3.1.22 \(\int \frac {1}{(1-\sqrt [3]{2} x) (1-x^3)^{2/3}} \, dx\) [22]

3.1.22.1 Optimal result

Integrand size = 24, antiderivative size = 159 \[ \int \frac {1}{\left (1-\sqrt [3]{2} x\right ) \left (1-x^3\right )^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2\ 2^{2/3}-2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (1-\sqrt [3]{2} x\right )}{2^{2/3}}+\frac {\log \left (-x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {3 \log \left (-2+\sqrt [3]{2} x+\sqrt [3]{2} \sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]

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

3.1.22.2 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]
 

3.1.22.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, 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.

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

\(\Big \downarrow \) 2579

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

input

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

output

-((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))
 

3.1.22.3.1 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]
 

3.1.22.4 Maple [C] (verified)

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

Time = 48.47 (sec) , antiderivative size = 3247, normalized size of antiderivative = 20.42

method	result	size
trager	\(\text {Expression too large to display}\)	\(3247\)

input

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

output

1/6*ln((1604954020235328*2^(1/3)*x^4+14712078518823840*x^3-384231172964755 
2*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(2/3)*(-x^3+1)^(1/3)*x^3+64719103531 
79844*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*(-x^3+1)^(2/3)*x^3+104986226 
07665136*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3)*(-x^3+1)^(1/3)*x^4+261388 
6855325640*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3)*(-x^3+1)^(2/3)*x-7759 
251414704196*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(2/3)*(-x^3+1)^(2/3)*x-8405 
05690860402*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3)*(-x^3+1)^(1/3)*x^2+3 
531674097632562*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(2/3)*(-x^3+1)^(1/3)*x^2 
[phone]*x^6+12498127505504256*2^(1/3)*(-x^3+1)^(1/3)-427987738729 
4208*x^5*2^(2/3)-11688730639030284*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(1/3) 
*(-x^3+1)^(1/3)*x[phone]*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)^2*2^(1/3) 
*(-x^3+1)^(2/3)*x^4-1203809884289286*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2)*2^(2/ 
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^5-3218487773589102*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^2) 
*2^(2/3)*(-x^3+1)^(1/3)*x^5-3775614346581480*RootOf(2^(2/3)+2^(1/3)*_Z+_Z^ 
2)^2*2^(2/3)*(-x^3+1)^(2/3)*x^2-2884227944870616*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+3919074648194 
292*2^(2/3)*(-x^3+1)^(2/3)*x^3+6964190009986188*RootOf(2^(2/3)+2^(1/3)*...
 

3.1.22.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (124) = 248\).

Time = 3.52 (sec) , antiderivative size = 720, normalized size of antiderivative = 4.53 \[ \int \frac {1}{\left (1-\sqrt [3]{2} x\right ) \left (1-x^3\right )^{2/3}} \, dx=\text {Too large to display} \]

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)*(442971093109 
30172741433829405399636654451725916403400759596345420183*x^16 - 4699117538 
77577297266687493361266274298219751726156511748796788210304*x^13 - 1686032 
19036433260440647021325346295645242325246375460547582960409424*x^10 + 1978 
806301182376573938292954227792627373330283397876582611558332893440*x^7 - 1 
440090891687177581422918763089301968602581036872213084389912370301872*x^4 
+ 2^(2/3)*(522710774531251076129959239776547583493948769228855528192099998 
66413*x^15 - 5906745478545485772932858207883407784932992812552133605939979 
94805172*x^12 + 3063142612229314316198873829666304230648222176902796253391 
978577817900*x^9 - 7331049558697577809008352571597039403457968857066730277 
786114959327080*x^6 + 7723244806756290443759770546780872971739444750173519 
635544186114816064*x^3 - 2911680898783900921956348574183551415589190446015 
106452608070501424800) + 6*2^(1/3)*(12601355996216322093314748679149120543 
302140685677058235520929344665*x^14 + 555869063001966513924627194919212678 
47820798890019850227115938089718*x^11 - 4503989201053205993076395360278839 
86131793624729303407436233610788504*x^8 + 72188870588094826143251705267039 
4106238338943844373553906510879866584*x^5 - 338668158068684373436309273067 
849464405691360751378507442472921774544*x^2) + 623676430454539792290217012 
35594440425380660140976292433240780519680*x)*(-x^3 + 1)^(2/3) + 1391001931 
8573948542*sqrt(3)*(202441513867627285828731764409166422760369138467219...
 

3.1.22.6 Sympy [F]

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

input

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

output

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

3.1.22.7 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)
 

3.1.22.8 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)
 

3.1.22.9 Mupad [F(-1)]

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

input

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

output

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