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

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

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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*}

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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]

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IntegrateAlgebraic [A]  time = 4.92, size = 146, normalized size = 1.00 \begin {gather*} \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 )} \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

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fricas [A]  time = 3.05, 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)

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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)

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maple [C]  time = 9.22, size = 1306, normalized size = 8.95

result too large to display

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)

[Out]

-1/2*x/(x^6-x^4+x^2+1)*(x*(x^6-x^4+1))^(1/3)+(1/6*ln(-(8*RootOf(4*_Z^2+2*_Z+1)^2*x^12+6*RootOf(4*_Z^2+2*_Z+1)*
x^12-16*RootOf(4*_Z^2+2*_Z+1)^2*x^10+x^12-12*RootOf(4*_Z^2+2*_Z+1)*x^10-2*x^10+4*RootOf(4*_Z^2+2*_Z+1)*x^8+6*(
x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(1/3)*RootOf(4*_Z^2+2*_Z+1)*x^6+24*RootOf(4*_Z^2+2*_Z+1)^2*x^6+x^8+14*RootOf
(4*_Z^2+2*_Z+1)*x^6-6*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(1/3)*RootOf(4*_Z^2+2*_Z+1)*x^4-16*RootOf(4*_Z^2+2*_Z
+1)^2*x^4+2*x^6-12*RootOf(4*_Z^2+2*_Z+1)*x^4-8*RootOf(4*_Z^2+2*_Z+1)^2*x^2-2*x^4+6*RootOf(4*_Z^2+2*_Z+1)*(x^14
-2*x^12+x^10+2*x^8-2*x^6+x^2)^(2/3)-2*RootOf(4*_Z^2+2*_Z+1)*x^2+3*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(2/3)+6*R
ootOf(4*_Z^2+2*_Z+1)*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(1/3)+8*RootOf(4*_Z^2+2*_Z+1)^2+6*RootOf(4*_Z^2+2*_Z+1
)+1)/(x^6-x^4+1)/(x^6-x^4+x^2+1))-1/6*ln((4*RootOf(4*_Z^2+2*_Z+1)^2*x^12-2*RootOf(4*_Z^2+2*_Z+1)*x^12-8*RootOf
(4*_Z^2+2*_Z+1)^2*x^10+4*RootOf(4*_Z^2+2*_Z+1)*x^10-2*RootOf(4*_Z^2+2*_Z+1)*x^8+12*RootOf(4*_Z^2+2*_Z+1)^2*x^6
+x^8-3*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(1/3)*x^6-4*RootOf(4*_Z^2+2*_Z+1)*x^6-8*RootOf(4*_Z^2+2*_Z+1)^2*x^4-
x^6+3*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(1/3)*x^4+4*RootOf(4*_Z^2+2*_Z+1)*x^4-4*RootOf(4*_Z^2+2*_Z+1)^2*x^2+6
*RootOf(4*_Z^2+2*_Z+1)*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(2/3)+4*RootOf(4*_Z^2+2*_Z+1)^2+x^2-3*(x^14-2*x^12+x
^10+2*x^8-2*x^6+x^2)^(1/3)-2*RootOf(4*_Z^2+2*_Z+1))/(x^6-x^4+1)/(x^6-x^4+x^2+1))-1/3*ln((4*RootOf(4*_Z^2+2*_Z+
1)^2*x^12-2*RootOf(4*_Z^2+2*_Z+1)*x^12-8*RootOf(4*_Z^2+2*_Z+1)^2*x^10+4*RootOf(4*_Z^2+2*_Z+1)*x^10-2*RootOf(4*
_Z^2+2*_Z+1)*x^8+12*RootOf(4*_Z^2+2*_Z+1)^2*x^6+x^8-3*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(1/3)*x^6-4*RootOf(4*
_Z^2+2*_Z+1)*x^6-8*RootOf(4*_Z^2+2*_Z+1)^2*x^4-x^6+3*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(1/3)*x^4+4*RootOf(4*_
Z^2+2*_Z+1)*x^4-4*RootOf(4*_Z^2+2*_Z+1)^2*x^2+6*RootOf(4*_Z^2+2*_Z+1)*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(2/3)
+4*RootOf(4*_Z^2+2*_Z+1)^2+x^2-3*(x^14-2*x^12+x^10+2*x^8-2*x^6+x^2)^(1/3)-2*RootOf(4*_Z^2+2*_Z+1))/(x^6-x^4+1)
/(x^6-x^4+x^2+1))*RootOf(4*_Z^2+2*_Z+1))*(x*(x^6-x^4+1))^(1/3)*(x^2*(x^6-x^4+1)^2)^(1/3)/x/(x^6-x^4+1)

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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)

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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)

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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((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

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