Optimal. Leaf size=72 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {(x+1) \left (\sqrt {2} x-\sqrt {2}\right )}{x^3+x^2-\sqrt {x^6+2 x^5+x^4-4 x^3-5 x^2+2 x+3}-x-1}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 67, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 7, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {6688, 6719, 1033, 724, 206, 688, 207} \begin {gather*} -\frac {\left (1-x^2\right ) \sqrt {x^2+2 x+3} \tanh ^{-1}\left (\frac {\sqrt {x^2+2 x+3}}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {\left (1-x^2\right )^2 \left (x^2+2 x+3\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 207
Rule 688
Rule 724
Rule 1033
Rule 6688
Rule 6719
Rubi steps
\begin {align*} \int \frac {1-x}{\sqrt {3+2 x-5 x^2-4 x^3+x^4+2 x^5+x^6}} \, dx &=\int \frac {1-x}{\sqrt {\left (-1+x^2\right )^2 \left (3+2 x+x^2\right )}} \, dx\\ &=\frac {\left (\left (-1+x^2\right ) \sqrt {3+2 x+x^2}\right ) \int \frac {1-x}{\left (-1+x^2\right ) \sqrt {3+2 x+x^2}} \, dx}{\sqrt {\left (-1+x^2\right )^2 \left (3+2 x+x^2\right )}}\\ &=-\frac {\left (\left (-1+x^2\right ) \sqrt {3+2 x+x^2}\right ) \int \frac {1}{(1+x) \sqrt {3+2 x+x^2}} \, dx}{\sqrt {\left (-1+x^2\right )^2 \left (3+2 x+x^2\right )}}\\ &=-\frac {\left (4 \left (-1+x^2\right ) \sqrt {3+2 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-8+4 x^2} \, dx,x,\sqrt {3+2 x+x^2}\right )}{\sqrt {\left (-1+x^2\right )^2 \left (3+2 x+x^2\right )}}\\ &=-\frac {\left (1-x^2\right ) \sqrt {3+2 x+x^2} \tanh ^{-1}\left (\frac {\sqrt {3+2 x+x^2}}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {\left (1-x^2\right )^2 \left (3+2 x+x^2\right )}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 1.00 \begin {gather*} -\frac {\left (x^2-1\right ) \sqrt {x^2+2 x+3} \left (\log (x+1)-\log \left (\sqrt {2} \sqrt {x^2+2 x+3}+2\right )\right )}{\sqrt {2} \sqrt {\left (x^2-1\right )^2 \left (x^2+2 x+3\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 72, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {(1+x) \left (-\sqrt {2}+\sqrt {2} x\right )}{-1-x+x^2+x^3-\sqrt {3+2 x-5 x^2-4 x^3+x^4+2 x^5+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 58, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (x^{2} - 1\right )} + \sqrt {x^{6} + 2 \, x^{5} + x^{4} - 4 \, x^{3} - 5 \, x^{2} + 2 \, x + 3}}{x^{3} + x^{2} - x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 90, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {2} \log \left (-\frac {{\left | -2 \, \sqrt {3} - 2 \, \sqrt {2} + 2 \, \sqrt {\frac {2}{x} + \frac {3}{x^{2}} + 1} - \frac {2 \, \sqrt {3}}{x} \right |}}{2 \, {\left (\sqrt {3} - \sqrt {2} - \sqrt {\frac {2}{x} + \frac {3}{x^{2}} + 1} + \frac {\sqrt {3}}{x}\right )}}\right )}{2 \, \mathrm {sgn}\left (-\frac {1}{x^{3}} + \frac {1}{x^{5}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 64, normalized size = 0.89
method | result | size |
default | \(\frac {\left (x^{2}-1\right ) \sqrt {x^{2}+2 x +3}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {2}}{\sqrt {x^{2}+2 x +3}}\right )}{2 \sqrt {x^{6}+2 x^{5}+x^{4}-4 x^{3}-5 x^{2}+2 x +3}}\) | \(64\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+\sqrt {x^{6}+2 x^{5}+x^{4}-4 x^{3}-5 x^{2}+2 x +3}+\RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (-1+x \right ) \left (1+x \right )^{2}}\right )}{2}\) | \(68\) |
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 {x - 1}{\sqrt {x^{6} + 2 \, x^{5} + x^{4} - 4 \, x^{3} - 5 \, x^{2} + 2 \, x + 3}}\,{d x} \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 {x-1}{\sqrt {x^6+2\,x^5+x^4-4\,x^3-5\,x^2+2\,x+3}} \,d x \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 {x}{\sqrt {x^{6} + 2 x^{5} + x^{4} - 4 x^{3} - 5 x^{2} + 2 x + 3}}\, dx - \int \left (- \frac {1}{\sqrt {x^{6} + 2 x^{5} + x^{4} - 4 x^{3} - 5 x^{2} + 2 x + 3}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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