3.23.85 \(\int \frac {(-2+x^3) (-1+x^3)^{2/3}}{x^3 (-1+2 x^3)} \, dx\) [2285]

Optimal. Leaf size=174 \[ -\frac {\left (-1+x^3\right )^{2/3}}{x^2}-\frac {1}{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^3}}\right )+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (x+\sqrt [3]{-1+x^3}\right )+\frac {1}{4} \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out]

-(x^3-1)^(2/3)/x^2-1/2*arctan(3^(1/2)*x/(-x+2*(x^3-1)^(1/3)))*3^(1/2)+1/6*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3))
)*3^(1/2)-1/6*ln(-x+(x^3-1)^(1/3))-1/2*ln(x+(x^3-1)^(1/3))+1/4*ln(x^2-x*(x^3-1)^(1/3)+(x^3-1)^(2/3))+1/12*ln(x
^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))

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Rubi [A]
time = 0.04, antiderivative size = 125, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {594, 544, 245, 384} \begin {gather*} \frac {1}{2} \sqrt {3} \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\frac {\text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (2 x^3-1\right )-\frac {3}{4} \log \left (-\sqrt [3]{x^3-1}-x\right )-\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\left (x^3-1\right )^{2/3}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((-1 + x^3)^(2/3)/x^2) + (Sqrt[3]*ArcTan[(1 - (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/2 + ArcTan[(1 + (2*x)/(-1 + x
^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Log[-1 + 2*x^3]/4 - (3*Log[-x - (-1 + x^3)^(1/3)])/4 - Log[-x + (-1 + x^3)^(
1/3)]/4

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 594

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx &=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}-\frac {1}{2} \int \frac {-2-2 x^3}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}+\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {3}{2} \int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{-1-x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {-2+x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{2} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{2} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 162, normalized size = 0.93 \begin {gather*} \frac {1}{12} \left (-\frac {12 \left (-1+x^3\right )^{2/3}}{x^2}+6 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^3}}\right )+2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (-x+\sqrt [3]{-1+x^3}\right )-6 \log \left (x+\sqrt [3]{-1+x^3}\right )+3 \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )+\log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*(-1 + x^3)^(2/3))/x^2 + 6*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(-1 + x^3)^(1/3))] + 2*Sqrt[3]*ArcTan[(Sqrt[
3]*x)/(x + 2*(-1 + x^3)^(1/3))] - 2*Log[-x + (-1 + x^3)^(1/3)] - 6*Log[x + (-1 + x^3)^(1/3)] + 3*Log[x^2 - x*(
-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)] + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/12

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 14.42, size = 836, normalized size = 4.80

method	result	size
trager	\(\text {Expression too large to display}\)	\(836\)
risch	\(\text {Expression too large to display}\)	\(1199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^3-1)^(2/3)/x^2-1/6*ln((-139696960898207711438078*(x^3-1)^(2/3)*x^10+419090882694623134314234*(x^3-1)^(2/3)
*x^4+3100410598970580503439069+28300298612496775796977664*RootOf(64*_Z^2-8*_Z+1)^2+18700889327124136369949940*
x^9-49803421050130594753656156*x^6+39403101898028224175982242*x^3+18845785825826111769772760*RootOf(64*_Z^2-8*
_Z+1)+279393921796415422876156*x*(x^3-1)^(2/3)+33016196884989482270840140*x^2*(x^3-1)^(1/3)+825404922124737056
7710035*x^11*(x^3-1)^(1/3)-24762147663742111703130105*x^8*(x^3-1)^(1/3)+55274020727532765228472000*RootOf(64*_
Z^2-8*_Z+1)^2*x^12-8026828278127257345063624*RootOf(64*_Z^2-8*_Z+1)*x^12-558046513265170797746653312*RootOf(64
*_Z^2-8*_Z+1)^2*x^9-102605316937845350087800336*RootOf(64*_Z^2-8*_Z+1)*x^9+1002891832080354492305395968*RootOf
(64*_Z^2-8*_Z+1)^2*x^6-611993957495242776609641984*RootOf(64*_Z^2-8*_Z+1)^2*x^3+129827490572386253096509632*Ro
otOf(64*_Z^2-8*_Z+1)*x^6-48860088729938611858127568*RootOf(64*_Z^2-8*_Z+1)*x^3-7250695686481463399577082*x^12+
66032393769978964541680280*(x^3-1)^(2/3)*RootOf(64*_Z^2-8*_Z+1)*x^10-64914818082793302850175656*(x^3-1)^(1/3)*
RootOf(64*_Z^2-8*_Z+1)*x^11+194744454248379908550526968*(x^3-1)^(1/3)*RootOf(64*_Z^2-8*_Z+1)*x^8-1980971813099
36893625040840*(x^3-1)^(2/3)*RootOf(64*_Z^2-8*_Z+1)*x^4-132064787539957929083360560*RootOf(64*_Z^2-8*_Z+1)*(x^
3-1)^(2/3)*x-259659272331173211400702624*(x^3-1)^(1/3)*RootOf(64*_Z^2-8*_Z+1)*x^2)/(2*x^3-1)^3)+4/3*RootOf(64*
_Z^2-8*_Z+1)*ln(-(-8114352260349162856271957*(x^3-1)^(2/3)*x^10+24343056781047488568815871*(x^3-1)^(2/3)*x^4-6
7191967266579353132732416*RootOf(64*_Z^2-8*_Z+1)^2-842928816094874669734198*x^9-5306305989843145461933312*x^6+
7033619137578544375323062*x^3+29664743373524966194481896*RootOf(64*_Z^2-8*_Z+1)+16228704520698325712543914*x*(
x^3-1)^(2/3)+33016196884989482270840140*x^2*(x^3-1)^(1/3)+8254049221247370567710035*x^11*(x^3-1)^(1/3)-2476214
7663742111703130105*x^8*(x^3-1)^(1/3)-131234311067537799087368000*RootOf(64*_Z^2-8*_Z+1)^2*x^12-48510529199351
077964254656*RootOf(64*_Z^2-8*_Z+1)*x^12+1324941604537861619586067328*RootOf(64*_Z^2-8*_Z+1)^2*x^9+53745228207
906738229672768*RootOf(64*_Z^2-8*_Z+1)*x^9-2381115340009405826641204992*RootOf(64*_Z^2-8*_Z+1)^2*x^6+145302629
2139778511495338496*RootOf(64*_Z^2-8*_Z+1)^2*x^3-226149429909913341237273120*RootOf(64*_Z^2-8*_Z+1)*x^6+210095
773353658826547145872*RootOf(64*_Z^2-8*_Z+1)*x^3-2190233071328485822178203*x^12+1748040905508223700350427+6603
2393769978964541680280*(x^3-1)^(2/3)*RootOf(64*_Z^2-8*_Z+1)*x^10-1117575687185661691504624*(x^3-1)^(1/3)*RootO
f(64*_Z^2-8*_Z+1)*x^11+3352727061556985074513872*(x^3-1)^(1/3)*RootOf(64*_Z^2-8*_Z+1)*x^8-19809718130993689362
5040840*(x^3-1)^(2/3)*RootOf(64*_Z^2-8*_Z+1)*x^4-132064787539957929083360560*RootOf(64*_Z^2-8*_Z+1)*(x^3-1)^(2
/3)*x-4470302748742646766018496*(x^3-1)^(1/3)*RootOf(64*_Z^2-8*_Z+1)*x^2)/(2*x^3-1)^3)

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

[Out]

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

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Fricas [A]
time = 3.29, size = 214, normalized size = 1.23 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {383838 \, \sqrt {3} {\left (x^{10} - 3 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 13468 \, \sqrt {3} {\left (x^{11} - 3 \, x^{8} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (198653 \, x^{12} + 393594 \, x^{9} + 5568 \, x^{6} - 400090 \, x^{3} - 198189\right )}}{3 \, {\left (185185 \, x^{12} + 370434 \, x^{9} - 96 \, x^{6} - 370322 \, x^{3} - 185193\right )}}\right ) - x^{2} \log \left (\frac {8 \, x^{9} - 12 \, x^{6} + 6 \, x^{3} - 3 \, {\left (x^{10} - 3 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{11} - 3 \, x^{8} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1}{8 \, x^{9} - 12 \, x^{6} + 6 \, x^{3} - 1}\right ) - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{12 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*sqrt(3)*x^2*arctan(1/3*(383838*sqrt(3)*(x^10 - 3*x^4 - 2*x)*(x^3 - 1)^(2/3) + 13468*sqrt(3)*(x^11 - 3*
x^8 + 4*x^2)*(x^3 - 1)^(1/3) + sqrt(3)*(198653*x^12 + 393594*x^9 + 5568*x^6 - 400090*x^3 - 198189))/(185185*x^
12 + 370434*x^9 - 96*x^6 - 370322*x^3 - 185193)) - x^2*log((8*x^9 - 12*x^6 + 6*x^3 - 3*(x^10 - 3*x^4 - 2*x)*(x
^3 - 1)^(2/3) + 3*(x^11 - 3*x^8 + 4*x^2)*(x^3 - 1)^(1/3) - 1)/(8*x^9 - 12*x^6 + 6*x^3 - 1)) - 12*(x^3 - 1)^(2/
3))/x^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} - 2\right )}{x^{3} \cdot \left (2 x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3-2\right )}{x^3\,\left (2\,x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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