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

Optimal. Leaf size=83 \[ -\frac {\sqrt [4]{2 x^4+2 x^2-1}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+2 x^2-1}}\right )}{2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+2 x^2-1}}\right )}{2^{3/4}} \]

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Rubi [F]  time = 0.51, 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^2\right ) \sqrt [4]{-1+2 x^2+2 x^4}}{x^2 \left (-1+2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-(((-1 + 2*x^2 + 2*x^4)^(1/4)*AppellF1[-1/2, -1/4, -1/4, 1/2, (-2*x^2)/(1 - Sqrt[3]), (-2*x^2)/(1 + Sqrt[3])])
/(x*(1 + (2*x^2)/(1 - Sqrt[3]))^(1/4)*(1 + (2*x^2)/(1 + Sqrt[3]))^(1/4))) + Defer[Int][(-1 + 2*x^2 + 2*x^4)^(1
/4)/(1 - 2*x^2), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{-1+2 x^2+2 x^4}}{x^2 \left (-1+2 x^2\right )} \, dx &=\int \left (\frac {\sqrt [4]{-1+2 x^2+2 x^4}}{x^2}+\frac {\sqrt [4]{-1+2 x^2+2 x^4}}{1-2 x^2}\right ) \, dx\\ &=\int \frac {\sqrt [4]{-1+2 x^2+2 x^4}}{x^2} \, dx+\int \frac {\sqrt [4]{-1+2 x^2+2 x^4}}{1-2 x^2} \, dx\\ &=\frac {\sqrt [4]{-1+2 x^2+2 x^4} \int \frac {\sqrt [4]{1+\frac {4 x^2}{2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^2}{2+2 \sqrt {3}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {4 x^2}{2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^2}{2+2 \sqrt {3}}}}+\int \frac {\sqrt [4]{-1+2 x^2+2 x^4}}{1-2 x^2} \, dx\\ &=-\frac {\sqrt [4]{-1+2 x^2+2 x^4} F_1\left (-\frac {1}{2};-\frac {1}{4},-\frac {1}{4};\frac {1}{2};-\frac {2 x^2}{1-\sqrt {3}},-\frac {2 x^2}{1+\sqrt {3}}\right )}{x \sqrt [4]{1+\frac {2 x^2}{1-\sqrt {3}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {3}}}}+\int \frac {\sqrt [4]{-1+2 x^2+2 x^4}}{1-2 x^2} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{-1+2 x^2+2 x^4}}{x^2 \left (-1+2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.22, size = 83, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{-1+2 x^2+2 x^4}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^2+2 x^4}}\right )}{2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^2+2 x^4}}\right )}{2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((-1 + 2*x^2 + 2*x^4)^(1/4)/x) - ArcTan[(2^(1/4)*x)/(-1 + 2*x^2 + 2*x^4)^(1/4)]/2^(3/4) + ArcTanh[(2^(1/4)*x)
/(-1 + 2*x^2 + 2*x^4)^(1/4)]/2^(3/4)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 126.12, size = 326, normalized size = 3.93

method	result	size
trager	\(-\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {1}{4}}}{x}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{4}+2 x^{2}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {2 x^{4}+2 x^{2}-1}\, x^{2}-4 \left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{2 x^{2}-1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{4}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{4}+2 x^{2}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \RootOf \left (\textit {\_Z}^{4}-2\right ) \sqrt {2 x^{4}+2 x^{2}-1}\, x^{2}-4 \left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{3}}{2 x^{2}-1}\right )}{4}\)	\(326\)
risch	\(-\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {1}{4}}}{x}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {16 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{12}+8 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} x^{9}+40 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{10}+16 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} x^{7}+12 x^{8} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-24 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{6}+8 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{6}-8 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{4}-2\right ) \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {3}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+8 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{4}+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} x +6 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{\left (2 x^{2}-1\right ) \left (2 x^{4}+2 x^{2}-1\right )^{2}}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-16 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{12}+8 \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{9}-40 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{10}+16 \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{7}-12 x^{8} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}+24 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{6}+8 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{6}-8 \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {3}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+8 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{4}+2 \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -6 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{\left (2 x^{2}-1\right ) \left (2 x^{4}+2 x^{2}-1\right )^{2}}\right )}{4}\right ) \left (\left (2 x^{4}+2 x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{4}}}\)	\(1057\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x^4+2*x^2-1)^(1/4)/x+1/4*RootOf(_Z^2+RootOf(_Z^4-2)^2)*ln(-(4*RootOf(_Z^4-2)^2*RootOf(_Z^2+RootOf(_Z^4-2)^
2)*x^4+4*RootOf(_Z^4-2)^2*(2*x^4+2*x^2-1)^(1/4)*x^3+2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^2-4*Roo
tOf(_Z^2+RootOf(_Z^4-2)^2)*(2*x^4+2*x^2-1)^(1/2)*x^2-4*(2*x^4+2*x^2-1)^(3/4)*x-RootOf(_Z^4-2)^2*RootOf(_Z^2+Ro
otOf(_Z^4-2)^2))/(2*x^2-1))-1/4*RootOf(_Z^4-2)*ln(-(4*RootOf(_Z^4-2)^3*x^4-4*RootOf(_Z^4-2)^2*(2*x^4+2*x^2-1)^
(1/4)*x^3+2*RootOf(_Z^4-2)^3*x^2+4*RootOf(_Z^4-2)*(2*x^4+2*x^2-1)^(1/2)*x^2-4*(2*x^4+2*x^2-1)^(3/4)*x-RootOf(_
Z^4-2)^3)/(2*x^2-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x - 1)*(x + 1)*(2*x**4 + 2*x**2 - 1)**(1/4)/(x**2*(2*x**2 - 1)), x)

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