Optimal. Leaf size=46 \[ \frac {\tanh ^{-1}\left (\frac {2 x+\sqrt {3}}{\sqrt {7}}\right )}{\sqrt {7}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3}-2 x}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1161, 618, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 x+\sqrt {3}}{\sqrt {7}}\right )}{\sqrt {7}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3}-2 x}{\sqrt {7}}\right )}{\sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {1-x^2}{1-5 x^2+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1}{-1-\sqrt {3} x+x^2} \, dx\right )-\frac {1}{2} \int \frac {1}{-1+\sqrt {3} x+x^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{7-x^2} \, dx,x,-\sqrt {3}+2 x\right )+\operatorname {Subst}\left (\int \frac {1}{7-x^2} \, dx,x,\sqrt {3}+2 x\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3}-2 x}{\sqrt {7}}\right )}{\sqrt {7}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3}+2 x}{\sqrt {7}}\right )}{\sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.87 \begin {gather*} \frac {\log \left (x^2+\sqrt {7} x+1\right )-\log \left (-x^2+\sqrt {7} x-1\right )}{2 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^2}{1-5 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.78, size = 39, normalized size = 0.85 \begin {gather*} \frac {1}{14} \, \sqrt {7} \log \left (\frac {x^{4} + 9 \, x^{2} + 2 \, \sqrt {7} {\left (x^{3} + x\right )} + 1}{x^{4} - 5 \, x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 39, normalized size = 0.85 \begin {gather*} -\frac {1}{14} \, \sqrt {7} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {7} + \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {7} + \frac {2}{x} \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 82, normalized size = 1.78 \begin {gather*} \frac {2 \left (-3+\sqrt {21}\right ) \sqrt {21}\, \arctanh \left (\frac {4 x}{2 \sqrt {7}-2 \sqrt {3}}\right )}{21 \left (2 \sqrt {7}-2 \sqrt {3}\right )}+\frac {2 \left (3+\sqrt {21}\right ) \sqrt {21}\, \arctanh \left (\frac {4 x}{2 \sqrt {7}+2 \sqrt {3}}\right )}{21 \left (2 \sqrt {7}+2 \sqrt {3}\right )} \end {gather*}
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^{2} - 1}{x^{4} - 5 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.39, size = 18, normalized size = 0.39 \begin {gather*} \frac {\sqrt {7}\,\mathrm {atanh}\left (\frac {\sqrt {7}\,x}{x^2+1}\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 39, normalized size = 0.85 \begin {gather*} - \frac {\sqrt {7} \log {\left (x^{2} - \sqrt {7} x + 1 \right )}}{14} + \frac {\sqrt {7} \log {\left (x^{2} + \sqrt {7} x + 1 \right )}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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