Optimal. Leaf size=144 \[ \frac {\sqrt {\left (x^2+1\right )^2} \left (\frac {5}{2} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^6+\text {$\#$1}^4-\text {$\#$1}^2+1\& ,\frac {\text {$\#$1}^6 \log (x-\text {$\#$1})-\text {$\#$1}^4 \log (x-\text {$\#$1})+\text {$\#$1}^2 \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{4 \text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3-\text {$\#$1}}\& \right ]+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{x^2+1}\right )\right )}{x^2+1} \]
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Rubi [F] time = 3.56, 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+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, 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 ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx &=\int \left (-\frac {4 x^2 \sqrt {1+2 x^2+x^4}}{1+x^4}+\frac {\sqrt {1+2 x^2+x^4} \left (-1+2 x^2-3 x^4+4 x^6\right )}{1-x^2+x^4-x^6+x^8}\right ) \, dx\\ &=-\left (4 \int \frac {x^2 \sqrt {1+2 x^2+x^4}}{1+x^4} \, dx\right )+\int \frac {\sqrt {1+2 x^2+x^4} \left (-1+2 x^2-3 x^4+4 x^6\right )}{1-x^2+x^4-x^6+x^8} \, dx\\ &=-\left (4 \int \left (-\frac {\sqrt {1+2 x^2+x^4}}{2 \left (i-x^2\right )}+\frac {\sqrt {1+2 x^2+x^4}}{2 \left (i+x^2\right )}\right ) \, dx\right )+\int \left (\frac {\sqrt {1+2 x^2+x^4}}{-1+x^2-x^4+x^6-x^8}+\frac {2 x^2 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8}-\frac {3 x^4 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8}+\frac {4 x^6 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+2 x^2+x^4}}{i-x^2} \, dx-2 \int \frac {\sqrt {1+2 x^2+x^4}}{i+x^2} \, dx+2 \int \frac {x^2 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8} \, dx-3 \int \frac {x^4 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8} \, dx+4 \int \frac {x^6 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8} \, dx+\int \frac {\sqrt {1+2 x^2+x^4}}{-1+x^2-x^4+x^6-x^8} \, dx\\ &=\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1+x^2}{i-x^2} \, dx}{1+x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1+x^2}{i+x^2} \, dx}{1+x^2}+\frac {\sqrt {1+2 x^2+x^4} \int \frac {2+2 x^2}{-1+x^2-x^4+x^6-x^8} \, dx}{2+2 x^2}+\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2 \left (2+2 x^2\right )}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (3 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4 \left (2+2 x^2\right )}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6 \left (2+2 x^2\right )}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}\\ &=-\frac {4 x \sqrt {1+2 x^2+x^4}}{1+x^2}-\frac {\left ((2-2 i) \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{i+x^2} \, dx}{1+x^2}+\frac {\left ((2+2 i) \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{i-x^2} \, dx}{1+x^2}+\frac {\sqrt {1+2 x^2+x^4} \int \left (-\frac {2}{1-x^2+x^4-x^6+x^8}-\frac {2 x^2}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2}+\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \left (\frac {2 x^2}{1-x^2+x^4-x^6+x^8}+\frac {2 x^4}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2}-\frac {\left (3 \sqrt {1+2 x^2+x^4}\right ) \int \left (\frac {2 x^4}{1-x^2+x^4-x^6+x^8}+\frac {2 x^6}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \left (2-\frac {2 \left (1-x^2+x^4-2 x^6\right )}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2}\\ &=-\frac {2 i \sqrt {2} \sqrt {1+2 x^2+x^4} \tan ^{-1}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {2 \sqrt {2} \sqrt {1+2 x^2+x^4} \tanh ^{-1}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1-x^2+x^4-2 x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}\\ &=-\frac {2 i \sqrt {2} \sqrt {1+2 x^2+x^4} \tan ^{-1}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {2 \sqrt {2} \sqrt {1+2 x^2+x^4} \tanh ^{-1}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \left (\frac {1}{1-x^2+x^4-x^6+x^8}-\frac {x^2}{1-x^2+x^4-x^6+x^8}+\frac {x^4}{1-x^2+x^4-x^6+x^8}-\frac {2 x^6}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2}\\ &=-\frac {2 i \sqrt {2} \sqrt {1+2 x^2+x^4} \tan ^{-1}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {2 \sqrt {2} \sqrt {1+2 x^2+x^4} \tanh ^{-1}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (16 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 161, normalized size = 1.12 \begin {gather*} \frac {\sqrt {\left (x^2+1\right )^2} \left (5 \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^6+\text {$\#$1}^4-\text {$\#$1}^2+1\&,\frac {\text {$\#$1}^6 \log (x-\text {$\#$1})-\text {$\#$1}^4 \log (x-\text {$\#$1})+\text {$\#$1}^2 \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{4 \text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3-\text {$\#$1}}\&\right ]+2 \sqrt {2} \left (\log \left (x^2+\sqrt {2} x+1\right )-\log \left (-x^2+\sqrt {2} x-1\right )\right )\right )}{2 \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.97, size = 134, normalized size = 0.93 \begin {gather*} 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+2 x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{2}-\frac {\sqrt {5}}{2}} x}{\sqrt {1+2 x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+2 x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 172, normalized size = 1.19 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (2 \, x^{2} + \sqrt {2} x \sqrt {\sqrt {5} + 5} + 2\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (2 \, x^{2} - \sqrt {2} x \sqrt {\sqrt {5} + 5} + 2\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (2 \, x^{2} + \sqrt {2} x \sqrt {-\sqrt {5} + 5} + 2\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (2 \, x^{2} - \sqrt {2} x \sqrt {-\sqrt {5} + 5} + 2\right ) + \sqrt {2} \log \left (\frac {x^{4} + 4 \, x^{2} + 2 \, \sqrt {2} {\left (x^{3} + x\right )} + 1}{x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 152, normalized size = 1.06 \begin {gather*} -\sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} + \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} + \frac {2}{x} \right |}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + \frac {1}{x} \right |}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + \frac {1}{x} \right |}\right ) - \frac {1}{4} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + \frac {1}{x} \right |}\right ) + \frac {1}{4} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + \frac {1}{x} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 110, normalized size = 0.76
method | result | size |
risch | \(\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \sqrt {2}\, \ln \left (x^{2}+\sqrt {2}\, x +1\right )}{x^{2}+1}-\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \sqrt {2}\, \ln \left (x^{2}-\sqrt {2}\, x +1\right )}{x^{2}+1}+\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-5 \textit {\_Z}^{2}+5\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R} x +x^{2}+1\right )\right )}{2 x^{2}+2}\) | \(110\) |
default | \(\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \left (\sqrt {2}\, \ln \left (-\frac {x^{2}+\sqrt {2}\, x +1}{\sqrt {2}\, x -x^{2}-1}\right )-\sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, x -x^{2}-1}{x^{2}+\sqrt {2}\, x +1}\right )+5 \left (\munderset {\textit {\_R} =\RootOf \left (125 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-5 \textit {\_R} x +x^{2}+1\right )\right )\right )}{2 x^{2}+2}\) | \(113\) |
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*} \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + 5 \, \int \frac {x^{6} - x^{4} + x^{2} - 1}{x^{8} - x^{6} + x^{4} - x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 202, normalized size = 1.40 \begin {gather*} 2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {420500000\,\sqrt {2}\,x}{420500000\,x^2+420500000}\right )+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {2125000\,\sqrt {2}\,x\,\sqrt {\sqrt {5}+5}}{1810000\,\sqrt {5}+1810000\,\sqrt {5}\,x^2-4250000\,x^2-4250000}-\frac {905000\,\sqrt {2}\,\sqrt {5}\,x\,\sqrt {\sqrt {5}+5}}{1810000\,\sqrt {5}+1810000\,\sqrt {5}\,x^2-4250000\,x^2-4250000}\right )\,\sqrt {\sqrt {5}+5}}{2}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {2125000\,\sqrt {2}\,x\,\sqrt {5-\sqrt {5}}}{1810000\,\sqrt {5}+1810000\,\sqrt {5}\,x^2+4250000\,x^2+4250000}+\frac {905000\,\sqrt {2}\,\sqrt {5}\,x\,\sqrt {5-\sqrt {5}}}{1810000\,\sqrt {5}+1810000\,\sqrt {5}\,x^2+4250000\,x^2+4250000}\right )\,\sqrt {5-\sqrt {5}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SympifyError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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