3.1.22 \(\int \frac {1}{(1-\sqrt [3]{2} x) (1-x^3)^{2/3}} \, dx\) [22]
Optimal. Leaf size=159 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2\ 2^{2/3}-2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (1-\sqrt [3]{2} x\right )}{2^{2/3}}+\frac {\log \left (-x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {3 \log \left (-2+\sqrt [3]{2} x+\sqrt [3]{2} \sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]
[Out]
1/2*ln(1-2^(1/3)*x)*2^(1/3)+1/4*ln(-x-(-x^3+1)^(1/3))*2^(1/3)-3/4*ln(-2+2^(1/3)*x+2^(1/3)*(-x^3+1)^(1/3))*2^(1
/3)+1/6*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)-1/2*arctan(1/3*(1+(2*2^(2/3)-2*x)/(-x^3+1)^
(1/3))*3^(1/2))*3^(1/2)*2^(1/3)
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2179}
\begin {gather*} -\frac {\sqrt {3} \text {ArcTan}\left (\frac {\frac {2\ 2^{2/3}-2 x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (-\sqrt [3]{1-x^3}-x\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{1-x^3}+\sqrt [3]{2} x-2\right )}{2\ 2^{2/3}}+\frac {\log \left (1-\sqrt [3]{2} x\right )}{2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((1 - 2^(1/3)*x)*(1 - x^3)^(2/3)),x]
[Out]
-((Sqrt[3]*ArcTan[(1 + (2*2^(2/3) - 2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/2^(2/3)) + ArcTan[(1 - (2*x)/(1 - x^3)^(1/
3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) + Log[1 - 2^(1/3)*x]/2^(2/3) + Log[-x - (1 - x^3)^(1/3)]/(2*2^(2/3)) - (3*Log[-
2 + 2^(1/3)*x + 2^(1/3)*(1 - x^3)^(1/3)])/(2*2^(2/3))
Rule 2179
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(2/3)), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[(-d)*(ArcTan[(
1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(2*Sqrt[3]*q^2*c^2)), x] + (Simp[Sqrt[3]*d*(ArcTan[(1 + 2*q*((2*c + d*
x)/(d*(a + b*x^3)^(1/3))))/Sqrt[3]]/(2*q^2*c^2)), x] - Simp[d*(Log[c + d*x]/(2*q^2*c^2)), x] - Simp[d*(Log[q*x
- (a + b*x^3)^(1/3)]/(4*q^2*c^2)), x] + Simp[3*d*(Log[q*(2*c + d*x) - d*(a + b*x^3)^(1/3)]/(4*q^2*c^2)), x])]
/; FreeQ[{a, b, c, d}, x] && EqQ[2*b*c^3 - a*d^3, 0]
Rubi steps
\begin {align*} \int \frac {1}{\left (1-\sqrt [3]{2} x\right ) \left (1-x^3\right )^{2/3}} \, dx &=\int \frac {1}{\left (1-\sqrt [3]{2} x\right ) \left (1-x^3\right )^{2/3}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 6.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (1-\sqrt [3]{2} x\right ) \left (1-x^3\right )^{2/3}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[1/((1 - 2^(1/3)*x)*(1 - x^3)^(2/3)),x]
[Out]
Integrate[1/((1 - 2^(1/3)*x)*(1 - x^3)^(2/3)), x]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 27.10, size = 3250, normalized size = 20.44
| | |
| method |
result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(3250\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1-2^(1/3)*x)/(-x^3+1)^(2/3),x,method=_RETURNVERBOSE)
[Out]
1/6*ln(-(-15559137585059152-936223178470608*x^6-14712078518823840*x^3-12498127505504256*2^(1/3)*(-x^3+1)^(1/3)
-1604954020235328*2^(1/3)*x^4+4279877387294208*x^5*2^(2/3)+23004340956706368*2^(1/3)*x-1604954020235328*2^(2/3
)*x^2+7959206999356368*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*x^2+2321435374812274*2^(2/3)*RootOf(2^(2/3)+2^(
1/3)*_Z+_Z^2)*x^6+1876782797064098*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3)*x^6-10197714008127436*2^(2/3)*Roo
tOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*x^3-5665414413224496*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*x^5-3532767618003008
*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3)*x^3+3217341937824168*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*x^4-
2081809489180344*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*x+2884227944870616*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*
(-x^3+1)^(2/3)*x^2+8123294120973864*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*(-x^3+1)^(1/3)*x^3-2613886855325640*RootOf
(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*(-x^3+1)^(2/3)*x^3-11138422684341672*2^(1/3)*(-x^3+1)^(2/3)*x^2-3919074648194292*2
^(2/3)*(-x^3+1)^(2/3)*x^3-1315592369000448*2^(2/3)*(-x^3+1)^(1/3)*x^4-6964190009986188*RootOf(2^(2/3)+2^(1/3)*
_Z+_Z^2)^2*(-x^3+1)^(1/3)*x^4-3288980922501120*2^(1/3)*(-x^3+1)^(1/3)*x^3+20062783627256832*2^(2/3)*(-x^3+1)^(
1/3)*x-7115580883942020*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*(-x^3+1)^(2/3)-2161300347926748*RootOf(2^(2/3)
+2^(1/3)*_Z+_Z^2)^2*(-x^3+1)^(1/3)*x-10107087250606332*2^(2/3)*(-x^3+1)^(2/3)-9125490357912936*RootOf(2^(2/3)+
2^(1/3)*_Z+_Z^2)*(-x^3+1)^(1/3)+3712807561447224*(-x^3+1)^(2/3)*x^4+2960082830251008*(-x^3+1)^(1/3)*x^5+325901
99706036744*(-x^3+1)^(2/3)*x-12827025597754368*(-x^3+1)^(1/3)*x^2+7959206999356368*RootOf(2^(2/3)+2^(1/3)*_Z+_
Z^2)*x^4+6434683875648336*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*x^2-5504178119120758*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^
2)*2^(2/3)+10391133689698608*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*x-6150800763487380*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2
)^2*x^5+3775614346581480*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*(-x^3+1)^(2/3)*x^2+3842311729647552*RootOf(
2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*(-x^3+1)^(1/3)*x^3-6471910353179844*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)
*(-x^3+1)^(2/3)*x^3-10498622607665136*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*(-x^3+1)^(1/3)*x^4+7759251414704
196*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(2/3)*(-x^3+1)^(2/3)*x-3531674097632562*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*
2^(2/3)*(-x^3+1)^(1/3)*x^2-2613886855325640*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3)*(-x^3+1)^(2/3)*x+8405056
90860402*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3)*(-x^3+1)^(1/3)*x^2+11688730639030284*RootOf(2^(2/3)+2^(1/3)
*_Z+_Z^2)*2^(1/3)*(-x^3+1)^(1/3)*x+1203809884289286*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(2/3)*(-x^3+1)^(2/3)*x^4
+3218487773589102*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(2/3)*(-x^3+1)^(1/3)*x^5-72607968203490*RootOf(2^(2/3)+2^(
1/3)*_Z+_Z^2)^2*2^(1/3)*(-x^3+1)^(2/3)*x^4+1560939140169318*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3)*(-x^3+1)
^(1/3)*x^5)/(2^(1/3)*x-1)^6)*2^(1/3)+1/6*ln(-(-15559137585059152-936223178470608*x^6-14712078518823840*x^3-124
98127505504256*2^(1/3)*(-x^3+1)^(1/3)-1604954020235328*2^(1/3)*x^4+4279877387294208*x^5*2^(2/3)+23004340956706
368*2^(1/3)*x-1604954020235328*2^(2/3)*x^2+7959206999356368*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*x^2+232143
5374812274*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*x^6+1876782797064098*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1
/3)*x^6-10197714008127436*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*x^3-5665414413224496*RootOf(2^(2/3)+2^(1/3)*
_Z+_Z^2)*2^(1/3)*x^5-3532767618003008*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3)*x^3+3217341937824168*RootOf(2^
(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*x^4-2081809489180344*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*x+288422794487
0616*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*(-x^3+1)^(2/3)*x^2+8123294120973864*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*(-x^3
+1)^(1/3)*x^3-2613886855325640*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*(-x^3+1)^(2/3)*x^3-11138422684341672*2^(1/3)*
(-x^3+1)^(2/3)*x^2-3919074648194292*2^(2/3)*(-x^3+1)^(2/3)*x^3-1315592369000448*2^(2/3)*(-x^3+1)^(1/3)*x^4-696
4190009986188*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*(-x^3+1)^(1/3)*x^4-3288980922501120*2^(1/3)*(-x^3+1)^(1/3)*x^3
+20062783627256832*2^(2/3)*(-x^3+1)^(1/3)*x-7115580883942020*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*(-x^3+1)^
(2/3)-2161300347926748*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*(-x^3+1)^(1/3)*x-10107087250606332*2^(2/3)*(-x^3+1)^(
2/3)-9125490357912936*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*(-x^3+1)^(1/3)+3712807561447224*(-x^3+1)^(2/3)*x^4+29600
82830251008*(-x^3+1)^(1/3)*x^5+32590199706036744*(-x^3+1)^(2/3)*x-12827025597754368*(-x^3+1)^(1/3)*x^2+7959206
999356368*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*x^4+6434683875648336*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*x^2-550417811
9120758*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(2/3)+10391133689698608*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*x-6150800763
487380*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*x^5+3775614346581480*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*(-x^3+
1)^(2/3)*x^2+3842311729647552*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*(-x^3+1)^(1/3)*x^3-6471910353179844*Ro
otOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*(-x^3+1)^...
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2^(1/3)*x)/(-x^3+1)^(2/3),x, algorithm="maxima")
[Out]
-integrate(1/((-x^3 + 1)^(2/3)*(2^(1/3)*x - 1)), x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 720 vs.
\(2 (124) = 248\).
time = 2.59, size = 720, normalized size = 4.53 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2^(1/3)*x)/(-x^3+1)^(2/3),x, algorithm="fricas")
[Out]
1/6*sqrt(3)*2^(1/3)*arctan(1/3*(13910019318573948542*sqrt(3)*(442971093109301727414338294053996366544517259164
03400759596345420183*x^16 - 469911753877577297266687493361266274298219751726156511748796788210304*x^13 - 16860
3219036433260440647021325346295645242325246375460547582960409424*x^10 + 19788063011823765739382929542277926273
73330283397876582611558332893440*x^7 - 1440090891687177581422918763089301968602581036872213084389912370301872*
x^4 + 2^(2/3)*(52271077453125107612995923977654758349394876922885552819209999866413*x^15 - 5906745478545485772
93285820788340778493299281255213360593997994805172*x^12 + 3063142612229314316198873829666304230648222176902796
253391978577817900*x^9 - 7331049558697577809008352571597039403457968857066730277786114959327080*x^6 + 77232448
06756290443759770546780872971739444750173519635544186114816064*x^3 - 29116808987839009219563485741835514155891
90446015106452608070501424800) + 6*2^(1/3)*(126013559962163220933147486791491205433021406856770582355209293446
65*x^14 + 55586906300196651392462719491921267847820798890019850227115938089718*x^11 - 450398920105320599307639
536027883986131793624729303407436233610788504*x^8 + 7218887058809482614325170526703941062383389438443735539065
10879866584*x^5 - 338668158068684373436309273067849464405691360751378507442472921774544*x^2) + 623676430454539
79229021701235594440425380660140976292433240780519680*x)*(-x^3 + 1)^(2/3) + 13910019318573948542*sqrt(3)*(2024
4151386762728582873176440916642276036913846721964342570319874272*x^17 - 74114613707883499096895869495695352578
6968216162791369141561079231342*x^14 + 2179843197271775401147438396101666875537043663345199103065290718350660*
x^11 - 2111024935028444803027635033172373996998638870275081528835019029426808*x^8 + 69058397930221264954184667
1752323578671762361564987198532372077617072*x^5 - 425604467193959940435036909294930892503769478498985960943870
69196992*x^2 - 2^(2/3)*(58175953016441250552894129028785848895343146706912452780410096144857*x^16 - 6033291234
40225928459512442880846367498086340467210508410170807919392*x^13 + 9932177244211605146408029249702161488721380
06799356417482692017634440*x^10 + 315373668616978600368729679828820826067145203897860799345951918357208*x^7 -
1535989781175898454904009764080477698123439140009523257833795294171024*x^4 + 774581653994506522185065060515457
999562469670838035710700279100960480*x) - 2*2^(1/3)*(442503373958626236413084321461052655915849816922169442468
72622437586*x^15 - 93730331994553087914588193029404165015738145719370012253256237142833*x^12 + 132185413165954
5520395638093435834861993288285254840631143087754453816*x^9 - 424770576770174688958921382572527816220243177376
0010908121531655858240*x^6 + 4593245463688643634993735851341621838359838170188285500151733185855040*x^3 - 1615
883737614789297142910770786922880950970969890530541101538638738800))*(-x^3 + 1)^(1/3) + sqrt(3)*(5808458566248
14138058536658925035752422341023745042657144110018434133971171392378653765*x^18 - 8512850211201658596320322423
507979436745037061604662252288106173984889011398391939493844*x^15 + 460376746342993998764649333533333936517987
14498861959697684952859181279514449172348801132*x^12 - 1000163483533668123579997239485409669524356118365804202
94833827058766585456463611215562912*x^9 + 91397758625366807679053421068886729440495107689602121025455736534255
642370122935700628112*x^6 - 2767920647122214781893234891470727140655412121614173478586396645113933854556904639
6842944*x^3 + 13910019318573948542*2^(2/3)*(384436668011412393857811958743841341080242882006615404045508535479
7*x^17 + 493131971154919078063173195983280278594703770406004388326552124793591*x^14 - 226365632973375052657523
9788393341804272268328404078377386979655411628*x^11 + 36032960889596430400658826061569773329427783689708679588
41266275405688*x^8 - 2375143924145462474790789297643082581023352457583644433698318090272160*x^5 + 538527827084
536759298395164308728360347336217790784309877024260129712*x^2) + 166920231822887382504*2^(1/3)*(13595892044042
828366275982006708049395032909698880004129949511339226*x^16 - 135133384885158250377179048595991346450771199327
236207956421113461903*x^13 + 402245899028058436823068109521885840258775610614711826343657868879359*x^10 - 5472
58710149879334691832999834525308297790387563356879645468036532966*x^7 + 36367419970364096388496001212438726310
6254909521640663154302302116404*x^4 - 97123895740704644005292055222464498011501842944639406026020532340120*x)
- 1800774080838794461192653903259802591850188394016866170707655609076236167687893936558400))/(4912705745775473
37465577862499678580919468289682240641599400002541818630173299555553387*x^18 - 1027777665853523188792896383051
7649364075160462302952752368573529738604577075128345830496*x^15 + 38053074604041164955598613382506657588718033
800015428848687354515819408113275280820067228*x^12 - 104552977375786496056156515228686393360634250389206134816
652347595105200990156089430013680*x^9 + 1937897787862171089238325621006741761898811317322806945023580588310752
31461508817660387440*x^6 - 17625077361521411327...
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{\sqrt [3]{2} x \left (1 - x^{3}\right )^{\frac {2}{3}} - \left (1 - x^{3}\right )^{\frac {2}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2**(1/3)*x)/(-x**3+1)**(2/3),x)
[Out]
-Integral(1/(2**(1/3)*x*(1 - x**3)**(2/3) - (1 - x**3)**(2/3)), x)
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2^(1/3)*x)/(-x^3+1)^(2/3),x, algorithm="giac")
[Out]
integrate(-1/((-x^3 + 1)^(2/3)*(2^(1/3)*x - 1)), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{{\left (1-x^3\right )}^{2/3}\,\left (2^{1/3}\,x-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-1/((1 - x^3)^(2/3)*(2^(1/3)*x - 1)),x)
[Out]
-int(1/((1 - x^3)^(2/3)*(2^(1/3)*x - 1)), x)
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