Optimal. Leaf size=94 \[ \frac {(1+2 x) \sqrt {1+2 x^2+4 x^3+x^4}}{2 \left (-1+2 x^2\right )}-\tanh ^{-1}\left (\frac {x (2+x) \left (7-x+27 x^2+33 x^3\right )}{\left (2+37 x^2+31 x^3\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \]
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Rubi [F]
time = 1.28, 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 {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx &=\int \left (\frac {9}{4 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}}+\frac {x^2}{2 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {-13-17 x}{2 \left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {15+2 x}{4 \left (-1+2 x^2\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx\\ &=\frac {1}{4} \int \frac {15+2 x}{\left (-1+2 x^2\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{2} \int \frac {-13-17 x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac {1}{4} \int \left (-\frac {15+\sqrt {2}}{2 \left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}}-\frac {15-\sqrt {2}}{2 \left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx+\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{2} \int \left (-\frac {13}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}-\frac {17 x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{2} \int \frac {1}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{2} \int \left (\frac {1}{2 \left (\sqrt {2}-2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {1}{2 \left (\sqrt {2}+2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {1}{\left (2-4 x^2\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}-2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}+2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{2} \int \frac {1}{\left (2-4 x^2\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}-2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}+2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{2} \int \left (\frac {1}{4 \left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {1}{4 \left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{8} \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{8} \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}-2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}+2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 16.13, size = 5137, normalized size = 54.65 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 1.34, size = 1197351, normalized size = 12737.78
method | result | size |
trager | \(\frac {\left (1+2 x \right ) \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}}{4 x^{2}-2}+\frac {\ln \left (-\frac {-1025 x^{10}+1023 \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}\, x^{8}-6138 x^{9}+4104 \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}\, x^{7}-12307 x^{8}+5084 \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}\, x^{6}-10188 x^{7}+2182 \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}\, x^{5}-4503 x^{6}+805 \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}\, x^{4}-3134 x^{5}+624 \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}\, x^{3}-1589 x^{4}+10 \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}\, x^{2}-140 x^{3}+28 \sqrt {x^{4}+4 x^{3}+2 x^{2}+1}\, x -176 x^{2}-2}{\left (2 x^{2}-1\right )^{5}}\right )}{2}\) | \(270\) |
risch | \(\text {Expression too large to display}\) | \(812620\) |
elliptic | \(\text {Expression too large to display}\) | \(891160\) |
default | \(\text {Expression too large to display}\) | \(1197351\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs.
\(2 (88) = 176\).
time = 1.47, size = 179, normalized size = 1.90 \begin {gather*} \frac {{\left (2 \, x^{2} - 1\right )} \log \left (\frac {1025 \, x^{10} + 6138 \, x^{9} + 12307 \, x^{8} + 10188 \, x^{7} + 4503 \, x^{6} + 3134 \, x^{5} + 1589 \, x^{4} + 140 \, x^{3} + 176 \, x^{2} - {\left (1023 \, x^{8} + 4104 \, x^{7} + 5084 \, x^{6} + 2182 \, x^{5} + 805 \, x^{4} + 624 \, x^{3} + 10 \, x^{2} + 28 \, x\right )} \sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} + 2}{32 \, x^{10} - 80 \, x^{8} + 80 \, x^{6} - 40 \, x^{4} + 10 \, x^{2} - 1}\right ) + \sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} {\left (2 \, x + 1\right )}}{2 \, {\left (2 \, x^{2} - 1\right )}} \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^{6} + 4 x^{5} + 7 x^{4} - 3 x^{3} - x^{2} - 8 x - 8}{\sqrt {\left (x + 1\right ) \left (x^{3} + 3 x^{2} - x + 1\right )} \left (2 x^{2} - 1\right )^{2}}\, 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 {-2\,x^6-4\,x^5-7\,x^4+3\,x^3+x^2+8\,x+8}{{\left (2\,x^2-1\right )}^2\,\sqrt {x^4+4\,x^3+2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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