∫ (−3+2x)(1−x+x3)2∕3 _____________________________________

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

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]{1-x+x^3}}\right )-\frac {1}{2} i \left (-i+\sqrt {3}\right ) \tan ^{-1}\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 ) \tan ^{-1}\left (\frac {\left (1+i \sqrt {3}\right ) x}{2 \sqrt [3]{1-x+x^3}}\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 = 6.34, 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 = 37.72, size = 3489, normalized size = 33.55


method	result	size

trager	\(\text {Expression too large to display}\)	\(3489\)

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]

-1/2*RootOf(_Z^2+1)*ln(-(3-6*x+28*RootOf(_Z^2+1)^2*x^3-6*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)+2

8*RootOf(_Z^2+1)^2*x^6-28*RootOf(_Z^2+1)^2*x^4-14*RootOf(_Z^2+1)*x+7*RootOf(_Z^2+1)*x^2+26*RootOf(_Z^2+1)*x^6-

26*RootOf(_Z^2+1)*x^4+26*RootOf(_Z^2+1)*x^3-42*x^2*(x^3-x+1)^(2/3)+6*x^6+3*x^2+6*x^3-6*x^4+96*RootOf(2*_Z*Root

Of(_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)*RootOf(_Z^2+1)^2*x^4-14*Root

Of(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)+7*RootOf(_Z^2+1)+224*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^4-224*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*Ro

otOf(_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-20*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*x^4+20*RootOf(2*_Z*Root

Of(_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

-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)+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^6-24*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*Roo

tOf(_Z^2+1)^2*x^6-224*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^6-6*RootOf(_Z^2+1)^2*(x^3-x+1)^(2/3)*x^4-28*Roo

tOf(_Z^2+1)^2*(x^3-x+1)^(1/3)*x^5+144*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(2/3)*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*RootOf(_Z^2+1)*(x^3-x+1)

^(1/3)*x^5+24*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(2/3)*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*RootOf(_Z^2+1)*(x^3-x+1)^(1/3)*x^3-24*RootOf(2*_Z

*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(2/3)*x-84*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(1/3)*x^2-6*Root

Of(_Z^2+1)*(x^3-x+1)^(2/3)*x-84*RootOf(_Z^2+1)*(x^3-x+1)^(1/3)*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+42*(x^3-x+1)^(2/3)*x^4+42*(x

^3-x+1)^(2/3)*x+48*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*RootOf(_Z^2+1)*(x^3-x+1)^(1/3)*x^5+12*RootOf(2*_Z*Ro

otOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)^2*(x^3-x+1)^(1/3)*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^4-104*(x^3-x+1)^(1/3)*RootOf(2*_Z*RootOf(_Z^2+1

)+4*_Z^2-1)*RootOf(_Z^2+1)*x^5)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1))-ln((3-6*x-28*RootOf(_Z^2+1)^2*x^3-28*RootOf(_Z^

2+1)^2*x^6+28*RootOf(_Z^2+1)^2*x^4+14*RootOf(_Z^2+1)*x-7*RootOf(_Z^2+1)*x^2-2*RootOf(_Z^2+1)*x^6+2*RootOf(_Z^2

+1)*x^4-2*RootOf(_Z^2+1)*x^3+42*x^2*(x^3-x+1)^(2/3)+6*x^6+3*x^2+6*x^3-6*x^4-28*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z

^2-1)-7*RootOf(_Z^2+1)+448*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^4-448*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)*Ro

otOf(_Z^2+1)*x^3+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*Root

Of(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-8*x^6*RootOf(2*_Z*RootOf(_Z^2+1

)+4*_Z^2-1)-224*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^6-448*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-

1)^2*x^6+6*RootOf(_Z^2+1)^2*(x^3-x+1)^(2/3)*x^4-14*RootOf(_Z^2+1)^2*(x^3-x+1)^(1/3)*x^5-144*RootOf(2*_Z*RootOf

(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(2/3)*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*RootOf(_Z^2+1)*(x^3-x+1)^(1/3)*x^5-24*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3

-x+1)^(2/3)*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*RootOf(_Z^2+1)*(x^3-x+1)^(1/3)*x^3+24*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(2/3)*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*RootOf(_Z^2+1)*(x^3-x

+1)^(1/3)*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^5-42*(x^3-x+1)^(2/3)*x^4-42*(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+48*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^4-112

*(x^3-x+1)^(1/3)*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^5)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1))*RootO

f(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)-1/2*ln((3-6*x-28*RootOf(_Z^2+1)^2*x^3-28*RootOf(_Z^2+1)^2*x^6+28*RootOf(_Z^2+1

)^2*x^4+14*RootOf(_Z^2+1)*x-7*RootOf(_Z^2+1)*x^2-2*RootOf(_Z^2+1)*x^6+2*RootOf(_Z^2+1)*x^4-2*RootOf(_Z^2+1)*x^

3+42*x^2*(x^3-x+1)^(2/3)+6*x^6+3*x^2+6*x^3-6*x^4-28*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)-7*RootOf(_Z^2+1)+448*

RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^4-448*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^3+224*RootOf(2*_Z*Root

Of(_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+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-8*x^6*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)-224*RootOf(2*_Z

*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^6-448*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)^2*x^6+6*RootOf(_Z^2+1)^2

*(x^3-x+1)^(2/3)*x^4-14*RootOf(_Z^2+1)^2*(x^3-x+1)^(1/3)*x^5-144*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1

)^(2/3)*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*RootOf(_Z^2+1)*(x^3-x+1)^(1/3)*x^5-24*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(2/3)*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*RootOf(_Z^2+1)*(x^3-

x+1)^(1/3)*x^3+24*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*(x^3-x+1)^(2/3)*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*RootOf(_Z^2+1)*(x^3-x+1)^(1/3)*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^

5-42*(x^3-x+1)^(2/3)*x^4-42*(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+48*RootOf(2*_Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*(x^3-x+1)^(2/3)*x^4-112*(x^3-x+1)^(1/3)*RootOf(2*_

Z*RootOf(_Z^2+1)+4*_Z^2-1)*RootOf(_Z^2+1)*x^5)/(2*x^6-2*x^4+2*x^3+x^2-2*x+1))*RootOf(_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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