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

Optimal. Leaf size=145 \[ -\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^4-1}-x\right )}{4\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^4-1}+x}\right )}{4\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^4-1} x+2^{2/3} \left (x^4-1\right )^{2/3}+x^2\right )}{8\ 2^{2/3}}+\frac {3 \left (x^4-1\right )^{2/3} \left (4 x^4-5 x^3-4\right )}{40 x^5} \]

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Rubi [F]  time = 1.41, 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 {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*(-1 + x^4)^(2/3))/(8*x^2) + (3*x^2)/(2*(1 + Sqrt[3] + (-1 + x^4)^(1/3))) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1
+ (-1 + x^4)^(1/3))*Sqrt[(1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]*Ellipt
icE[ArcSin[(1 - Sqrt[3] + (-1 + x^4)^(1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))], -7 - 4*Sqrt[3]])/(4*x^2*Sqrt[(1
+ (-1 + x^4)^(1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]) + (3^(3/4)*(1 + (-1 + x^4)^(1/3))*Sqrt[(1 - (-1 + x^
4)^(1/3) + (-1 + x^4)^(2/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] + (-1 + x^4)^(1
/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))], -7 - 4*Sqrt[3]])/(Sqrt[2]*x^2*Sqrt[(1 + (-1 + x^4)^(1/3))/(1 + Sqrt[3]
+ (-1 + x^4)^(1/3))^2]) - (3*(-1 + x^4)^(2/3)*Hypergeometric2F1[-5/4, -2/3, -1/4, x^4])/(10*x^5*(1 - x^4)^(2/
3)) - ((-1 + x^4)^(2/3)*Hypergeometric2F1[-2/3, -1/4, 3/4, x^4])/(2*x*(1 - x^4)^(2/3)) + (3*Defer[Int][(-1 + x
^4)^(2/3)/(-2 - x^3 + 2*x^4), x])/4 - 2*Defer[Int][(x*(-1 + x^4)^(2/3))/(-2 - x^3 + 2*x^4), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx &=\int \left (\frac {3 \left (-1+x^4\right )^{2/3}}{2 x^6}+\frac {3 \left (-1+x^4\right )^{2/3}}{4 x^3}+\frac {\left (-1+x^4\right )^{2/3}}{2 x^2}+\frac {(3-8 x) \left (-1+x^4\right )^{2/3}}{4 \left (-2-x^3+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {(3-8 x) \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx+\frac {1}{2} \int \frac {\left (-1+x^4\right )^{2/3}}{x^2} \, dx+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx+\frac {3}{2} \int \frac {\left (-1+x^4\right )^{2/3}}{x^6} \, dx\\ &=\frac {1}{4} \int \left (\frac {3 \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4}-\frac {8 x \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4}\right ) \, dx+\frac {3}{8} \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )+\frac {\left (-1+x^4\right )^{2/3} \int \frac {\left (1-x^4\right )^{2/3}}{x^2} \, dx}{2 \left (1-x^4\right )^{2/3}}+\frac {\left (3 \left (-1+x^4\right )^{2/3}\right ) \int \frac {\left (1-x^4\right )^{2/3}}{x^6} \, dx}{2 \left (1-x^4\right )^{2/3}}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{8 x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{10 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{2 x \left (1-x^4\right )^{2/3}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx-2 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{8 x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{10 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{2 x \left (1-x^4\right )^{2/3}}+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx-2 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx+\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{4 x^2}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{8 x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{10 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{2 x \left (1-x^4\right )^{2/3}}+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx-2 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx+\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{4 x^2}+\frac {\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{2 x^2}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{8 x^2}+\frac {3 x^2}{2 \left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{4 x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}+\frac {3^{3/4} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{\sqrt {2} x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{10 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{2 x \left (1-x^4\right )^{2/3}}+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx-2 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.30, size = 145, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^4\right )^{2/3} \left (-4-5 x^3+4 x^4\right )}{40 x^5}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^4}}\right )}{4\ 2^{2/3}}-\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^4}\right )}{4\ 2^{2/3}}+\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^4}+2^{2/3} \left (-1+x^4\right )^{2/3}\right )}{8\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + x^4)^(2/3)*(-4 - 5*x^3 + 4*x^4))/(40*x^5) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^4)^(1/
3))])/(4*2^(2/3)) - Log[-x + 2^(1/3)*(-1 + x^4)^(1/3)]/(4*2^(2/3)) + Log[x^2 + 2^(1/3)*x*(-1 + x^4)^(1/3) + 2^
(2/3)*(-1 + x^4)^(2/3)]/(8*2^(2/3))

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fricas [B]  time = 129.21, size = 433, normalized size = 2.99 \begin {gather*} -\frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (2 \, x^{9} + x^{8} - x^{7} - 4 \, x^{5} - x^{4} + 2 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} - 12 \, \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{10} + 14 \, x^{9} + x^{8} - 8 \, x^{6} - 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} {\left (8 \, x^{12} + 60 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 120 \, x^{7} - 24 \, x^{6} + 24 \, x^{4} + 60 \, x^{3} - 8\right )}\right )}}{6 \, {\left (8 \, x^{12} - 12 \, x^{11} - 48 \, x^{10} - x^{9} - 24 \, x^{8} + 24 \, x^{7} + 48 \, x^{6} + 24 \, x^{4} - 12 \, x^{3} - 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{4} - x^{3} - 2\right )} - 12 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x}{2 \, x^{4} - x^{3} - 2}\right ) + 5 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} + x^{4} - x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{8} + 14 \, x^{7} + x^{6} - 8 \, x^{4} - 14 \, x^{3} + 4\right )} - 6 \, {\left (4 \, x^{6} + x^{5} - 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{4 \, x^{8} - 4 \, x^{7} + x^{6} - 8 \, x^{4} + 4 \, x^{3} + 4}\right ) - 36 \, {\left (4 \, x^{4} - 5 \, x^{3} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{480 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/480*(20*4^(1/6)*sqrt(3)*(-1)^(1/3)*x^5*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(-1)^(2/3)*(2*x^9 + x^8 - x^7
- 4*x^5 - x^4 + 2*x)*(x^4 - 1)^(2/3) - 12*(-1)^(1/3)*(4*x^10 + 14*x^9 + x^8 - 8*x^6 - 14*x^5 + 4*x^2)*(x^4 -
1)^(1/3) + 4^(1/3)*(8*x^12 + 60*x^11 + 24*x^10 - x^9 - 24*x^8 - 120*x^7 - 24*x^6 + 24*x^4 + 60*x^3 - 8))/(8*x^
12 - 12*x^11 - 48*x^10 - x^9 - 24*x^8 + 24*x^7 + 48*x^6 + 24*x^4 - 12*x^3 - 8)) - 10*4^(2/3)*(-1)^(1/3)*x^5*lo
g((6*4^(1/3)*(-1)^(2/3)*(x^4 - 1)^(1/3)*x^2 - 4^(2/3)*(-1)^(1/3)*(2*x^4 - x^3 - 2) - 12*(x^4 - 1)^(2/3)*x)/(2*
x^4 - x^3 - 2)) + 5*4^(2/3)*(-1)^(1/3)*x^5*log(-(6*4^(2/3)*(-1)^(1/3)*(x^5 + x^4 - x)*(x^4 - 1)^(2/3) - 4^(1/3
)*(-1)^(2/3)*(4*x^8 + 14*x^7 + x^6 - 8*x^4 - 14*x^3 + 4) - 6*(4*x^6 + x^5 - 4*x^2)*(x^4 - 1)^(1/3))/(4*x^8 - 4
*x^7 + x^6 - 8*x^4 + 4*x^3 + 4)) - 36*(4*x^4 - 5*x^3 - 4)*(x^4 - 1)^(2/3))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 95.83, size = 669, normalized size = 4.61

method	result	size
risch	\(\frac {\frac {3}{10} x^{8}-\frac {3}{5} x^{4}+\frac {3}{10}-\frac {3}{8} x^{7}+\frac {3}{8} x^{3}}{x^{5} \left (x^{4}-1\right )^{\frac {1}{3}}}+\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) \ln \left (\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+2\right )^{3} x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+2\right )^{2} x^{3}+2 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+2\right )^{2} x +\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+2\right )^{2} x^{2}+4 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+2\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{3}+2\right ) x^{4}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}+2\right )-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{4}-x^{3}-2}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{3}+2\right ) \ln \left (-\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+2\right )^{3} x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+2\right )^{2} x^{3}+2 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+2\right )^{2} x +\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+2\right )^{2} x^{2}-2 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+2\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}+2\right ) x^{4}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) x^{4}-\RootOf \left (\textit {\_Z}^{3}+2\right ) x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-4 \left (x^{4}-1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{4}-x^{3}-2}\right )}{8}\)	\(669\)
trager	\(\text {Expression too large to display}\)	\(1471\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*(4*x^8-5*x^7-8*x^4+5*x^3+4)/x^5/(x^4-1)^(1/3)+1/4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*ln(
(RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3+2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootO
f(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+2*(x^4-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*R
ootOf(_Z^3+2)^2*x+(x^4-1)^(1/3)*RootOf(_Z^3+2)^2*x^2+4*(x^4-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+
2)+4*_Z^2)*RootOf(_Z^3+2)*x^2+2*RootOf(_Z^3+2)*x^4+4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^4+2
*(x^4-1)^(2/3)*x-2*RootOf(_Z^3+2)-4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(2*x^4-x^3-2))+1/8*Ro
otOf(_Z^3+2)*ln(-(RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3+2*RootOf(RootOf(_Z^
3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+2*(x^4-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(
_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x+(x^4-1)^(1/3)*RootOf(_Z^3+2)^2*x^2-2*(x^4-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+
2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x^2-2*RootOf(_Z^3+2)*x^4-4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3
+2)+4*_Z^2)*x^4-RootOf(_Z^3+2)*x^3-2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^3-4*(x^4-1)^(2/3)*x
+2*RootOf(_Z^3+2)+4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(2*x^4-x^3-2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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