3.10.24 \(\int \frac {(-2+x^6) (1+x^6) \sqrt [4]{1-x^4+x^6}}{x^6 (1-2 x^4+x^6)} \, dx\) [924]

Optimal. Leaf size=70 \[ \frac {2 \sqrt [4]{1-x^4+x^6} \left (1+9 x^4+x^6\right )}{5 x^5}+2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right ) \]

[Out]

2/5*(x^6-x^4+1)^(1/4)*(x^6+9*x^4+1)/x^5+2*arctan(x/(x^6-x^4+1)^(1/4))-2*arctanh(x/(x^6-x^4+1)^(1/4))

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Rubi [F]
time = 1.26, 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 (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][(1 - x^4 + x^6)^(1/4), x] + Defer[Int][(1 - x^4 + x^6)^(1/4)/(-1 - x), x] + Defer[Int][(1 - x^4 + x
^6)^(1/4)/(-1 + x), x] - 2*Defer[Int][(1 - x^4 + x^6)^(1/4)/x^6, x] - 4*Defer[Int][(1 - x^4 + x^6)^(1/4)/x^2,
x] + ((2*I)*Defer[Int][(1 - x^4 + x^6)^(1/4)/(I*Sqrt[-1 + Sqrt[5]] - Sqrt[2]*x), x])/Sqrt[-1 + Sqrt[5]] - (2*D
efer[Int][(1 - x^4 + x^6)^(1/4)/(Sqrt[1 + Sqrt[5]] - Sqrt[2]*x), x])/Sqrt[1 + Sqrt[5]] + ((2*I)*Defer[Int][(1
- x^4 + x^6)^(1/4)/(I*Sqrt[-1 + Sqrt[5]] + Sqrt[2]*x), x])/Sqrt[-1 + Sqrt[5]] - (2*Defer[Int][(1 - x^4 + x^6)^
(1/4)/(Sqrt[1 + Sqrt[5]] + Sqrt[2]*x), x])/Sqrt[1 + Sqrt[5]]

Rubi steps

\begin {align*} \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx &=\int \left (\sqrt [4]{1-x^4+x^6}+\frac {\sqrt [4]{1-x^4+x^6}}{-1-x}+\frac {\sqrt [4]{1-x^4+x^6}}{-1+x}-\frac {2 \sqrt [4]{1-x^4+x^6}}{x^6}-\frac {4 \sqrt [4]{1-x^4+x^6}}{x^2}+\frac {2 \left (-1+2 x^2\right ) \sqrt [4]{1-x^4+x^6}}{-1-x^2+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )+2 \int \frac {\left (-1+2 x^2\right ) \sqrt [4]{1-x^4+x^6}}{-1-x^2+x^4} \, dx-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )+2 \int \left (\frac {2 \sqrt [4]{1-x^4+x^6}}{-1-\sqrt {5}+2 x^2}+\frac {2 \sqrt [4]{1-x^4+x^6}}{-1+\sqrt {5}+2 x^2}\right ) \, dx-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+4 \int \frac {\sqrt [4]{1-x^4+x^6}}{-1-\sqrt {5}+2 x^2} \, dx+4 \int \frac {\sqrt [4]{1-x^4+x^6}}{-1+\sqrt {5}+2 x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+4 \int \left (\frac {i \sqrt [4]{1-x^4+x^6}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {i \sqrt [4]{1-x^4+x^6}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx+4 \int \left (\frac {\sqrt {1+\sqrt {5}} \sqrt [4]{1-x^4+x^6}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {\sqrt {1+\sqrt {5}} \sqrt [4]{1-x^4+x^6}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\frac {(2 i) \int \frac {\sqrt [4]{1-x^4+x^6}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {-1+\sqrt {5}}}+\frac {(2 i) \int \frac {\sqrt [4]{1-x^4+x^6}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {-1+\sqrt {5}}}-\frac {2 \int \frac {\sqrt [4]{1-x^4+x^6}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {1+\sqrt {5}}}-\frac {2 \int \frac {\sqrt [4]{1-x^4+x^6}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {1+\sqrt {5}}}+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.49, size = 70, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{1-x^4+x^6} \left (1+9 x^4+x^6\right )}{5 x^5}+2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 - x^4 + x^6)^(1/4)*(1 + 9*x^4 + x^6))/(5*x^5) + 2*ArcTan[x/(1 - x^4 + x^6)^(1/4)] - 2*ArcTanh[x/(1 - x^4
+ x^6)^(1/4)]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 14.41, size = 218, normalized size = 3.11

method	result	size
trager	\(\frac {2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} \left (x^{6}+9 x^{4}+1\right )}{5 x^{5}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}-x^{4}+1}\, x^{2}+2 \left (x^{6}-x^{4}+1\right )^{\frac {3}{4}} x -2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right ) \left (-1+x \right ) \left (x^{4}-x^{2}-1\right )}\right )+\ln \left (-\frac {-x^{6}+2 \left (x^{6}-x^{4}+1\right )^{\frac {3}{4}} x -2 \sqrt {x^{6}-x^{4}+1}\, x^{2}+2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} x^{3}-1}{\left (1+x \right ) \left (-1+x \right ) \left (x^{4}-x^{2}-1\right )}\right )\)	\(218\)
risch	\(\frac {\frac {2}{5} x^{12}+\frac {16}{5} x^{10}+\frac {4}{5} x^{6}-\frac {18}{5} x^{8}+\frac {16}{5} x^{4}+\frac {2}{5}}{x^{5} \left (x^{6}-x^{4}+1\right )^{\frac {3}{4}}}+\frac {\left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{18}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{16}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{14}+2 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{13}-3 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{12}-4 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{11}+2 \sqrt {x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{10}+2 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{9}-2 \sqrt {x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+4 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{7}-3 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {3}{4}} x^{3}-4 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+2 \sqrt {x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (x^{6}-x^{4}+1\right )^{2} \left (1+x \right ) \left (-1+x \right ) \left (x^{4}-x^{2}-1\right )}\right )+\ln \left (-\frac {-x^{18}+2 x^{16}+2 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{13}-x^{14}-4 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{11}-3 x^{12}-2 \sqrt {x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1}\, x^{8}+2 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+4 x^{10}+2 \sqrt {x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1}\, x^{6}+4 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{7}-x^{8}+2 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {3}{4}} x^{3}-4 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x^{5}-3 x^{6}-2 \sqrt {x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1}\, x^{2}+2 x^{4}+2 \left (x^{18}-3 x^{16}+3 x^{14}+2 x^{12}-6 x^{10}+3 x^{8}+3 x^{6}-3 x^{4}+1\right )^{\frac {1}{4}} x -1}{\left (x^{6}-x^{4}+1\right )^{2} \left (1+x \right ) \left (-1+x \right ) \left (x^{4}-x^{2}-1\right )}\right )\right ) \left (\left (x^{6}-x^{4}+1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{6}-x^{4}+1\right )^{\frac {3}{4}}}\)	\(1246\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x^6-x^4+1)^(1/4)*(x^6+9*x^4+1)/x^5-RootOf(_Z^2+1)*ln(-(-RootOf(_Z^2+1)*x^6+2*RootOf(_Z^2+1)*(x^6-x^4+1)^(
1/2)*x^2+2*(x^6-x^4+1)^(3/4)*x-2*(x^6-x^4+1)^(1/4)*x^3-RootOf(_Z^2+1))/(1+x)/(-1+x)/(x^4-x^2-1))+ln(-(-x^6+2*(
x^6-x^4+1)^(3/4)*x-2*(x^6-x^4+1)^(1/2)*x^2+2*(x^6-x^4+1)^(1/4)*x^3-1)/(1+x)/(-1+x)/(x^4-x^2-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (62) = 124\).
time = 46.23, size = 154, normalized size = 2.20 \begin {gather*} \frac {5 \, x^{5} \arctan \left (\frac {2 \, {\left ({\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{6} - 2 \, x^{4} + 1}\right ) + 5 \, x^{5} \log \left (\frac {x^{6} - 2 \, {\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{6} - x^{4} + 1} x^{2} - 2 \, {\left (x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} - 2 \, x^{4} + 1}\right ) + 2 \, {\left (x^{6} + 9 \, x^{4} + 1\right )} {\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(5*x^5*arctan(2*((x^6 - x^4 + 1)^(1/4)*x^3 + (x^6 - x^4 + 1)^(3/4)*x)/(x^6 - 2*x^4 + 1)) + 5*x^5*log((x^6
- 2*(x^6 - x^4 + 1)^(1/4)*x^3 + 2*sqrt(x^6 - x^4 + 1)*x^2 - 2*(x^6 - x^4 + 1)^(3/4)*x + 1)/(x^6 - 2*x^4 + 1))
+ 2*(x^6 + 9*x^4 + 1)*(x^6 - x^4 + 1)^(1/4))/x^5

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Sympy [F(-1)] Timed out
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**6-2)*(x**6+1)*(x**6-x**4+1)**(1/4)/x**6/(x**6-2*x**4+1),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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