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3.15.95 \(\int \frac {(-3+2 x) (1-x+x^3)^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx\) [1495]

Optimal. Leaf size=104 \[ -\text {ArcTan}\left (\frac {x}{\sqrt [3]{1-x+x^3}}\right )-\frac {1}{2} \text {ArcTan}\left (\frac {x \sqrt [3]{1-x+x^3}}{-x^2+\left (1-x+x^3\right )^{2/3}}\right )-\frac {1}{2} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x \sqrt [3]{1-x+x^3}}{x^2+\left (1-x+x^3\right )^{2/3}}\right ) \]

[Out]

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

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Rubi [F]
time = 0.35, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-3 + 2*x)*(1 - x + x^3)^(2/3))/(1 - 2*x + x^2 + 2*x^3 - 2*x^4 + 2*x^6),x]

[Out]

-3*Defer[Int][(1 - x + x^3)^(2/3)/(1 - 2*x + x^2 + 2*x^3 - 2*x^4 + 2*x^6), x] + 2*Defer[Int][(x*(1 - x + x^3)^
(2/3))/(1 - 2*x + x^2 + 2*x^3 - 2*x^4 + 2*x^6), x]

Rubi steps

\begin {align*} \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx &=\int \left (-\frac {3 \left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6}+\frac {2 x \left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6}\right ) \, dx\\ &=2 \int \frac {x \left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx-3 \int \frac {\left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.36, size = 106, normalized size = 1.02 \begin {gather*} -\text {ArcTan}\left (\frac {x}{\sqrt [3]{1-x+x^3}}\right )-\frac {1}{2} i \left (-i+\sqrt {3}\right ) \text {ArcTan}\left (\frac {\left (1-i \sqrt {3}\right ) x}{2 \sqrt [3]{1-x+x^3}}\right )+\frac {1}{2} i \left (i+\sqrt {3}\right ) \text {ArcTan}\left (\frac {\left (1+i \sqrt {3}\right ) x}{2 \sqrt [3]{1-x+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-3 + 2*x)*(1 - x + x^3)^(2/3))/(1 - 2*x + x^2 + 2*x^3 - 2*x^4 + 2*x^6),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 23.26, size = 3735, normalized size = 35.91

method result size
trager \(\text {Expression too large to display}\) \(3735\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3+2*x)*(x^3-x+1)^(2/3)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

4*ln((-3+6*x+384*RootOf(16*_Z^4-4*_Z^2+1)^6*(x^3-x+1)^(2/3)*x^4-896*RootOf(16*_Z^4-4*_Z^2+1)^6*(x^3-x+1)^(1/3)
*x^5-192*RootOf(16*_Z^4-4*_Z^2+1)^5*(x^3-x+1)^(1/3)*x^5-384*RootOf(16*_Z^4-4*_Z^2+1)^4*(x^3-x+1)^(2/3)*x^4+224
*RootOf(16*_Z^4-4*_Z^2+1)^4*(x^3-x+1)^(1/3)*x^5-288*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(2/3)*x^4-240*RootOf(
16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(1/3)*x^5+96*RootOf(16*_Z^4-4*_Z^2+1)^2*(x^3-x+1)^(2/3)*x^4+40*RootOf(16*_Z^4-4*
_Z^2+1)^2*(x^3-x+1)^(1/3)*x^5-48*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(2/3)*x^2+144*RootOf(16*_Z^4-4*_Z^2+1)*(
x^3-x+1)^(2/3)*x^4+336*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(1/3)*x^3-84*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(1
/3)*x^5+48*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(2/3)*x-336*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(1/3)*x^2+72*
RootOf(16*_Z^4-4*_Z^2+1)^2*(x^3-x+1)^(1/3)*x^3+24*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(2/3)*x^2-72*RootOf(16*_Z
^4-4*_Z^2+1)^2*(x^3-x+1)^(1/3)*x^2+84*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(1/3)*x^3-24*RootOf(16*_Z^4-4*_Z^2+1)
*(x^3-x+1)^(2/3)*x-84*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(1/3)*x^2+3584*RootOf(16*_Z^4-4*_Z^2+1)^6*x^6-6*x^6-3
*x^2-6*x^3+6*x^4+112*RootOf(16*_Z^4-4*_Z^2+1)^3+12*RootOf(16*_Z^4-4*_Z^2+1)^2+248*x^6*RootOf(16*_Z^4-4*_Z^2+1)
^2+20*x^6*RootOf(16*_Z^4-4*_Z^2+1)-64*RootOf(16*_Z^4-4*_Z^2+1)^3*x^6-248*RootOf(16*_Z^4-4*_Z^2+1)^2*x^4+248*Ro
otOf(16*_Z^4-4*_Z^2+1)^2*x^3+12*RootOf(16*_Z^4-4*_Z^2+1)^2*x^2-24*RootOf(16*_Z^4-4*_Z^2+1)^2*x+64*RootOf(16*_Z
^4-4*_Z^2+1)^3*x^4-64*RootOf(16*_Z^4-4*_Z^2+1)^3*x^3+112*RootOf(16*_Z^4-4*_Z^2+1)^3*x^2-20*RootOf(16*_Z^4-4*_Z
^2+1)*x^4-224*RootOf(16*_Z^4-4*_Z^2+1)^3*x+20*RootOf(16*_Z^4-4*_Z^2+1)*x^3-14*RootOf(16*_Z^4-4*_Z^2+1)*x^2+28*
RootOf(16*_Z^4-4*_Z^2+1)*x+3584*RootOf(16*_Z^4-4*_Z^2+1)^6*x^3+384*RootOf(16*_Z^4-4*_Z^2+1)^5*x^3-2240*RootOf(
16*_Z^4-4*_Z^2+1)^4*x^3-14*RootOf(16*_Z^4-4*_Z^2+1)+384*RootOf(16*_Z^4-4*_Z^2+1)^5*x^6-3584*RootOf(16*_Z^4-4*_
Z^2+1)^6*x^4-2240*RootOf(16*_Z^4-4*_Z^2+1)^4*x^6-384*RootOf(16*_Z^4-4*_Z^2+1)^5*x^4+2240*RootOf(16*_Z^4-4*_Z^2
+1)^4*x^4+42*(x^3-x+1)^(2/3)*x^2-42*(x^3-x+1)^(2/3)*x-42*(x^3-x+1)^(2/3)*x^4)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1))*R
ootOf(16*_Z^4-4*_Z^2+1)^3-ln((-3+6*x+384*RootOf(16*_Z^4-4*_Z^2+1)^6*(x^3-x+1)^(2/3)*x^4-896*RootOf(16*_Z^4-4*_
Z^2+1)^6*(x^3-x+1)^(1/3)*x^5-192*RootOf(16*_Z^4-4*_Z^2+1)^5*(x^3-x+1)^(1/3)*x^5-384*RootOf(16*_Z^4-4*_Z^2+1)^4
*(x^3-x+1)^(2/3)*x^4+224*RootOf(16*_Z^4-4*_Z^2+1)^4*(x^3-x+1)^(1/3)*x^5-288*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+
1)^(2/3)*x^4-240*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(1/3)*x^5+96*RootOf(16*_Z^4-4*_Z^2+1)^2*(x^3-x+1)^(2/3)*
x^4+40*RootOf(16*_Z^4-4*_Z^2+1)^2*(x^3-x+1)^(1/3)*x^5-48*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(2/3)*x^2+144*Ro
otOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(2/3)*x^4+336*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(1/3)*x^3-84*RootOf(16*_Z^
4-4*_Z^2+1)*(x^3-x+1)^(1/3)*x^5+48*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(2/3)*x-336*RootOf(16*_Z^4-4*_Z^2+1)^3
*(x^3-x+1)^(1/3)*x^2+72*RootOf(16*_Z^4-4*_Z^2+1)^2*(x^3-x+1)^(1/3)*x^3+24*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(
2/3)*x^2-72*RootOf(16*_Z^4-4*_Z^2+1)^2*(x^3-x+1)^(1/3)*x^2+84*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(1/3)*x^3-24*
RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(2/3)*x-84*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(1/3)*x^2+3584*RootOf(16*_Z^4
-4*_Z^2+1)^6*x^6-6*x^6-3*x^2-6*x^3+6*x^4+112*RootOf(16*_Z^4-4*_Z^2+1)^3+12*RootOf(16*_Z^4-4*_Z^2+1)^2+248*x^6*
RootOf(16*_Z^4-4*_Z^2+1)^2+20*x^6*RootOf(16*_Z^4-4*_Z^2+1)-64*RootOf(16*_Z^4-4*_Z^2+1)^3*x^6-248*RootOf(16*_Z^
4-4*_Z^2+1)^2*x^4+248*RootOf(16*_Z^4-4*_Z^2+1)^2*x^3+12*RootOf(16*_Z^4-4*_Z^2+1)^2*x^2-24*RootOf(16*_Z^4-4*_Z^
2+1)^2*x+64*RootOf(16*_Z^4-4*_Z^2+1)^3*x^4-64*RootOf(16*_Z^4-4*_Z^2+1)^3*x^3+112*RootOf(16*_Z^4-4*_Z^2+1)^3*x^
2-20*RootOf(16*_Z^4-4*_Z^2+1)*x^4-224*RootOf(16*_Z^4-4*_Z^2+1)^3*x+20*RootOf(16*_Z^4-4*_Z^2+1)*x^3-14*RootOf(1
6*_Z^4-4*_Z^2+1)*x^2+28*RootOf(16*_Z^4-4*_Z^2+1)*x+3584*RootOf(16*_Z^4-4*_Z^2+1)^6*x^3+384*RootOf(16*_Z^4-4*_Z
^2+1)^5*x^3-2240*RootOf(16*_Z^4-4*_Z^2+1)^4*x^3-14*RootOf(16*_Z^4-4*_Z^2+1)+384*RootOf(16*_Z^4-4*_Z^2+1)^5*x^6
-3584*RootOf(16*_Z^4-4*_Z^2+1)^6*x^4-2240*RootOf(16*_Z^4-4*_Z^2+1)^4*x^6-384*RootOf(16*_Z^4-4*_Z^2+1)^5*x^4+22
40*RootOf(16*_Z^4-4*_Z^2+1)^4*x^4+42*(x^3-x+1)^(2/3)*x^2-42*(x^3-x+1)^(2/3)*x-42*(x^3-x+1)^(2/3)*x^4)/(2*x^6-2
*x^4+2*x^3+x^2-2*x+1))*RootOf(16*_Z^4-4*_Z^2+1)+RootOf(16*_Z^4-4*_Z^2+1)*ln((192*RootOf(16*_Z^4-4*_Z^2+1)^6*(x
^3-x+1)^(2/3)*x^4+896*RootOf(16*_Z^4-4*_Z^2+1)^6*(x^3-x+1)^(1/3)*x^5-96*RootOf(16*_Z^4-4*_Z^2+1)^5*(x^3-x+1)^(
1/3)*x^5-192*RootOf(16*_Z^4-4*_Z^2+1)^4*(x^3-x+1)^(2/3)*x^4-560*RootOf(16*_Z^4-4*_Z^2+1)^4*(x^3-x+1)^(1/3)*x^5
+144*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(2/3)*x^4-264*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^(1/3)*x^5+48*Root
Of(16*_Z^4-4*_Z^2+1)^2*(x^3-x+1)^(2/3)*x^4+92*RootOf(16*_Z^4-4*_Z^2+1)^2*(x^3-x+1)^(1/3)*x^5+24*RootOf(16*_Z^4
-4*_Z^2+1)^3*(x^3-x+1)^(2/3)*x^2-72*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(2/3)*x^4+336*RootOf(16*_Z^4-4*_Z^2+1)^
3*(x^3-x+1)^(1/3)*x^3+30*RootOf(16*_Z^4-4*_Z^2+1)*(x^3-x+1)^(1/3)*x^5-24*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1)^
(2/3)*x-336*RootOf(16*_Z^4-4*_Z^2+1)^3*(x^3-x+1...

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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((-3+2*x)*(x^3-x+1)^(2/3)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1),x, algorithm="maxima")

[Out]

integrate((x^3 - x + 1)^(2/3)*(2*x - 3)/(2*x^6 - 2*x^4 + 2*x^3 + x^2 - 2*x + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2090 vs. \(2 (87) = 174\).
time = 3.60, size = 2090, normalized size = 20.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*x)*(x^3-x+1)^(2/3)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1),x, algorithm="fricas")

[Out]

-1/8*sqrt(3)*log(8*(2*x^6 - 2*x^4 + 2*x^3 + x^2 + 2*(3*x^4 + sqrt(3)*(x^4 - x^2 + x))*(x^3 - x + 1)^(2/3) + 4*
sqrt(3)*(x^6 - x^4 + x^3) + 2*(sqrt(3)*x^5 + 3*x^5 - 3*x^3 + 3*x^2)*(x^3 - x + 1)^(1/3) - 2*x + 1)/(2*x^6 - 2*
x^4 + 2*x^3 + x^2 - 2*x + 1)) + 1/8*sqrt(3)*log(8*(2*x^6 - 2*x^4 + 2*x^3 + x^2 + 2*(3*x^4 - sqrt(3)*(x^4 - x^2
 + x))*(x^3 - x + 1)^(2/3) - 4*sqrt(3)*(x^6 - x^4 + x^3) - 2*(sqrt(3)*x^5 - 3*x^5 + 3*x^3 - 3*x^2)*(x^3 - x +
1)^(1/3) - 2*x + 1)/(2*x^6 - 2*x^4 + 2*x^3 + x^2 - 2*x + 1)) - 1/2*arctan((24*x^12 - 16*x^10 + 16*x^9 - 32*x^8
 + 64*x^7 - 8*x^6 - 72*x^5 + 70*x^4 - 16*x^3 - 12*x^2 - sqrt(2)*(116*x^12 - 300*x^10 + 300*x^9 + 240*x^8 - 480
*x^7 + 186*x^6 + 162*x^5 - 163*x^4 + 58*x^3 - 6*x^2 + 4*(22*x^10 - 30*x^8 + 30*x^7 + 13*x^6 - 26*x^5 + 11*x^4
+ 6*x^3 - 6*x^2 - sqrt(3)*(14*x^10 - 20*x^8 + 20*x^7 + 3*x^6 - 6*x^5 + 4*x^4 - 3*x^3 + 3*x^2 - x) + 2*x)*(x^3
- x + 1)^(2/3) - sqrt(3)*(36*x^12 - 36*x^10 + 36*x^9 - 4*x^8 + 8*x^7 - 2*x^6 - 6*x^5 + 7*x^4 - 6*x^3 + 6*x^2 -
 4*x + 1) + 2*(28*x^11 - 2*x^9 + 2*x^8 - 40*x^7 + 80*x^6 - 25*x^5 - 45*x^4 + 45*x^3 - 15*x^2 - sqrt(3)*(32*x^1
1 - 70*x^9 + 70*x^8 + 46*x^7 - 92*x^6 + 37*x^5 + 27*x^4 - 27*x^3 + 9*x^2))*(x^3 - x + 1)^(1/3) + 4*x - 1)*sqrt
((2*x^6 - 2*x^4 + 2*x^3 + x^2 + 2*(3*x^4 + sqrt(3)*(x^4 - x^2 + x))*(x^3 - x + 1)^(2/3) + 4*sqrt(3)*(x^6 - x^4
 + x^3) + 2*(sqrt(3)*x^5 + 3*x^5 - 3*x^3 + 3*x^2)*(x^3 - x + 1)^(1/3) - 2*x + 1)/(2*x^6 - 2*x^4 + 2*x^3 + x^2
- 2*x + 1)) + 4*(18*x^10 - 46*x^8 + 46*x^7 + 23*x^6 - 46*x^5 + 21*x^4 + 6*x^3 - 6*x^2 - sqrt(3)*(2*x^10 + 4*x^
8 - 4*x^7 - 3*x^6 + 6*x^5 - 2*x^4 - 3*x^3 + 3*x^2 - x) + 2*x)*(x^3 - x + 1)^(2/3) - sqrt(3)*(20*x^12 - 40*x^10
 + 40*x^9 + 24*x^8 - 48*x^7 + 20*x^6 + 12*x^5 - 11*x^4 + 6*x^2 - 4*x + 1) + 4*(2*x^11 + 16*x^9 - 16*x^8 - 27*x
^7 + 54*x^6 - 20*x^5 - 21*x^4 + 21*x^3 - 7*x^2 - sqrt(3)*(6*x^11 - 14*x^9 + 14*x^8 + 13*x^7 - 26*x^6 + 9*x^5 +
 12*x^4 - 12*x^3 + 4*x^2))*(x^3 - x + 1)^(1/3) + 8*x - 2)/(52*x^12 - 232*x^10 + 232*x^9 + 248*x^8 - 496*x^7 +
180*x^6 + 204*x^5 - 203*x^4 + 64*x^3 + 6*x^2 - 4*x + 1)) + 1/2*arctan(-(24*x^12 - 16*x^10 + 16*x^9 - 32*x^8 +
64*x^7 - 8*x^6 - 72*x^5 + 70*x^4 - 16*x^3 - 12*x^2 - sqrt(2)*(116*x^12 - 300*x^10 + 300*x^9 + 240*x^8 - 480*x^
7 + 186*x^6 + 162*x^5 - 163*x^4 + 58*x^3 - 6*x^2 + 4*(22*x^10 - 30*x^8 + 30*x^7 + 13*x^6 - 26*x^5 + 11*x^4 + 6
*x^3 - 6*x^2 + sqrt(3)*(14*x^10 - 20*x^8 + 20*x^7 + 3*x^6 - 6*x^5 + 4*x^4 - 3*x^3 + 3*x^2 - x) + 2*x)*(x^3 - x
 + 1)^(2/3) + sqrt(3)*(36*x^12 - 36*x^10 + 36*x^9 - 4*x^8 + 8*x^7 - 2*x^6 - 6*x^5 + 7*x^4 - 6*x^3 + 6*x^2 - 4*
x + 1) + 2*(28*x^11 - 2*x^9 + 2*x^8 - 40*x^7 + 80*x^6 - 25*x^5 - 45*x^4 + 45*x^3 - 15*x^2 + sqrt(3)*(32*x^11 -
 70*x^9 + 70*x^8 + 46*x^7 - 92*x^6 + 37*x^5 + 27*x^4 - 27*x^3 + 9*x^2))*(x^3 - x + 1)^(1/3) + 4*x - 1)*sqrt((2
*x^6 - 2*x^4 + 2*x^3 + x^2 + 2*(3*x^4 - sqrt(3)*(x^4 - x^2 + x))*(x^3 - x + 1)^(2/3) - 4*sqrt(3)*(x^6 - x^4 +
x^3) - 2*(sqrt(3)*x^5 - 3*x^5 + 3*x^3 - 3*x^2)*(x^3 - x + 1)^(1/3) - 2*x + 1)/(2*x^6 - 2*x^4 + 2*x^3 + x^2 - 2
*x + 1)) + 4*(18*x^10 - 46*x^8 + 46*x^7 + 23*x^6 - 46*x^5 + 21*x^4 + 6*x^3 - 6*x^2 + sqrt(3)*(2*x^10 + 4*x^8 -
 4*x^7 - 3*x^6 + 6*x^5 - 2*x^4 - 3*x^3 + 3*x^2 - x) + 2*x)*(x^3 - x + 1)^(2/3) + sqrt(3)*(20*x^12 - 40*x^10 +
40*x^9 + 24*x^8 - 48*x^7 + 20*x^6 + 12*x^5 - 11*x^4 + 6*x^2 - 4*x + 1) + 4*(2*x^11 + 16*x^9 - 16*x^8 - 27*x^7
+ 54*x^6 - 20*x^5 - 21*x^4 + 21*x^3 - 7*x^2 + sqrt(3)*(6*x^11 - 14*x^9 + 14*x^8 + 13*x^7 - 26*x^6 + 9*x^5 + 12
*x^4 - 12*x^3 + 4*x^2))*(x^3 - x + 1)^(1/3) + 8*x - 2)/(52*x^12 - 232*x^10 + 232*x^9 + 248*x^8 - 496*x^7 + 180
*x^6 + 204*x^5 - 203*x^4 + 64*x^3 + 6*x^2 - 4*x + 1)) - 1/2*arctan((6*x^6 - 4*x^4 + 4*x^3 - x^2 + 4*(3*x^4 - x
^2 + x)*(x^3 - x + 1)^(2/3) - 4*(x^5 - 2*x^3 + 2*x^2)*(x^3 - x + 1)^(1/3) + 2*x - 1)/(14*x^6 - 16*x^4 + 16*x^3
 + x^2 - 2*x + 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x - 3\right ) \left (x^{3} - x + 1\right )^{\frac {2}{3}}}{2 x^{6} - 2 x^{4} + 2 x^{3} + x^{2} - 2 x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*x)*(x**3-x+1)**(2/3)/(2*x**6-2*x**4+2*x**3+x**2-2*x+1),x)

[Out]

Integral((2*x - 3)*(x**3 - x + 1)**(2/3)/(2*x**6 - 2*x**4 + 2*x**3 + x**2 - 2*x + 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((-3+2*x)*(x^3-x+1)^(2/3)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1),x, algorithm="giac")

[Out]

integrate((x^3 - x + 1)^(2/3)*(2*x - 3)/(2*x^6 - 2*x^4 + 2*x^3 + x^2 - 2*x + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x-3\right )\,{\left (x^3-x+1\right )}^{2/3}}{2\,x^6-2\,x^4+2\,x^3+x^2-2\,x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x - 3)*(x^3 - x + 1)^(2/3))/(x^2 - 2*x + 2*x^3 - 2*x^4 + 2*x^6 + 1),x)

[Out]

int(((2*x - 3)*(x^3 - x + 1)^(2/3))/(x^2 - 2*x + 2*x^3 - 2*x^4 + 2*x^6 + 1), x)

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