3.14.3 \(\int \frac {(-3+2 x) (1-x+x^3)^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx\)

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

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Rubi [F]  time = 0.52, 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 [F]  time = 0.18, size = 0, normalized size = 0.00 \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]

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]

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]

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IntegrateAlgebraic [C]  time = 0.43, size = 106, normalized size = 1.02 \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt [3]{x^3-x+1}}\right )-\frac {1}{2} i \left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\left (1-i \sqrt {3}\right ) x}{2 \sqrt [3]{x^3-x+1}}\right )+\frac {1}{2} i \left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {\left (1+i \sqrt {3}\right ) x}{2 \sqrt [3]{x^3-x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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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fricas [B]  time = 5.14, size = 2090, normalized size = 20.10

result too large to display

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

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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maple [C]  time = 40.64, size = 3488, normalized size = 33.54 \begin {gather*} \text {output too large to display} \end {gather*}

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)

[Out]

1/2*RootOf(_Z^2+1)*ln((-3+6*x-224*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^4-6*RootOf(-2*_Z*RootOf(_Z^2+1)+4*
_Z^2-1)*RootOf(_Z^2+1)-26*RootOf(_Z^2+1)*x^4+26*RootOf(_Z^2+1)*x^6+7*RootOf(_Z^2+1)*x^2+26*RootOf(_Z^2+1)*x^3-
6*x^6-3*x^2-6*x^3+6*x^4-28*RootOf(_Z^2+1)^2*x^6+224*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^6+44*RootOf(-2*_
Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^6-96*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*RootOf(_Z^2+1)*x^3+24
*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)^2*x^3-20*x^6*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)-28*Ro
otOf(_Z^2+1)^2*x^3+28*RootOf(_Z^2+1)^2*x^4+7*RootOf(_Z^2+1)-14*RootOf(_Z^2+1)*x+14*RootOf(-2*_Z*RootOf(_Z^2+1)
+4*_Z^2-1)+20*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^4-20*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^3+14*RootOf
(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^2-28*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x+224*RootOf(-2*_Z*RootOf(_Z^2+1)
+4*_Z^2-1)^2*x^3-44*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^4+44*RootOf(-2*_Z*RootOf(_Z^2+1)+4*
_Z^2-1)*RootOf(_Z^2+1)*x^3-6*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^2+12*RootOf(-2*_Z*RootOf(_
Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x-42*(x^3-x+1)^(2/3)*x^4+42*(x^3-x+1)^(2/3)*x^2-42*(x^3-x+1)^(2/3)*x-96*RootOf
(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*RootOf(_Z^2+1)*x^6+24*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)^2
*x^6+96*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*RootOf(_Z^2+1)*x^4-24*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*Ro
otOf(_Z^2+1)^2*x^4-104*(x^3-x+1)^(1/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^5+48*RootOf(-2*_
Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)*x^5-12*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-
x+1)^(1/3)*RootOf(_Z^2+1)^2*x^5-36*(x^3-x+1)^(1/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^3+36
*(x^3-x+1)^(1/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^2-48*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^
2-1)*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^4+6*(x^3-x+1)^(2/3)*RootOf(_Z^2+1)^2*x^4+28*(x^3-x+1)^(1/3)*RootOf(_Z^2+
1)^2*x^5-144*(x^3-x+1)^(2/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^4+84*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1
)*(x^3-x+1)^(1/3)*x^5+36*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^4-84*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)*x^5-24*(x^3-x+1)
^(2/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^2-84*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(1/3)*x^3+
6*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^2+84*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)*x^3+24*(x^3-x+1)^(2/3)*RootOf(-2*_Z*Roo
tOf(_Z^2+1)+4*_Z^2-1)*x+84*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(1/3)*x^2-6*RootOf(_Z^2+1)*(x^3-x+1
)^(2/3)*x-84*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)*x^2+96*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*(x^3-x+1)^(2/3)*x^4
+112*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*(x^3-x+1)^(1/3)*x^5)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1))+1/2*ln((3-6*x
+448*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^4-2*RootOf(_Z^2+1)*x^4+2*RootOf(_Z^2+1)*x^6+7*RootOf(_Z^2+1)*x^
2+2*RootOf(_Z^2+1)*x^3+6*x^6+3*x^2+6*x^3-6*x^4-28*RootOf(_Z^2+1)^2*x^6-448*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-
1)^2*x^6+224*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^6-8*x^6*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2
-1)-28*RootOf(_Z^2+1)^2*x^3+28*RootOf(_Z^2+1)^2*x^4+7*RootOf(_Z^2+1)-14*RootOf(_Z^2+1)*x-28*RootOf(-2*_Z*RootO
f(_Z^2+1)+4*_Z^2-1)+8*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^4-8*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^3-28
*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^2+56*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x-448*RootOf(-2*_Z*RootOf(
_Z^2+1)+4*_Z^2-1)^2*x^3-224*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^4+224*RootOf(-2*_Z*RootOf(_
Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^3-42*(x^3-x+1)^(2/3)*x^4+42*(x^3-x+1)^(2/3)*x^2-42*(x^3-x+1)^(2/3)*x+18*(x^3
-x+1)^(1/3)*x^5-18*(x^3-x+1)^(1/3)*x^3+18*(x^3-x+1)^(1/3)*x^2+112*(x^3-x+1)^(1/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+
4*_Z^2-1)*RootOf(_Z^2+1)*x^5-48*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^4+6*(x^
3-x+1)^(2/3)*RootOf(_Z^2+1)^2*x^4-14*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)^2*x^5-144*(x^3-x+1)^(2/3)*RootOf(-2*_Z*Roo
tOf(_Z^2+1)+4*_Z^2-1)*x^4-144*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(1/3)*x^5+36*RootOf(_Z^2+1)*(x^3
-x+1)^(2/3)*x^4+36*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)*x^5-24*(x^3-x+1)^(2/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)
*x^2+168*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(1/3)*x^3+6*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^2-42*(x^
3-x+1)^(1/3)*RootOf(_Z^2+1)*x^3+24*(x^3-x+1)^(2/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x-168*RootOf(-2*_Z*Ro
otOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(1/3)*x^2-6*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x+42*(x^3-x+1)^(1/3)*RootOf(_Z^2+1
)*x^2+96*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*(x^3-x+1)^(2/3)*x^4-224*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)
^2*(x^3-x+1)^(1/3)*x^5)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1))*RootOf(_Z^2+1)-ln((3-6*x+448*RootOf(-2*_Z*RootOf(_Z^2+1
)+4*_Z^2-1)^2*x^4-2*RootOf(_Z^2+1)*x^4+2*RootOf(_Z^2+1)*x^6+7*RootOf(_Z^2+1)*x^2+2*RootOf(_Z^2+1)*x^3+6*x^6+3*
x^2+6*x^3-6*x^4-28*RootOf(_Z^2+1)^2*x^6-448*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^6+224*RootOf(-2*_Z*RootO
f(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^6-8*x^6*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)-28*RootOf(_Z^2+1)^2*x^3+28*
RootOf(_Z^2+1)^2*x^4+7*RootOf(_Z^2+1)-14*RootOf(_Z^2+1)*x-28*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)+8*RootOf(-2
*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^4-8*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^3-28*RootOf(-2*_Z*RootOf(_Z^2+1)+4*
_Z^2-1)*x^2+56*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x-448*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^3-224*Roo
tOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^4+224*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)
*x^3-42*(x^3-x+1)^(2/3)*x^4+42*(x^3-x+1)^(2/3)*x^2-42*(x^3-x+1)^(2/3)*x+18*(x^3-x+1)^(1/3)*x^5-18*(x^3-x+1)^(1
/3)*x^3+18*(x^3-x+1)^(1/3)*x^2+112*(x^3-x+1)^(1/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^5-48
*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^4+6*(x^3-x+1)^(2/3)*RootOf(_Z^2+1)^2*x
^4-14*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)^2*x^5-144*(x^3-x+1)^(2/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^4-144*R
ootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(1/3)*x^5+36*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^4+36*(x^3-x+1)^(1
/3)*RootOf(_Z^2+1)*x^5-24*(x^3-x+1)^(2/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^2+168*RootOf(-2*_Z*RootOf(_Z
^2+1)+4*_Z^2-1)*(x^3-x+1)^(1/3)*x^3+6*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^2-42*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)*x^3
+24*(x^3-x+1)^(2/3)*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x-168*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1
)^(1/3)*x^2-6*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x+42*(x^3-x+1)^(1/3)*RootOf(_Z^2+1)*x^2+96*RootOf(-2*_Z*RootOf(_Z
^2+1)+4*_Z^2-1)^2*(x^3-x+1)^(2/3)*x^4-224*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*(x^3-x+1)^(1/3)*x^5)/(2*x^6-
2*x^4+2*x^3+x^2-2*x+1))*RootOf(-2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)

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

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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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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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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