3.21 \(\int \frac{1}{(1+\sqrt [3]{2} x) (1+x^3)^{2/3}} \, dx\)
Optimal. Leaf size=147 \[ -\frac{\log \left (x-\sqrt [3]{x^3+1}\right )}{2\ 2^{2/3}}+\frac{3 \log \left (-\sqrt [3]{2} \sqrt [3]{x^3+1}+\sqrt [3]{2} x+2\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \left (x+2^{2/3}\right )}{\sqrt [3]{x^3+1}}+1}{\sqrt{3}}\right )}{2^{2/3}}-\frac{\log \left (\sqrt [3]{2} x+1\right )}{2^{2/3}} \]
[Out]
-(ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3])) + (Sqrt[3]*ArcTan[(1 + (2*(2^(2/3) + x))/(1 +
x^3)^(1/3))/Sqrt[3]])/2^(2/3) - 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))
________________________________________________________________________________________
Rubi [F] time = 0.0854677, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((1 + 2^(1/3)*x)*(1 + x^3)^(2/3)),x]
[Out]
Defer[Int][1/((1 + 2^(1/3)*x)*(1 + x^3)^(2/3)), x]
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 = 0.0724752, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx \]
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 [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{1+\sqrt [3]{2}x} \left ({x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1+2^(1/3)*x)/(x^3+1)^(2/3),x)
[Out]
int(1/(1+2^(1/3)*x)/(x^3+1)^(2/3),x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}^{\frac{2}{3}}{\left (2^{\frac{1}{3}} x + 1\right )}}\,{d x} \end{align*}
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] time = 20.1788, size = 7992, normalized size = 54.37 \begin{align*} \text{result too large to display} \end{align*}
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)*(44297109310930172741433829405399636654451725916
403400759596345420183*x^16 + 469911753877577297266687493361266274298219751726156511748796788210304*x^13 - 1686
03219036433260440647021325346295645242325246375460547582960409424*x^10 - 1978806301182376573938292954227792627
373330283397876582611558332893440*x^7 - 1440090891687177581422918763089301968602581036872213084389912370301872
*x^4 - 2^(2/3)*(52271077453125107612995923977654758349394876922885552819209999866413*x^15 + 590674547854548577
293285820788340778493299281255213360593997994805172*x^12 + 306314261222931431619887382966630423064822217690279
6253391978577817900*x^9 + 7331049558697577809008352571597039403457968857066730277786114959327080*x^6 + 7723244
806756290443759770546780872971739444750173519635544186114816064*x^3 + 2911680898783900921956348574183551415589
190446015106452608070501424800) + 6*2^(1/3)*(12601355996216322093314748679149120543302140685677058235520929344
665*x^14 - 55586906300196651392462719491921267847820798890019850227115938089718*x^11 - 45039892010532059930763
9536027883986131793624729303407436233610788504*x^8 - 721888705880948261432517052670394106238338943844373553906
510879866584*x^5 - 338668158068684373436309273067849464405691360751378507442472921774544*x^2) - 62367643045453
979229021701235594440425380660140976292433240780519680*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)*(58084585662481
4138058536658925035752422341023745042657144110018434133971171392378653765*x^18 + 85128502112016585963203224235
07979436745037061604662252288106173984889011398391939493844*x^15 + 4603767463429939987646493335333339365179871
4498861959697684952859181279514449172348801132*x^12 + 10001634835336681235799972394854096695243561183658042029
4833827058766585456463611215562912*x^9 + 913977586253668076790534210688867294404951076896021210254557365342556
42370122935700628112*x^6 + 27679206471222147818932348914707271406554121216141734785863966451139338545569046396
842944*x^3 - 13910019318573948542*2^(2/3)*(3844366680114123938578119587438413410802428820066154040455085354797
*x^17 - 493131971154919078063173195983280278594703770406004388326552124793591*x^14 - 2263656329733750526575239
788393341804272268328404078377386979655411628*x^11 - 360329608895964304006588260615697733294277836897086795884
1266275405688*x^8 - 2375143924145462474790789297643082581023352457583644433698318090272160*x^5 - 5385278270845
36759298395164308728360347336217790784309877024260129712*x^2) + 166920231822887382504*2^(1/3)*(135958920440428
28366275982006708049395032909698880004129949511339226*x^16 + 1351333848851582503771790485959913464507711993272
36207956421113461903*x^13 + 402245899028058436823068109521885840258775610614711826343657868879359*x^10 + 54725
8710149879334691832999834525308297790387563356879645468036532966*x^7 + 363674199703640963884960012124387263106
254909521640663154302302116404*x^4 + 97123895740704644005292055222464498011501842944639406026020532340120*x) -
1800774080838794461192653903259802591850188394016866170707655609076236167687893936558400))/(49127057457754733
7465577862499678580919468289682240641599400002541818630173299555553387*x^18 + 10277776658535231887928963830517
649364075160462302952752368573529738604577075128345830496*x^15 + 380530746040411649555986133825066575887180338
00015428848687354515819408113275280820067228*x^12 + 1045529773757864960561565152286863933606342503892061348166
52347595105200990156089430013680*x^9 + 19378977878621710892383256210067417618988113173228069450235805883107523
1461508817660387440*x^6 + 176250773615214113270216364768545940531543731282577338989973916134409349945587251955
701568*x^3 + 58729517358150193708087322484283950706773934182867349449322904141070201590185330889048000)) + 1/1
2*2^(1/3)*log((6048*x^16 + 6048*x^13 - 9072*x^10 - 12204*x^7 - 2808*x^4 + 2^(2/3)*(352*x^18 - 5136*x^15 - 1063
2*x^12 - 3224*x^9 + 3390*x^6 + 1434*x^3 - 35) + 3*(2032*x^14 + 752*x^11 - 3000*x^8 - 1576*x^5 + 172*x^2 + 2^(2
/3)*(112*x^16 - 1760*x^13 - 2228*x^10 + 356*x^7 + 707*x^4 - 22*x) - 2*2^(1/3)*(352*x^15 - 728*x^12 - 1736*x^9
- 451*x^6 + 215*x^3 - 1))*(x^3 + 1)^(2/3) - 18*2^(1/3)*(112*x^17 - 192*x^14 - 820*x^11 - 586*x^8 - 21*x^5 + 49
*x^2) + 3*(2096*x^15 + 1664*x^12 - 2680*x^9 - 2492*x^6 - 224*x^3 + 2^(2/3)*(112*x^17 - 1760*x^14 - 2996*x^11 -
472*x^8 + 779*x^5 + 125*x^2) - 2*2^(1/3)*(336*x^16 - 664*x^13 - 2132*x^10 - 1107*x^7 + 55*x^4 + 29*x) + 16)*(
x^3 + 1)^(1/3) + 324*x)/(64*x^18 + 192*x^15 + 240*x^12 + 160*x^9 + 60*x^6 + 12*x^3 + 1))
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac{2}{3}} \left (\sqrt [3]{2} x + 1\right )}\, dx \end{align*}
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/(((x + 1)*(x**2 - x + 1))**(2/3)*(2**(1/3)*x + 1)), x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}^{\frac{2}{3}}{\left (2^{\frac{1}{3}} x + 1\right )}}\,{d x} \end{align*}
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)