Integrand size = 18, antiderivative size = 46 \[ \int \frac {1+x^2}{1-5 x^2+x^4} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {7}-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {7}+2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1175, 632, 212} \[ \int \frac {1+x^2}{1-5 x^2+x^4} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {7}-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {2 x+\sqrt {7}}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rule 212
Rule 632
Rule 1175
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {1}{1-\sqrt {7} x+x^2} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {7} x+x^2} \, dx \\ & = -\text {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,-\sqrt {7}+2 x\right )-\text {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\sqrt {7}+2 x\right ) \\ & = \frac {\tanh ^{-1}\left (\frac {\sqrt {7}-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {7}+2 x}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87 \[ \int \frac {1+x^2}{1-5 x^2+x^4} \, dx=\frac {\log \left (1+\sqrt {3} x-x^2\right )-\log \left (-1+\sqrt {3} x+x^2\right )}{2 \sqrt {3}} \]
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Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {\sqrt {3}\, \ln \left (x^{2}-x \sqrt {3}-1\right )}{6}-\frac {\sqrt {3}\, \ln \left (x^{2}+x \sqrt {3}-1\right )}{6}\) | \(35\) |
default | \(-\frac {2 \sqrt {21}\, \left (7+\sqrt {21}\right ) \operatorname {arctanh}\left (\frac {4 x}{2 \sqrt {7}+2 \sqrt {3}}\right )}{21 \left (2 \sqrt {7}+2 \sqrt {3}\right )}-\frac {2 \left (-7+\sqrt {21}\right ) \sqrt {21}\, \operatorname {arctanh}\left (\frac {4 x}{2 \sqrt {7}-2 \sqrt {3}}\right )}{21 \left (2 \sqrt {7}-2 \sqrt {3}\right )}\) | \(82\) |
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Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1+x^2}{1-5 x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {3} {\left (x^{3} - x\right )} + 1}{x^{4} - 5 \, x^{2} + 1}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1+x^2}{1-5 x^2+x^4} \, dx=\frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x - 1 \right )}}{6} - \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x - 1 \right )}}{6} \]
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\[ \int \frac {1+x^2}{1-5 x^2+x^4} \, dx=\int { \frac {x^{2} + 1}{x^{4} - 5 \, x^{2} + 1} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1+x^2}{1-5 x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {3} - \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {3} - \frac {2}{x} \right |}}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.39 \[ \int \frac {1+x^2}{1-5 x^2+x^4} \, dx=-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,x}{x^2-1}\right )}{3} \]
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