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

Optimal. Leaf size=123 \[ \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4+x^2-1}}{\sqrt {x^4+x^2-1}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4+x^2-1}}{x^2+\sqrt {x^4+x^2-1}}\right )+\frac {2 \sqrt [4]{x^4+x^2-1} \left (9 x^4-x^2+1\right )}{5 x^5} \]

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Rubi [F]  time = 1.04, 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^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*(-1 + x^2 + x^4)^(1/4)*AppellF1[-1/2, -1/4, -1/4, 1/2, (-2*x^2)/(1 - Sqrt[5]), (-2*x^2)/(1 + Sqrt[5])])/(x*
(1 + (2*x^2)/(1 - Sqrt[5]))^(1/4)*(1 + (2*x^2)/(1 + Sqrt[5]))^(1/4)) - ((1 + (2*x^2)/(1 + Sqrt[5]))^(5/4)*(-1
+ x^2 + x^4)^(1/4)*Hypergeometric2F1[-3/2, -1/4, -1/2, (-2*(x^2/(1 - Sqrt[5]) - x^2/(1 + Sqrt[5])))/(1 + (2*x^
2)/(1 + Sqrt[5]))])/(3*x^3*(1 + (2*x^2)/(1 - Sqrt[5]))^(1/4)) - (4*(1 + (2*x^2)/(1 + Sqrt[5]))*(-1 + x^2 + x^4
)^(1/4)*((3*(1 + Sqrt[5]) - (13 + 3*Sqrt[5])*x^2 + 2*(1 + Sqrt[5])*x^4)*Gamma[-1/4]*Hypergeometric2F1[-1/4, 1,
-1/2, (-2*Sqrt[5]*x^2)/(2 - (1 + Sqrt[5])*x^2)] - 4*x^2*(5 + Sqrt[5] + 2*Sqrt[5]*x^2)*Gamma[3/4]*Hypergeometr
ic2F1[3/4, 2, 1/2, (-2*Sqrt[5]*x^2)/(2 - (1 + Sqrt[5])*x^2)]))/(15*(3 + Sqrt[5])*x^5*(1 - Sqrt[5] + 2*x^2)*Gam
ma[-1/4]) + 2*Defer[Int][(-1 + x^2 + x^4)^(1/4)/(1 + x^2), x] + 4*Defer[Int][(-1 + x^2 + x^4)^(1/4)/(-1 + 2*x^
2), x]

Rubi steps

\begin {align*} \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx &=\int \left (-\frac {2 \sqrt [4]{-1+x^2+x^4}}{x^6}+\frac {\sqrt [4]{-1+x^2+x^4}}{x^4}-\frac {4 \sqrt [4]{-1+x^2+x^4}}{x^2}+\frac {2 \sqrt [4]{-1+x^2+x^4}}{1+x^2}+\frac {4 \sqrt [4]{-1+x^2+x^4}}{-1+2 x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{x^6} \, dx\right )+2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx-4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx+\int \frac {\sqrt [4]{-1+x^2+x^4}}{x^4} \, dx\\ &=2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx+\frac {\sqrt [4]{-1+x^2+x^4} \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^4} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (2 \sqrt [4]{-1+x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^6} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (4 \sqrt [4]{-1+x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}\\ &=\frac {4 \sqrt [4]{-1+x^2+x^4} F_1\left (-\frac {1}{2};-\frac {1}{4},-\frac {1}{4};\frac {1}{2};-\frac {2 x^2}{1-\sqrt {5}},-\frac {2 x^2}{1+\sqrt {5}}\right )}{x \sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (1+\frac {2 x^2}{1+\sqrt {5}}\right )^{5/4} \sqrt [4]{-1+x^2+x^4} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};-\frac {1}{2};-\frac {2 \left (\frac {x^2}{1-\sqrt {5}}-\frac {x^2}{1+\sqrt {5}}\right )}{1+\frac {2 x^2}{1+\sqrt {5}}}\right )}{3 x^3 \sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}}}-\frac {4 \left (1+\frac {2 x^2}{1+\sqrt {5}}\right ) \sqrt [4]{-1+x^2+x^4} \left (\left (3 \left (1+\sqrt {5}\right )-\left (13+3 \sqrt {5}\right ) x^2+2 \left (1+\sqrt {5}\right ) x^4\right ) \Gamma \left (-\frac {1}{4}\right ) \, _2F_1\left (-\frac {1}{4},1;-\frac {1}{2};-\frac {2 \sqrt {5} x^2}{2-\left (1+\sqrt {5}\right ) x^2}\right )-4 x^2 \left (5+\sqrt {5}+2 \sqrt {5} x^2\right ) \Gamma \left (\frac {3}{4}\right ) \, _2F_1\left (\frac {3}{4},2;\frac {1}{2};-\frac {2 \sqrt {5} x^2}{2-\left (1+\sqrt {5}\right ) x^2}\right )\right )}{15 \left (3+\sqrt {5}\right ) x^5 \left (1-\sqrt {5}+2 x^2\right ) \Gamma \left (-\frac {1}{4}\right )}+2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.38, size = 123, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{-1+x^2+x^4} \left (1-x^2+9 x^4\right )}{5 x^5}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{-x^2+\sqrt {-1+x^2+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{x^2+\sqrt {-1+x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 8.94, size = 559, normalized size = 4.54 \begin {gather*} \frac {20 \, \sqrt {2} x^{5} \arctan \left (\frac {\sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x - {\left (2 \, x^{4} - \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} - \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + x^{2} - 1\right )} \sqrt {\frac {2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + x^{2} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + x^{2} - 1}{2 \, x^{4} + x^{2} - 1}}}{x^{2} - 1}\right ) + 20 \, \sqrt {2} x^{5} \arctan \left (\frac {\sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + {\left (2 \, x^{4} + \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} + \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + x^{2} - 1\right )} \sqrt {\frac {2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + x^{2} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + x^{2} - 1}{2 \, x^{4} + x^{2} - 1}}}{x^{2} - 1}\right ) - 5 \, \sqrt {2} x^{5} \log \left (\frac {2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + x^{2} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + x^{2} - 1}{2 \, x^{4} + x^{2} - 1}\right ) + 5 \, \sqrt {2} x^{5} \log \left (\frac {2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + x^{2} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + x^{2} - 1}{2 \, x^{4} + x^{2} - 1}\right ) + 8 \, {\left (9 \, x^{4} - x^{2} + 1\right )} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}}}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(20*sqrt(2)*x^5*arctan((sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x - (2*x^4 - sq
rt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 2*sqrt(x^4 + x^2 - 1)*x^2 - sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x + x^2 - 1)*sqrt(
(2*x^4 + 2*sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 4*sqrt(x^4 + x^2 - 1)*x^2 + 2*sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x +
x^2 - 1)/(2*x^4 + x^2 - 1)))/(x^2 - 1)) + 20*sqrt(2)*x^5*arctan((sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + sqrt(2)*
(x^4 + x^2 - 1)^(3/4)*x + (2*x^4 + sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 2*sqrt(x^4 + x^2 - 1)*x^2 + sqrt(2)*(x^
4 + x^2 - 1)^(3/4)*x + x^2 - 1)*sqrt((2*x^4 - 2*sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 4*sqrt(x^4 + x^2 - 1)*x^2
- 2*sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x + x^2 - 1)/(2*x^4 + x^2 - 1)))/(x^2 - 1)) - 5*sqrt(2)*x^5*log((2*x^4 + 2*s
qrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 4*sqrt(x^4 + x^2 - 1)*x^2 + 2*sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x + x^2 - 1)/(2
*x^4 + x^2 - 1)) + 5*sqrt(2)*x^5*log((2*x^4 - 2*sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 4*sqrt(x^4 + x^2 - 1)*x^2
- 2*sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x + x^2 - 1)/(2*x^4 + x^2 - 1)) + 8*(9*x^4 - x^2 + 1)*(x^4 + x^2 - 1)^(1/4))
/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 8.98, size = 239, normalized size = 1.94

method	result	size
trager	\(\frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \left (9 x^{4}-x^{2}+1\right )}{5 x^{5}}-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{4}+x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \sqrt {x^{4}+x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )\)	\(239\)
risch	\(\frac {\frac {18}{5} x^{8}+\frac {16}{5} x^{6}-\frac {18}{5} x^{4}+\frac {4}{5} x^{2}-\frac {2}{5}}{x^{5} \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}}}+\frac {\left (-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{8}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{6}-4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x +3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{2}-1\right )^{2} \left (2 x^{2}-1\right ) \left (x^{2}+1\right )}\right )+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{10}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{9}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{8}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{6}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{5}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{4}+4 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{2}-1\right )^{2} \left (2 x^{2}-1\right ) \left (x^{2}+1\right )}\right )\right ) \left (\left (x^{4}+x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{4}+x^{2}-1\right )^{\frac {3}{4}}}\)	\(849\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((x^2 - 1)*(x^2 - 2)*(x^2 + x^4 - 1)^(1/4))/(x^6*(x^2 + 2*x^4 - 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 ) \left (x^{2} - 2\right ) \sqrt [4]{x^{4} + x^{2} - 1}}{x^{6} \left (x^{2} + 1\right ) \left (2 x^{2} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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