3.24.48 \(\int \frac {(1+x^2) (1-3 x^2+x^4)}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} (1-x-3 x^2+x^3+x^4)} \, dx\)

Optimal. Leaf size=186 \[ \frac {\sqrt {\frac {2 x^2+x-2}{x^2+x-1}} \left (x^2+x-1\right )}{2 x}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {\frac {2 x^2+x-2}{x^2+x-1}}}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {1}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {1}{2}+\frac {1}{2 \sqrt {5}}} \sqrt {\frac {2 x^2+x-2}{x^2+x-1}}\right )+\frac {1}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {5}}} \sqrt {\frac {2 x^2+x-2}{x^2+x-1}}\right ) \]

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Rubi [F]  time = 6.14, 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 ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 6.66, size = 26530, normalized size = 142.63 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.69, size = 188, normalized size = 1.01 \begin {gather*} \frac {\left (-1+x+x^2\right ) \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}}{2 x}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {1}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {1}{2}-\frac {1}{2 \sqrt {5}}} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}\right )+\frac {1}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {1}{2}+\frac {1}{2 \sqrt {5}}} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x + x^2)*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)])/(2*x) - (3*ArcTanh[Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)
]/Sqrt[2]])/(2*Sqrt[2]) + (Sqrt[2*(5 + Sqrt[5])]*ArcTanh[Sqrt[1/2 - 1/(2*Sqrt[5])]*Sqrt[(-2 + x + 2*x^2)/(-1 +
x + x^2)]])/5 + (Sqrt[2*(5 - Sqrt[5])]*ArcTanh[Sqrt[1/2 + 1/(2*Sqrt[5])]*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)
]])/5

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fricas [B]  time = 0.99, size = 723, normalized size = 3.89 \begin {gather*} \frac {4 \, x \sqrt {2 \, \sqrt {5} + 10} \log \left (\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} - \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {2 \, \sqrt {5} + 10} + 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} - \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) - 4 \, x \sqrt {2 \, \sqrt {5} + 10} \log \left (-\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} - \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {2 \, \sqrt {5} + 10} - 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} - \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) + 4 \, x \sqrt {-2 \, \sqrt {5} + 10} \log \left (\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} + \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {-2 \, \sqrt {5} + 10} + 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} + \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) - 4 \, x \sqrt {-2 \, \sqrt {5} + 10} \log \left (-\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} + \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {-2 \, \sqrt {5} + 10} - 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} + \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) + 15 \, \sqrt {2} x \log \left (-\frac {32 \, x^{4} + 48 \, x^{3} - 47 \, x^{2} - 4 \, \sqrt {2} {\left (4 \, x^{4} + 7 \, x^{3} - 5 \, x^{2} - 7 \, x + 4\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}} - 48 \, x + 32}{x^{2}}\right ) + 40 \, {\left (x^{2} + x - 1\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{80 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2-x+1),x, algorithm="fricas")

[Out]

1/80*(4*x*sqrt(2*sqrt(5) + 10)*log(((20575*x^4 + 50235*x^3 - 15795*x^2 - sqrt(5)*(10237*x^4 + 22677*x^3 - 9661
*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(2*sqrt(5) + 10) + 20*(1627*x^4 + 4593*x^3 - 288*x^2 - sqrt(5)*
(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3
- 3*x^2 - x + 1)) - 4*x*sqrt(2*sqrt(5) + 10)*log(-((20575*x^4 + 50235*x^3 - 15795*x^2 - sqrt(5)*(10237*x^4 +
22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(2*sqrt(5) + 10) - 20*(1627*x^4 + 4593*x^3 - 28
8*x^2 - sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 + x - 2)/(x^2 + x -
1)))/(x^4 + x^3 - 3*x^2 - x + 1)) + 4*x*sqrt(-2*sqrt(5) + 10)*log(((20575*x^4 + 50235*x^3 - 15795*x^2 + sqrt(
5)*(10237*x^4 + 22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(-2*sqrt(5) + 10) + 20*(1627*x^
4 + 4593*x^3 - 288*x^2 + sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 +
x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) - 4*x*sqrt(-2*sqrt(5) + 10)*log(-((20575*x^4 + 50235*x^3 -
15795*x^2 + sqrt(5)*(10237*x^4 + 22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(-2*sqrt(5) +
10) - 20*(1627*x^4 + 4593*x^3 - 288*x^2 + sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 16
27)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) + 15*sqrt(2)*x*log(-(32*x^4 + 48*x^3 - 4
7*x^2 - 4*sqrt(2)*(4*x^4 + 7*x^3 - 5*x^2 - 7*x + 4)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)) - 48*x + 32)/x^2) + 40
*(x^2 + x - 1)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 3 \, x^{2} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{3} - 3 \, x^{2} - x + 1\right )} x^{2} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2-x+1),x, algorithm="giac")

[Out]

integrate((x^4 - 3*x^2 + 1)*(x^2 + 1)/((x^4 + x^3 - 3*x^2 - x + 1)*x^2*sqrt((2*x^2 + x - 2)/(x^2 + x - 1))), x
)

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maple [C]  time = 6.13, size = 1372, normalized size = 7.38

method	result	size
trager	\(\frac {\left (x^{2}+x -1\right ) \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}}{2 x}-\frac {\RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right ) \ln \left (\frac {614375 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{5} x^{2}+497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x^{2}-614375 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{5}-2032100 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3} x^{2}+497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x -1081300 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3} x -497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2}-293952 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, x^{2}+2032100 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3}+981920 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right ) x^{2}-293952 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, x +635360 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right ) x +293952 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}-981920 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )}{25 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x^{2}-25 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2}-12 x^{2}+4 x +12}\right )}{2}+\frac {25 \ln \left (-\frac {413125 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{5} x^{2}+497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x^{2}-413125 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{5}+693200 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3} x^{2}+497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x +727100 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3} x -497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2}-104128 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, x^{2}-693200 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3}-192000 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right ) x^{2}-104128 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, x -176000 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right ) x +104128 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}+192000 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )}{25 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x^{2}-25 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2}-8 x^{2}-4 x +8}\right ) \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3}}{8}-\frac {3 \ln \left (-\frac {413125 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{5} x^{2}+497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x^{2}-413125 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{5}+693200 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3} x^{2}+497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x +727100 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3} x -497600 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2}-104128 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, x^{2}-693200 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{3}-192000 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right ) x^{2}-104128 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, x -176000 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right ) x +104128 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}+192000 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )}{25 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2} x^{2}-25 \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )^{2}-8 x^{2}-4 x +8}\right ) \RootOf \left (125 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+16\right )}{2}+\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-4 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -4 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}\, x -4 \RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {-\frac {-2 x^{2}-x +2}{x^{2}+x -1}}}{x}\right )}{8}\)	\(1372\)
risch	\(\text {Expression too large to display}\)	\(1635\)
default	\(\text {Expression too large to display}\)	\(14835\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2-x+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(x^2+x-1)/x*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)-1/2*RootOf(125*_Z^4-100*_Z^2+16)*ln((614375*RootOf(125*_Z^4-10
0*_Z^2+16)^5*x^2+497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf(125*_Z^4-100*_Z^2+16)^2*x^2-614375*RootOf(125*_
Z^4-100*_Z^2+16)^5-2032100*RootOf(125*_Z^4-100*_Z^2+16)^3*x^2+497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf(12
5*_Z^4-100*_Z^2+16)^2*x-1081300*RootOf(125*_Z^4-100*_Z^2+16)^3*x-497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf
(125*_Z^4-100*_Z^2+16)^2-293952*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*x^2+2032100*RootOf(125*_Z^4-100*_Z^2+16)^3+981
920*RootOf(125*_Z^4-100*_Z^2+16)*x^2-293952*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*x+635360*RootOf(125*_Z^4-100*_Z^2+
16)*x+293952*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)-981920*RootOf(125*_Z^4-100*_Z^2+16))/(25*RootOf(125*_Z^4-100*_Z^2
+16)^2*x^2-25*RootOf(125*_Z^4-100*_Z^2+16)^2-12*x^2+4*x+12))+25/8*ln(-(413125*RootOf(125*_Z^4-100*_Z^2+16)^5*x
^2+497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf(125*_Z^4-100*_Z^2+16)^2*x^2-413125*RootOf(125*_Z^4-100*_Z^2+1
6)^5+693200*RootOf(125*_Z^4-100*_Z^2+16)^3*x^2+497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf(125*_Z^4-100*_Z^2
+16)^2*x+727100*RootOf(125*_Z^4-100*_Z^2+16)^3*x-497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf(125*_Z^4-100*_Z
^2+16)^2-104128*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*x^2-693200*RootOf(125*_Z^4-100*_Z^2+16)^3-192000*RootOf(125*_Z
^4-100*_Z^2+16)*x^2-104128*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*x-176000*RootOf(125*_Z^4-100*_Z^2+16)*x+104128*(-(-
2*x^2-x+2)/(x^2+x-1))^(1/2)+192000*RootOf(125*_Z^4-100*_Z^2+16))/(25*RootOf(125*_Z^4-100*_Z^2+16)^2*x^2-25*Roo
tOf(125*_Z^4-100*_Z^2+16)^2-8*x^2-4*x+8))*RootOf(125*_Z^4-100*_Z^2+16)^3-3/2*ln(-(413125*RootOf(125*_Z^4-100*_
Z^2+16)^5*x^2+497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf(125*_Z^4-100*_Z^2+16)^2*x^2-413125*RootOf(125*_Z^4
-100*_Z^2+16)^5+693200*RootOf(125*_Z^4-100*_Z^2+16)^3*x^2+497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf(125*_Z
^4-100*_Z^2+16)^2*x+727100*RootOf(125*_Z^4-100*_Z^2+16)^3*x-497600*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*RootOf(125*
_Z^4-100*_Z^2+16)^2-104128*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*x^2-693200*RootOf(125*_Z^4-100*_Z^2+16)^3-192000*Ro
otOf(125*_Z^4-100*_Z^2+16)*x^2-104128*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*x-176000*RootOf(125*_Z^4-100*_Z^2+16)*x+
104128*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)+192000*RootOf(125*_Z^4-100*_Z^2+16))/(25*RootOf(125*_Z^4-100*_Z^2+16)^2
*x^2-25*RootOf(125*_Z^4-100*_Z^2+16)^2-8*x^2-4*x+8))*RootOf(125*_Z^4-100*_Z^2+16)+3/8*RootOf(_Z^2-2)*ln((4*Roo
tOf(_Z^2-2)*x^2-4*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*x^2+3*RootOf(_Z^2-2)*x-4*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2)*x-4
*RootOf(_Z^2-2)+4*(-(-2*x^2-x+2)/(x^2+x-1))^(1/2))/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 3 \, x^{2} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{3} - 3 \, x^{2} - x + 1\right )} x^{2} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2-x+1),x, algorithm="maxima")

[Out]

integrate((x^4 - 3*x^2 + 1)*(x^2 + 1)/((x^4 + x^3 - 3*x^2 - x + 1)*x^2*sqrt((2*x^2 + x - 2)/(x^2 + x - 1))), 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^4-3\,x^2+1\right )}{x^2\,\sqrt {\frac {2\,x^2+x-2}{x^2+x-1}}\,\left (x^4+x^3-3\,x^2-x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 + 1)*(x^4 - 3*x^2 + 1))/(x^2*((x + 2*x^2 - 2)/(x + x^2 - 1))^(1/2)*(x^3 - 3*x^2 - x + x^4 + 1)),x)

[Out]

int(((x^2 + 1)*(x^4 - 3*x^2 + 1))/(x^2*((x + 2*x^2 - 2)/(x + x^2 - 1))^(1/2)*(x^3 - 3*x^2 - x + x^4 + 1)), 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((x**2+1)*(x**4-3*x**2+1)/x**2/((2*x**2+x-2)/(x**2+x-1))**(1/2)/(x**4+x**3-3*x**2-x+1),x)

[Out]

Timed out

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