3.21.43 \(\int \frac {(-1-x^4+2 x^6) \sqrt [3]{x-x^5+x^7}}{(1+x^2-x^4+x^6)^2} \, dx\)

Optimal. Leaf size=146 \[ \frac {1}{6} \log \left (\sqrt [3]{x^7-x^5+x}+x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^7-x^5+x}}{\sqrt [3]{x^7-x^5+x}-2 x}\right )}{2 \sqrt {3}}-\frac {1}{12} \log \left (x^2-\sqrt [3]{x^7-x^5+x} x+\left (x^7-x^5+x\right )^{2/3}\right )-\frac {\sqrt [3]{x^7-x^5+x} x}{2 \left (x^6-x^4+x^2+1\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 2.23, 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+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 - x^4 + 2*x^6)*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6)^2,x]

[Out]

(-9*(x - x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 - x^6 + x^9)^(1/3))/(1 + x^3 - x^6 + x^9)^2, x], x, x^
(2/3)])/(2*x^(1/3)*(1 - x^4 + x^6)^(1/3)) - (3*(x - x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][(x^4*(1 - x^6 + x
^9)^(1/3))/(1 + x^3 - x^6 + x^9)^2, x], x, x^(2/3)])/(x^(1/3)*(1 - x^4 + x^6)^(1/3)) + (3*(x - x^5 + x^7)^(1/3
)*Defer[Subst][Defer[Int][(x^7*(1 - x^6 + x^9)^(1/3))/(1 + x^3 - x^6 + x^9)^2, x], x, x^(2/3)])/(2*x^(1/3)*(1
- x^4 + x^6)^(1/3)) + (3*(x - x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 - x^6 + x^9)^(1/3))/(1 + x^3 - x^
6 + x^9), x], x, x^(2/3)])/(x^(1/3)*(1 - x^4 + x^6)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx &=\frac {\sqrt [3]{x-x^5+x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{1-x^4+x^6} \left (-1-x^4+2 x^6\right )}{\left (1+x^2-x^4+x^6\right )^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^{12}+x^{18}} \left (-1-x^{12}+2 x^{18}\right )}{\left (1+x^6-x^{12}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9} \left (-1-x^6+2 x^9\right )}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {x \left (-3-2 x^3+x^6\right ) \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}+\frac {2 x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \left (-3-2 x^3+x^6\right ) \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}-\frac {2 x^4 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}+\frac {x^7 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^7 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}-\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}-\frac {\left (9 \sqrt [3]{x-x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 1.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 - x^4 + 2*x^6)*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6)^2,x]

[Out]

Integrate[((-1 - x^4 + 2*x^6)*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6)^2, x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 4.92, size = 146, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt [3]{x-x^5+x^7}}{2 \left (1+x^2-x^4+x^6\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-x^5+x^7}}{-2 x+\sqrt [3]{x-x^5+x^7}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (x+\sqrt [3]{x-x^5+x^7}\right )-\frac {1}{12} \log \left (x^2-x \sqrt [3]{x-x^5+x^7}+\left (x-x^5+x^7\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 - x^4 + 2*x^6)*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6)^2,x]

[Out]

-1/2*(x*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6) - ArcTan[(Sqrt[3]*(x - x^5 + x^7)^(1/3))/(-2*x + (x - x^5
+ x^7)^(1/3))]/(2*Sqrt[3]) + Log[x + (x - x^5 + x^7)^(1/3)]/6 - Log[x^2 - x*(x - x^5 + x^7)^(1/3) + (x - x^5
+ x^7)^(2/3)]/12

________________________________________________________________________________________

fricas [A]  time = 3.31, size = 198, normalized size = 1.36 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (x^{6} - x^{4} + x^{2} + 1\right )} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} - x^{4} - x^{2} + 1\right )} - 2 \, \sqrt {3} {\left (x^{7} - x^{5} + x\right )}^{\frac {2}{3}}}{x^{6} - x^{4} + x^{2} + 1}\right ) - {\left (x^{6} - x^{4} + x^{2} + 1\right )} \log \left (\frac {x^{6} - x^{4} + x^{2} + 3 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{6} - x^{4} + x^{2} + 1}\right ) + 6 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x}{12 \, {\left (x^{6} - x^{4} + x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6-x^4-1)*(x^7-x^5+x)^(1/3)/(x^6-x^4+x^2+1)^2,x, algorithm="fricas")

[Out]

-1/12*(2*sqrt(3)*(x^6 - x^4 + x^2 + 1)*arctan(-(2*sqrt(3)*(x^7 - x^5 + x)^(1/3)*x + sqrt(3)*(x^6 - x^4 - x^2 +
1) - 2*sqrt(3)*(x^7 - x^5 + x)^(2/3))/(x^6 - x^4 + x^2 + 1)) - (x^6 - x^4 + x^2 + 1)*log((x^6 - x^4 + x^2 + 3
*(x^7 - x^5 + x)^(1/3)*x + 3*(x^7 - x^5 + x)^(2/3) + 1)/(x^6 - x^4 + x^2 + 1)) + 6*(x^7 - x^5 + x)^(1/3)*x)/(x
^6 - x^4 + x^2 + 1)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - x^{4} - 1\right )}}{{\left (x^{6} - x^{4} + x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6-x^4-1)*(x^7-x^5+x)^(1/3)/(x^6-x^4+x^2+1)^2,x, algorithm="giac")

[Out]

integrate((x^7 - x^5 + x)^(1/3)*(2*x^6 - x^4 - 1)/(x^6 - x^4 + x^2 + 1)^2, x)

________________________________________________________________________________________

maple [C]  time = 17.27, size = 774, normalized size = 5.30

method	result	size
trager	\(-\frac {x \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}}{2 \left (x^{6}-x^{4}+x^{2}+1\right )}+2 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {19423596460464 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{6}-19423596460464 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-1424454519036 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}-19423596460464 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}-1230275999700 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-x^{5}+x \right )^{\frac {2}{3}}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}} x -3043087557408 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}[phone] \left (x^{7}-x^{5}+x \right )^{\frac {2}{3}}-151067629809 x \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}-118704543253 x^{2}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )}{x^{6}-x^{4}+x^{2}+1}\right )-\frac {\ln \left (-\frac {9711798230232 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{6}-9711798230232 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}[phone] x^{6}-906405778854 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}-9711798230232 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}-8090771639 x^{4}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-x^{5}+x \right )^{\frac {2}{3}}-615137999850 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}} x -97089259668 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}[phone] \left (x^{7}-x^{5}+x \right )^{\frac {2}{3}}-126795314892 x \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )[phone]}{x^{6}-x^{4}+x^{2}+1}\right )}{6}-2 \ln \left (-\frac {9711798230232 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{6}-9711798230232 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}[phone] x^{6}-906405778854 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}-9711798230232 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}-8090771639 x^{4}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-x^{5}+x \right )^{\frac {2}{3}}-615137999850 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}} x -97089259668 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}[phone] \left (x^{7}-x^{5}+x \right )^{\frac {2}{3}}-126795314892 x \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )[phone]}{x^{6}-x^{4}+x^{2}+1}\right ) \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )\)	\(774\)
risch	\(-\frac {x \left (x \left (x^{6}-x^{4}+1\right )\right )^{\frac {1}{3}}}{2 \left (x^{6}-x^{4}+x^{2}+1\right )}+\frac {\left (-\frac {\ln \left (\frac {4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{12}-8 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{10}-x^{12}+2 x^{10}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{8}+12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}+6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{6}-x^{8}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}+3 \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{6}-8 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{4}-2 x^{6}-3 \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{4}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+2 x^{4}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {2}{3}}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}}-3 \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {2}{3}}+3 \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}}-1}{\left (x^{6}-x^{4}+1\right ) \left (x^{6}-x^{4}+x^{2}+1\right )}\right )}{6}-\frac {\ln \left (\frac {4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{12}-8 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{10}-x^{12}+2 x^{10}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{8}+12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}+6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{6}-x^{8}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}+3 \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{6}-8 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{4}-2 x^{6}-3 \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{4}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+2 x^{4}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {2}{3}}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}}-3 \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {2}{3}}+3 \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}}-1}{\left (x^{6}-x^{4}+1\right ) \left (x^{6}-x^{4}+x^{2}+1\right )}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{3}+\frac {\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{12}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{12}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{10}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{10}-\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{8}+6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}-3 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{6}-x^{8}+7 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}+3 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}} x^{4}+x^{6}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {2}{3}}-3 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-3 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{14}-2 x^{12}+x^{10}+2 x^{8}-2 x^{6}+x^{2}\right )^{\frac {1}{3}}-x^{2}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{\left (x^{6}-x^{4}+1\right ) \left (x^{6}-x^{4}+x^{2}+1\right )}\right )}{3}\right ) \left (x \left (x^{6}-x^{4}+1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{6}-x^{4}+1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{6}-x^{4}+1\right )}\)	\(1528\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6-x^4-1)*(x^7-x^5+x)^(1/3)/(x^6-x^4+x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*x/(x^6-x^4+x^2+1)*(x^7-x^5+x)^(1/3)+2*RootOf(144*_Z^2+12*_Z+1)*ln(-(19423596460464*RootOf(144*_Z^2+12*_Z+
1)^2*x^6+1424454519036*RootOf(144*_Z^2+12*_Z+1)*x^6-19423596460464*RootOf(144*_Z^2+12*_Z+1)^2*x^4-142445451903
6*RootOf(144*_Z^2+12*_Z+1)*x^4-19423596460464*RootOf(144*_Z^2+12*_Z+1)^2*x^2-1230275999700*RootOf(144*_Z^2+12*
_Z+1)*(x^7-x^5+x)^(2/3)[phone]*RootOf(144*_Z^2+12*_Z+1)*(x^7-x^5+x)^(1/3)*x-3043087557408*RootOf(144*_Z
^2+12*_Z+1)*x^2+19423596460464*RootOf(144*_Z^2+12*_Z+1)^2+151067629809*(x^7-x^5+x)^(2/3)-151067629809*x*(x^7-x
^5+x)^(1/3)-118704543253*x^2+1424454519036*RootOf(144*_Z^2+12*_Z+1))/(x^6-x^4+x^2+1))-1/6*ln(-(9711798230232*R
ootOf(144*_Z^2+12*_Z+1)^2*x^6+906405778854*RootOf(144*_Z^2+12*_Z+1)*x^6-9711798230232*RootOf(144*_Z^2+12*_Z+1)
^2*x^4+8090771639*x^6-906405778854*RootOf(144*_Z^2+12*_Z+1)*x^4-9711798230232*RootOf(144*_Z^2+12*_Z+1)^2*x^2-8
090771639*x^4+615137999850*RootOf(144*_Z^2+12*_Z+1)*(x^7-x^5+x)^(2/3)-615137999850*RootOf(144*_Z^2+12*_Z+1)*(x
^7-x^5+x)^(1/3)*x-97089259668*RootOf(144*_Z^2+12*_Z+1)*x^2+9711798230232*RootOf(144*_Z^2+12*_Z+1)^2+1267953148
92*(x^7-x^5+x)^(2/3)-126795314892*x*(x^7-x^5+x)^(1/3)[phone]*RootOf(144*_Z^2+12*_Z+1)[phone])/(x^6-x
^4+x^2+1))-2*ln(-(9711798230232*RootOf(144*_Z^2+12*_Z+1)^2*x^6+906405778854*RootOf(144*_Z^2+12*_Z+1)*x^6-97117
98230232*RootOf(144*_Z^2+12*_Z+1)^2*x^4+8090771639*x^6-906405778854*RootOf(144*_Z^2+12*_Z+1)*x^4-9711798230232
*RootOf(144*_Z^2+12*_Z+1)^2*x^2-8090771639*x^4+615137999850*RootOf(144*_Z^2+12*_Z+1)*(x^7-x^5+x)^(2/3)-6151379
99850*RootOf(144*_Z^2+12*_Z+1)*(x^7-x^5+x)^(1/3)*x-97089259668*RootOf(144*_Z^2+12*_Z+1)*x^2+9711798230232*Root
Of(144*_Z^2+12*_Z+1)^2+126795314892*(x^7-x^5+x)^(2/3)-126795314892*x*(x^7-x^5+x)^(1/3)[phone]*RootOf(144
*_Z^2+12*_Z+1)[phone])/(x^6-x^4+x^2+1))*RootOf(144*_Z^2+12*_Z+1)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - x^{4} - 1\right )}}{{\left (x^{6} - x^{4} + x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6-x^4-1)*(x^7-x^5+x)^(1/3)/(x^6-x^4+x^2+1)^2,x, algorithm="maxima")

[Out]

integrate((x^7 - x^5 + x)^(1/3)*(2*x^6 - x^4 - 1)/(x^6 - x^4 + x^2 + 1)^2, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (-2\,x^6+x^4+1\right )\,{\left (x^7-x^5+x\right )}^{1/3}}{{\left (x^6-x^4+x^2+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^4 - 2*x^6 + 1)*(x - x^5 + x^7)^(1/3))/(x^2 - x^4 + x^6 + 1)^2,x)

[Out]

int(-((x^4 - 2*x^6 + 1)*(x - x^5 + x^7)^(1/3))/(x^2 - x^4 + x^6 + 1)^2, x)

________________________________________________________________________________________

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((2*x**6-x**4-1)*(x**7-x**5+x)**(1/3)/(x**6-x**4+x**2+1)**2,x)

[Out]

Timed out

________________________________________________________________________________________