Optimal. Leaf size=95 \[ \text {ArcTan}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\text {ArcTan}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right )-\tanh ^{-1}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right ) \]
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Rubi [F]
time = 0.04, 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 (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx &=\int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx\\ \end {align*}
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Mathematica [A]
time = 6.13, size = 95, normalized size = 1.00 \begin {gather*} \text {ArcTan}\left (\frac {-i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\text {ArcTan}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{i+x}\right )-\tanh ^{-1}\left (\frac {-i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{i+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.06, size = 232, normalized size = 2.44
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x -\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x}{\left (-1+x \right ) \left (1+x \right )}\right )}{2}+\frac {\ln \left (\frac {\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x -\sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}-x^{4}+\sqrt {x^{4}-6 x^{2}+1}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x +5 x^{2}}{\left (1+x \right ) \left (-1+x \right )}\right )}{2}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {2 x^{2}}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx - \int \frac {x^{4}}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx - \int \frac {1}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (x^2+1\right )}^2}{\left (x^2-1\right )\,{\left (x^4-6\,x^2+1\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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