3.1.21 \(\int \frac {1}{(1+\sqrt [3]{2} x) (1+x^3)^{2/3}} \, dx\) [21]

3.1.21.1 Optimal result

Integrand size = 21, antiderivative size = 147 \[ \int \frac {1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \left (2^{2/3}+x\right )}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (1+\sqrt [3]{2} x\right )}{2^{2/3}}-\frac {\log \left (x-\sqrt [3]{1+x^3}\right )}{2\ 2^{2/3}}+\frac {3 \log \left (2+\sqrt [3]{2} x-\sqrt [3]{2} \sqrt [3]{1+x^3}\right )}{2\ 2^{2/3}} \]

output

-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)+x)/(x^3+1)^(1/3))*3^ 
(1/2))*3^(1/2)*2^(1/3)
 

3.1.21.2 Mathematica [F]

\[ \int \frac {1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx=\int \frac {1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx \]

input

Integrate[1/((1 + 2^(1/3)*x)*(1 + x^3)^(2/3)),x]
 

output

Integrate[1/((1 + 2^(1/3)*x)*(1 + x^3)^(2/3)), x]
 

3.1.21.3 Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2579}

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.

\(\displaystyle \int \frac {1}{\left (\sqrt [3]{2} x+1\right ) \left (x^3+1\right )^{2/3}} \, dx\)

\(\Big \downarrow \) 2579

\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \left (x+2^{2/3}\right )}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (x-\sqrt [3]{x^3+1}\right )}{2\ 2^{2/3}}+\frac {3 \log \left (-\sqrt [3]{2} \sqrt [3]{x^3+1}+\sqrt [3]{2} x+2\right )}{2\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2} x+1\right )}{2^{2/3}}\)

input

Int[1/((1 + 2^(1/3)*x)*(1 + x^3)^(2/3)),x]
 

output

-(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))
 

3.1.21.3.1 Defintions of rubi rules used

rule 2579

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]
 

3.1.21.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 50.89 (sec) , antiderivative size = 3064, normalized size of antiderivative = 20.84

method	result	size
trager	\(\text {Expression too large to display}\)	\(3064\)

input

int(1/(1+2^(1/3)*x)/(x^3+1)^(2/3),x,method=_RETURNVERBOSE)
 

output

-1/6*ln(-(-15559137585059152-1604954020235328*2^(1/3)*x^4+1471207851882384 
0*x^3-936223178470608*x^6-12498127505504256*2^(1/3)*(x^3+1)^(1/3)-42798773 
87294208*x^5*2^(2/3)+6471910353179844*2^(1/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z 
^2)*(x^3+1)^(2/3)*x^3+1203809884289286*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_ 
Z^2)*(x^3+1)^(2/3)*x^4-3218487773589102*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+ 
_Z^2)*(x^3+1)^(1/3)*x^5-72607968203490*2^(1/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_ 
Z^2)^2*(x^3+1)^(2/3)*x^4-23004340956706368*2^(1/3)*x-1604954020235328*2^(2 
/3)*x^2-1560939140169318*2^(1/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*(x^3+1) 
^(1/3)*x^5+3775614346581480*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*(x^3 
+1)^(2/3)*x^2-3842311729647552*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*( 
x^3+1)^(1/3)*x^3-10498622607665136*2^(1/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2) 
*(x^3+1)^(1/3)*x^4-7759251414704196*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2 
)*(x^3+1)^(2/3)*x-3531674097632562*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2) 
*(x^3+1)^(1/3)*x^2+2613886855325640*2^(1/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2 
)^2*(x^3+1)^(2/3)*x[phone]*2^(1/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2 
)^2*(x^3+1)^(1/3)*x^2-11688730639030284*2^(1/3)*RootOf(2^(2/3)+2^(1/3)*_Z+ 
_Z^2)*(x^3+1)^(1/3)*x-5504178119120758*2^(2/3)*RootOf(2^(2/3)+2^(1/3)*_Z+_ 
Z^2)+7959206999356368*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*x^4-1039113368969860 
8*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*x+6434683875648336*RootOf(2^(2/3)+2^(1/3 
)*_Z+_Z^2)^2*x^2+6150800763487380*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*x^5...
 

3.1.21.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (112) = 224\).

Time = 3.74 (sec) , antiderivative size = 712, normalized size of antiderivative = 4.84 \[ \int \frac {1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx=\text {Too large to display} \]

input

integrate(1/(1+2^(1/3)*x)/(x^3+1)^(2/3),x, algorithm="fricas")
 

output

1/6*sqrt(3)*2^(1/3)*arctan(-1/3*(13910019318573948542*sqrt(3)*(44297109310 
930172741433829405399636654451725916403400759596345420183*x^16 + 469911753 
877577297266687493361266274298219751726156511748796788210304*x^13 - 168603 
219036433260440647021325346295645242325246375460547582960409424*x^10 - 197 
8806301182376573938292954227792627373330283397876582611558332893440*x^7 - 
1440090891687177581422918763089301968602581036872213084389912370301872*x^4 
 - 2^(2/3)*(52271077453125107612995923977654758349394876922885552819209999 
866413*x^15 + 590674547854548577293285820788340778493299281255213360593997 
994805172*x^12 + 306314261222931431619887382966630423064822217690279625339 
1978577817900*x^9 + 733104955869757780900835257159703940345796885706673027 
7786114959327080*x^6 + 772324480675629044375977054678087297173944475017351 
9635544186114816064*x^3 + 291168089878390092195634857418355141558919044601 
5106452608070501424800) + 6*2^(1/3)*(1260135599621632209331474867914912054 
3302140685677058235520929344665*x^14 - 55586906300196651392462719491921267 
847820798890019850227115938089718*x^11 - 450398920105320599307639536027883 
986131793624729303407436233610788504*x^8 - 7218887058809482614325170526703 
94106238338943844373553906510879866584*x^5 - 33866815806868437343630927306 
7849464405691360751378507442472921774544*x^2) - 62367643045453979229021701 
235594440425380660140976292433240780519680*x)*(x^3 + 1)^(2/3) - 1391001931 
8573948542*sqrt(3)*(202441513867627285828731764409166422760369138467219...
 

3.1.21.6 Sympy [F]

\[ \int \frac {1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx=\int \frac {1}{\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \cdot \left (\sqrt [3]{2} x + 1\right )}\, dx \]

input

integrate(1/(1+2**(1/3)*x)/(x**3+1)**(2/3),x)
 

output

Integral(1/(((x + 1)*(x**2 - x + 1))**(2/3)*(2**(1/3)*x + 1)), x)
 

3.1.21.7 Maxima [F]

\[ \int \frac {1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (2^{\frac {1}{3}} x + 1\right )}} \,d x } \]

input

integrate(1/(1+2^(1/3)*x)/(x^3+1)^(2/3),x, algorithm="maxima")
 

output

integrate(1/((x^3 + 1)^(2/3)*(2^(1/3)*x + 1)), x)
 

3.1.21.8 Giac [F]

\[ \int \frac {1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (2^{\frac {1}{3}} x + 1\right )}} \,d x } \]

input

integrate(1/(1+2^(1/3)*x)/(x^3+1)^(2/3),x, algorithm="giac")
 

output

integrate(1/((x^3 + 1)^(2/3)*(2^(1/3)*x + 1)), x)
 

3.1.21.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (1+\sqrt [3]{2} x\right ) \left (1+x^3\right )^{2/3}} \, dx=\int \frac {1}{{\left (x^3+1\right )}^{2/3}\,\left (2^{1/3}\,x+1\right )} \,d x \]

input

int(1/((x^3 + 1)^(2/3)*(2^(1/3)*x + 1)),x)
 

output

int(1/((x^3 + 1)^(2/3)*(2^(1/3)*x + 1)), x)