3.16.41 \(\int \frac {(1+x^6) (-1+2 x^6) (-1+x^4+2 x^6)^{5/4}}{x^{10} (-1-x^4+2 x^6)} \, dx\) [1541]

Optimal. Leaf size=106 \[ \frac {\sqrt [4]{-1+x^4+2 x^6} \left (5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right )}{45 x^9}+\sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right ) \]

[Out]

1/45*(2*x^6+x^4-1)^(1/4)*(20*x^12+38*x^10+104*x^8-20*x^6-19*x^4+5)/x^9+2^(1/4)*arctan(2^(1/4)*x/(2*x^6+x^4-1)^
(1/4))-2^(1/4)*arctanh(2^(1/4)*x/(2*x^6+x^4-1)^(1/4))

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Rubi [F]
time = 1.88, 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^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^6)*(-1 + 2*x^6)*(-1 + x^4 + 2*x^6)^(5/4))/(x^10*(-1 - x^4 + 2*x^6)),x]

[Out]

((3 + I*Sqrt[7])*Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/(Sqrt[-1 - I*Sqrt[7]] - 2*x), x])/(2*Sqrt[-1 - I*Sqrt[7]]
) + ((3 - I*Sqrt[7])*Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/(Sqrt[-1 + I*Sqrt[7]] - 2*x), x])/(2*Sqrt[-1 + I*Sqrt
[7]]) + Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/(-1 + x), x]/4 + Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/x^10, x] - De
fer[Int][(-1 + x^4 + 2*x^6)^(5/4)/x^6, x] + Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/x^4, x] + Defer[Int][(-1 + x^4
+ 2*x^6)^(5/4)/x^2, x] - Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/(1 + x), x]/4 + ((3 + I*Sqrt[7])*Defer[Int][(-1
+ x^4 + 2*x^6)^(5/4)/(Sqrt[-1 - I*Sqrt[7]] + 2*x), x])/(2*Sqrt[-1 - I*Sqrt[7]]) + ((3 - I*Sqrt[7])*Defer[Int][
(-1 + x^4 + 2*x^6)^(5/4)/(Sqrt[-1 + I*Sqrt[7]] + 2*x), x])/(2*Sqrt[-1 + I*Sqrt[7]])

Rubi steps

\begin {align*} \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}}-\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (-1+x^2\right )}+\frac {\left (-5-6 x^2\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+x^2+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x^2} \, dx+\frac {1}{2} \int \frac {\left (-5-6 x^2\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1+x^2+2 x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 (-1+x)}-\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 (1+x)}\right ) \, dx+\frac {1}{2} \int \left (\frac {\left (-6+2 i \sqrt {7}\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1-i \sqrt {7}+4 x^2}+\frac {\left (-6-2 i \sqrt {7}\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1+i \sqrt {7}+4 x^2}\right ) \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\left (-3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+i \sqrt {7}+4 x^2} \, dx+\left (-3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1-i \sqrt {7}+4 x^2} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\left (-3-i \sqrt {7}\right ) \int \left (\frac {\sqrt {-1-i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+i \sqrt {7}\right ) \left (\sqrt {-1-i \sqrt {7}}-2 x\right )}+\frac {\sqrt {-1-i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+i \sqrt {7}\right ) \left (\sqrt {-1-i \sqrt {7}}+2 x\right )}\right ) \, dx+\left (-3+i \sqrt {7}\right ) \int \left (\frac {\sqrt {-1+i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1-i \sqrt {7}\right ) \left (\sqrt {-1+i \sqrt {7}}-2 x\right )}+\frac {\sqrt {-1+i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1-i \sqrt {7}\right ) \left (\sqrt {-1+i \sqrt {7}}+2 x\right )}\right ) \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}-2 x} \, dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}+2 x} \, dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}-2 x} \, dx}{2 \sqrt {-1-i \sqrt {7}}}+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}+2 x} \, dx}{2 \sqrt {-1-i \sqrt {7}}}+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.45, size = 106, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-1+x^4+2 x^6} \left (5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right )}{45 x^9}+\sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^6)*(-1 + 2*x^6)*(-1 + x^4 + 2*x^6)^(5/4))/(x^10*(-1 - x^4 + 2*x^6)),x]

[Out]

((-1 + x^4 + 2*x^6)^(1/4)*(5 - 19*x^4 - 20*x^6 + 104*x^8 + 38*x^10 + 20*x^12))/(45*x^9) + 2^(1/4)*ArcTan[(2^(1
/4)*x)/(-1 + x^4 + 2*x^6)^(1/4)] - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(-1 + x^4 + 2*x^6)^(1/4)]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 118.29, size = 363, normalized size = 3.42

method	result	size
trager	\(\frac {\left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{12}+38 x^{10}+104 x^{8}-20 x^{6}-19 x^{4}+5\right )}{45 x^{9}}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{6}+3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{4}-2\right ) \sqrt {2 x^{6}+x^{4}-1}\, x^{2}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{6}+3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {2 x^{6}+x^{4}-1}\, x^{2}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}\)	\(363\)
risch	\(\text {Expression too large to display}\)	\(1580\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)*(2*x^6-1)*(2*x^6+x^4-1)^(5/4)/x^10/(2*x^6-x^4-1),x,method=_RETURNVERBOSE)

[Out]

1/45*(2*x^6+x^4-1)^(1/4)*(20*x^12+38*x^10+104*x^8-20*x^6-19*x^4+5)/x^9-1/2*RootOf(_Z^4-2)*ln((2*RootOf(_Z^4-2)
^3*x^6+3*RootOf(_Z^4-2)^3*x^4+4*RootOf(_Z^4-2)^2*(2*x^6+x^4-1)^(1/4)*x^3+4*RootOf(_Z^4-2)*(2*x^6+x^4-1)^(1/2)*
x^2+4*(2*x^6+x^4-1)^(3/4)*x-RootOf(_Z^4-2)^3)/(-1+x)/(1+x)/(2*x^4+x^2+1))+1/2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*ln
((2*RootOf(_Z^4-2)^2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x^6+3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^4-4*
RootOf(_Z^4-2)^2*(2*x^6+x^4-1)^(1/4)*x^3-4*RootOf(_Z^2+RootOf(_Z^4-2)^2)*(2*x^6+x^4-1)^(1/2)*x^2+4*(2*x^6+x^4-
1)^(3/4)*x-RootOf(_Z^4-2)^2*RootOf(_Z^2+RootOf(_Z^4-2)^2))/(-1+x)/(1+x)/(2*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+1)*(2*x^6-1)*(2*x^6+x^4-1)^(5/4)/x^10/(2*x^6-x^4-1),x, algorithm="maxima")

[Out]

integrate((2*x^6 + x^4 - 1)^(5/4)*(2*x^6 - 1)*(x^6 + 1)/((2*x^6 - x^4 - 1)*x^10), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (90) = 180\).
time = 70.28, size = 345, normalized size = 3.25 \begin {gather*} -\frac {180 \cdot 8^{\frac {3}{4}} x^{9} \arctan \left (-\frac {16 \cdot 8^{\frac {1}{4}} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 8^{\frac {3}{4}} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x - 2^{\frac {3}{4}} {\left (8 \cdot 8^{\frac {1}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} + 8^{\frac {3}{4}} {\left (2 \, x^{6} + 3 \, x^{4} - 1\right )}\right )}}{8 \, {\left (2 \, x^{6} - x^{4} - 1\right )}}\right ) + 45 \cdot 8^{\frac {3}{4}} x^{9} \log \left (\frac {4 \, \sqrt {2} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {3}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} + 4 \, {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x + 8^{\frac {1}{4}} {\left (2 \, x^{6} + 3 \, x^{4} - 1\right )}}{2 \, x^{6} - x^{4} - 1}\right ) - 45 \cdot 8^{\frac {3}{4}} x^{9} \log \left (\frac {4 \, \sqrt {2} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 8^{\frac {3}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} + 4 \, {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x - 8^{\frac {1}{4}} {\left (2 \, x^{6} + 3 \, x^{4} - 1\right )}}{2 \, x^{6} - x^{4} - 1}\right ) - 16 \, {\left (20 \, x^{12} + 38 \, x^{10} + 104 \, x^{8} - 20 \, x^{6} - 19 \, x^{4} + 5\right )} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}}}{720 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(2*x^6-1)*(2*x^6+x^4-1)^(5/4)/x^10/(2*x^6-x^4-1),x, algorithm="fricas")

[Out]

-1/720*(180*8^(3/4)*x^9*arctan(-1/8*(16*8^(1/4)*(2*x^6 + x^4 - 1)^(1/4)*x^3 + 4*8^(3/4)*(2*x^6 + x^4 - 1)^(3/4
)*x - 2^(3/4)*(8*8^(1/4)*sqrt(2*x^6 + x^4 - 1)*x^2 + 8^(3/4)*(2*x^6 + 3*x^4 - 1)))/(2*x^6 - x^4 - 1)) + 45*8^(
3/4)*x^9*log((4*sqrt(2)*(2*x^6 + x^4 - 1)^(1/4)*x^3 + 8^(3/4)*sqrt(2*x^6 + x^4 - 1)*x^2 + 4*(2*x^6 + x^4 - 1)^
(3/4)*x + 8^(1/4)*(2*x^6 + 3*x^4 - 1))/(2*x^6 - x^4 - 1)) - 45*8^(3/4)*x^9*log((4*sqrt(2)*(2*x^6 + x^4 - 1)^(1
/4)*x^3 - 8^(3/4)*sqrt(2*x^6 + x^4 - 1)*x^2 + 4*(2*x^6 + x^4 - 1)^(3/4)*x - 8^(1/4)*(2*x^6 + 3*x^4 - 1))/(2*x^
6 - x^4 - 1)) - 16*(20*x^12 + 38*x^10 + 104*x^8 - 20*x^6 - 19*x^4 + 5)*(2*x^6 + x^4 - 1)^(1/4))/x^9

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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+1)*(2*x**6-1)*(2*x**6+x**4-1)**(5/4)/x**10/(2*x**6-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+1)*(2*x^6-1)*(2*x^6+x^4-1)^(5/4)/x^10/(2*x^6-x^4-1),x, algorithm="giac")

[Out]

integrate((2*x^6 + x^4 - 1)^(5/4)*(2*x^6 - 1)*(x^6 + 1)/((2*x^6 - x^4 - 1)*x^10), 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 (2\,x^6-1\right )\,{\left (2\,x^6+x^4-1\right )}^{5/4}}{x^{10}\,\left (-2\,x^6+x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^6 + 1)*(2*x^6 - 1)*(x^4 + 2*x^6 - 1)^(5/4))/(x^10*(x^4 - 2*x^6 + 1)),x)

[Out]

int(-((x^6 + 1)*(2*x^6 - 1)*(x^4 + 2*x^6 - 1)^(5/4))/(x^10*(x^4 - 2*x^6 + 1)), x)

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