{16}}{4722366482869645213696}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6568408355712890625/295147905179352825856*x^15*ln(1/4*x^2)^8+164210208892822265625/295147905179352825856*x
^15*ln(1/4*x^2)^7+1793175481109619140625/295147905179352825856*x^15*ln(1/4*x^2)^6+11172862613067626953125/2951
47905179352825856*x^15*ln(1/4*x^2)^5+43450021273040771484375/295147905179352825856*x^15*ln(1/4*x^2)^4+10800433
8592987060546875/295147905179352825856*x^15*ln(1/4*x^2)^3+167592939196014404296875/295147905179352825856*x^15*
ln(1/4*x^2)^2+148439460430755615234375/295147905179352825856*x^15*ln(1/4*x^2)+14365109073944091796875/73786976
294838206464*x^15,x,method=_RETURNVERBOSE)

[Out]

6568408355712890625/4722366482869645213696*x^16*ln(1/4*x^2)^8+19705225067138671875/590295810358705651712*x^16*
ln(1/4*x^2)^7+413809726409912109375/1180591620717411303424*x^16*ln(1/4*x^2)^6+1241429179229736328125/590295810
358705651712*x^16*ln(1/4*x^2)^5+18621437688446044921875/2361183241434822606848*x^16*ln(1/4*x^2)^4+111728626130
67626953125/590295810358705651712*x^16*ln(1/4*x^2)^3+33518587839202880859375/1180591620717411303424*x^16*ln(1/
4*x^2)^2+14365109073944091796875/590295810358705651712*x^16*ln(1/4*x^2)+43095327221832275390625/47223664828696
45213696*x^16

________________________________________________________________________________________

maxima [B]  time = 0.37, size = 108, normalized size = 5.68 \begin {gather*} \frac {6568408355712890625}{4722366482869645213696} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{8} + \frac {19705225067138671875}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{7} + \frac {413809726409912109375}{1180591620717411303424} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{6} + \frac {1241429179229736328125}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{5} + \frac {18621437688446044921875}{2361183241434822606848} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{4} + \frac {11172862613067626953125}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{3} + \frac {33518587839202880859375}{1180591620717411303424} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right )^{2} + \frac {14365109073944091796875}{590295810358705651712} \, x^{16} \log \left (\frac {1}{4} \, x^{2}\right ) + \frac {43095327221832275390625}{4722366482869645213696} \, x^{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6568408355712890625/295147905179352825856*x^15*log(1/4*x^2)^8+164210208892822265625/2951479051793528
25856*x^15*log(1/4*x^2)^7+1793175481109619140625/295147905179352825856*x^15*log(1/4*x^2)^6+1117286261306762695
3125/295147905179352825856*x^15*log(1/4*x^2)^5+43450021273040771484375/295147905179352825856*x^15*log(1/4*x^2)
^4+108004338592987060546875/295147905179352825856*x^15*log(1/4*x^2)^3+167592939196014404296875/295147905179352
825856*x^15*log(1/4*x^2)^2+148439460430755615234375/295147905179352825856*x^15*log(1/4*x^2)+143651090739440917
96875/73786976294838206464*x^15,x, algorithm="maxima")

[Out]

6568408355712890625/4722366482869645213696*x^16*log(1/4*x^2)^8 + 19705225067138671875/590295810358705651712*x^
16*log(1/4*x^2)^7 + 413809726409912109375/1180591620717411303424*x^16*log(1/4*x^2)^6 + 1241429179229736328125/
590295810358705651712*x^16*log(1/4*x^2)^5 + 18621437688446044921875/2361183241434822606848*x^16*log(1/4*x^2)^4
+ 11172862613067626953125/590295810358705651712*x^16*log(1/4*x^2)^3 + 33518587839202880859375/118059162071741
1303424*x^16*log(1/4*x^2)^2 + 14365109073944091796875/590295810358705651712*x^16*log(1/4*x^2) + 43095327221832
275390625/4722366482869645213696*x^16

________________________________________________________________________________________

mupad [B]  time = 0.54, size = 108, normalized size = 5.68 \begin {gather*} \frac {6568408355712890625\,x^{16}\,{\ln \left (\frac {x^2}{4}\right )}^8}{4722366482869645213696}+\frac {19705225067138671875\,x^{16}\,{\ln \left (\frac {x^2}{4}\right )}^7}{590295810358705651712}+\frac {413809726409912109375\,x^{16}\,{\ln \left (\frac {x^2}{4}\right )}^6}{1180591620717411303424}+\frac {1241429179229736328125\,x^{16}\,{\ln \left (\frac {x^2}{4}\right )}^5}{590295810358705651712}+\frac {18621437688446044921875\,x^{16}\,{\ln \left (\frac {x^2}{4}\right )}^4}{2361183241434822606848}+\frac {11172862613067626953125\,x^{16}\,{\ln \left (\frac {x^2}{4}\right )}^3}{590295810358705651712}+\frac {33518587839202880859375\,x^{16}\,{\ln \left (\frac {x^2}{4}\right )}^2}{1180591620717411303424}+\frac {14365109073944091796875\,x^{16}\,\ln \left (\frac {x^2}{4}\right )}{590295810358705651712}+\frac {43095327221832275390625\,x^{16}}{4722366482869645213696} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((167592939196014404296875*x^15*log(x^2/4)^2)/295147905179352825856 + (108004338592987060546875*x^15*log(x^
2/4)^3)/295147905179352825856 + (43450021273040771484375*x^15*log(x^2/4)^4)/295147905179352825856 + (111728626
13067626953125*x^15*log(x^2/4)^5)/295147905179352825856 + (1793175481109619140625*x^15*log(x^2/4)^6)/295147905
179352825856 + (164210208892822265625*x^15*log(x^2/4)^7)/295147905179352825856 + (6568408355712890625*x^15*log
(x^2/4)^8)/295147905179352825856 + (14365109073944091796875*x^15)/73786976294838206464 + (14843946043075561523
4375*x^15*log(x^2/4))/295147905179352825856,x)

[Out]

(33518587839202880859375*x^16*log(x^2/4)^2)/1180591620717411303424 + (11172862613067626953125*x^16*log(x^2/4)^
3)/590295810358705651712 + (18621437688446044921875*x^16*log(x^2/4)^4)/2361183241434822606848 + (1241429179229
736328125*x^16*log(x^2/4)^5)/590295810358705651712 + (413809726409912109375*x^16*log(x^2/4)^6)/118059162071741
1303424 + (19705225067138671875*x^16*log(x^2/4)^7)/590295810358705651712 + (6568408355712890625*x^16*log(x^2/4
)^8)/4722366482869645213696 + (43095327221832275390625*x^16)/4722366482869645213696 + (14365109073944091796875
*x^16*log(x^2/4))/590295810358705651712

________________________________________________________________________________________

sympy [B]  time = 0.32, size = 126, normalized size = 6.63 \begin {gather*} \frac {6568408355712890625 x^{16} \log {\left (\frac {x^{2}}{4} \right )}^{8}}{4722366482869645213696} + \frac {19705225067138671875 x^{16} \log {\left (\frac {x^{2}}{4} \right )}^{7}}{590295810358705651712} + \frac {413809726409912109375 x^{16} \log {\left (\frac {x^{2}}{4} \right )}^{6}}{1180591620717411303424} + \frac {1241429179229736328125 x^{16} \log {\left (\frac {x^{2}}{4} \right )}^{5}}{590295810358705651712} + \frac {18621437688446044921875 x^{16} \log {\left (\frac {x^{2}}{4} \right )}^{4}}{2361183241434822606848} + \frac {11172862613067626953125 x^{16} \log {\left (\frac {x^{2}}{4} \right )}^{3}}{590295810358705651712} + \frac {33518587839202880859375 x^{16} \log {\left (\frac {x^{2}}{4} \right )}^{2}}{1180591620717411303424} + \frac {14365109073944091796875 x^{16} \log {\left (\frac {x^{2}}{4} \right )}}{590295810358705651712} + \frac {43095327221832275390625 x^{16}}{4722366482869645213696} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6568408355712890625/295147905179352825856*x**15*ln(1/4*x**2)**8+164210208892822265625/29514790517935
2825856*x**15*ln(1/4*x**2)**7+1793175481109619140625/295147905179352825856*x**15*ln(1/4*x**2)**6+1117286261306
7626953125/295147905179352825856*x**15*ln(1/4*x**2)**5+43450021273040771484375/295147905179352825856*x**15*ln(
1/4*x**2)**4+108004338592987060546875/295147905179352825856*x**15*ln(1/4*x**2)**3+167592939196014404296875/295
147905179352825856*x**15*ln(1/4*x**2)**2+148439460430755615234375/295147905179352825856*x**15*ln(1/4*x**2)+143
65109073944091796875/73786976294838206464*x**15,x)

[Out]

6568408355712890625*x**16*log(x**2/4)**8/4722366482869645213696 + 19705225067138671875*x**16*log(x**2/4)**7/59
0295810358705651712 + 413809726409912109375*x**16*log(x**2/4)**6/1180591620717411303424 + 12414291792297363281
25*x**16*log(x**2/4)**5/590295810358705651712 + 18621437688446044921875*x**16*log(x**2/4)**4/23611832414348226
06848 + 11172862613067626953125*x**16*log(x**2/4)**3/590295810358705651712 + 33518587839202880859375*x**16*log
(x**2/4)**2/1180591620717411303424 + 14365109073944091796875*x**16*log(x**2/4)/590295810358705651712 + 4309532
7221832275390625*x**16/4722366482869645213696

________________________________________________________________________________________