Optimal. Leaf size=260 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^7+12 \text {$\#$1}^6-4 \text {$\#$1}^5-10 \text {$\#$1}^4+4 \text {$\#$1}^3+12 \text {$\#$1}^2+4 \text {$\#$1}+1\& ,\frac {\text {$\#$1}^6 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )-2 \text {$\#$1}^5 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )-7 \text {$\#$1}^4 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )+12 \text {$\#$1}^3 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )+7 \text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )-2 \text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )-\log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )}{2 \text {$\#$1}^7-7 \text {$\#$1}^6+18 \text {$\#$1}^5-5 \text {$\#$1}^4-10 \text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+1}\& \right ] \]
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Rubi [F] time = 1.55, 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 (x+x^3\right )}{\sqrt {1+x^4} \left (1-2 x^2+4 x^4-2 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 )^2 \left (x+x^3\right )}{\sqrt {1+x^4} \left (1-2 x^2+4 x^4-2 x^6+x^8\right )} \, dx &=\int \frac {x \left (-1+x^2\right )^2 \left (1+x^2\right )}{\sqrt {1+x^4} \left (1-2 x^2+4 x^4-2 x^6+x^8\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+x)^2 (1+x)}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )}-\frac {x}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )}-\frac {x^2}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )}+\frac {x^3}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )\\ \end {align*}
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Mathematica [C] time = 7.48, size = 5697, normalized size = 21.91 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.40, size = 274, normalized size = 1.05 \begin {gather*} \frac {1}{4} \text {RootSum}\left [1-4 \text {$\#$1}+12 \text {$\#$1}^2-4 \text {$\#$1}^3-10 \text {$\#$1}^4+4 \text {$\#$1}^5+12 \text {$\#$1}^6+4 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-\log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right )+2 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}+7 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^2-12 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^3-7 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^5+\log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^6}{-1+6 \text {$\#$1}-3 \text {$\#$1}^2-10 \text {$\#$1}^3+5 \text {$\#$1}^4+18 \text {$\#$1}^5+7 \text {$\#$1}^6+2 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 455, normalized size = 1.75 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {x^{4} - x^{2} + \sqrt {2} {\left (x^{4} - x^{2} + 1\right )} - \sqrt {2 \, x^{8} - 2 \, x^{6} + 4 \, x^{4} - \sqrt {2} {\left (x^{4} + x^{2} + 1\right )} - {\left (2 \, x^{6} - 2 \, x^{4} + 3 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{4} + 1} + 2} {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )} + 1\right )} - \sqrt {x^{4} + 1} {\left (x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} - 1\right )} + 1}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {x^{4} - x^{2} - \sqrt {2} {\left (x^{4} - x^{2} + 1\right )} - \sqrt {2 \, x^{8} - 2 \, x^{6} + 4 \, x^{4} + \sqrt {2} {\left (x^{4} + x^{2} + 1\right )} - {\left (2 \, x^{6} - 2 \, x^{4} + 3 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{4} + 1} + 2} {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 1\right )} - \sqrt {x^{4} + 1} {\left (x^{2} - \sqrt {2} {\left (x^{2} - 1\right )} - 1\right )} + 1}{x^{2}}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (8 \, x^{8} - 8 \, x^{6} + 16 \, x^{4} + 4 \, \sqrt {2} {\left (x^{4} + x^{2} + 1\right )} - 4 \, {\left (2 \, x^{6} - 2 \, x^{4} + 3 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{4} + 1} + 8\right ) + \frac {1}{8} \, \sqrt {2} \log \left (8 \, x^{8} - 8 \, x^{6} + 16 \, x^{4} - 4 \, \sqrt {2} {\left (x^{4} + x^{2} + 1\right )} - 4 \, {\left (2 \, x^{6} - 2 \, x^{4} + 3 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{4} + 1} + 8\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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.
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maple [B] time = 2.25, size = 67, normalized size = 0.26
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{3}-\textit {\_R}^{2}+\textit {\_R} +1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )-\textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{4}\) | \(67\) |
elliptic | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{3}-\textit {\_R}^{2}+\textit {\_R} +1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )-\textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{4}\) | \(67\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+x^{2} \sqrt {x^{4}+1}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-\sqrt {x^{4}+1}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+x^{4}-x^{2}+1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-x^{2} \sqrt {x^{4}+1}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+\sqrt {x^{4}+1}}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-x^{4}+x^{2}-1}\right )}{4}\) | \(275\) |
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*} \int \frac {{\left (x^{3} + x\right )} {\left (x^{2} - 1\right )}^{2}}{{\left (x^{8} - 2 \, x^{6} + 4 \, x^{4} - 2 \, x^{2} + 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.77, size = 475, normalized size = 1.83 \begin {gather*} \left (\sum _{k=1}^4\frac {{\left (\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )-1\right )}^2\,\left (\ln \left (-\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )+x^2\right )-\ln \left (\sqrt {\left ({\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+1\right )\,\left (x^4+1\right )}+\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )\,x^2+1\right )\right )\,\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}{4\,\left (4\,\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )-3\,{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+2\,{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^3-1\right )\,\sqrt {{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+1}}\right )+\left (\sum _{k=1}^4\frac {{\left (\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )-1\right )}^2\,\left (\ln \left (\sqrt {\left (x^4+1\right )\,\left ({\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+1\right )}+x^2\,\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )+1\right )-\ln \left (x^2-\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )\right )\right )}{\sqrt {{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+1}\,\left (-8\,{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^3+12\,{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2-16\,\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )+4\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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