Integrand size = 20, antiderivative size = 46 \[ \int \frac {1-x^2}{1-5 x^2+x^4} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {3}-2 x}{\sqrt {7}}\right )}{\sqrt {7}}+\frac {\text {arctanh}\left (\frac {\sqrt {3}+2 x}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1175, 632, 212} \[ \int \frac {1-x^2}{1-5 x^2+x^4} \, dx=\frac {\text {arctanh}\left (\frac {2 x+\sqrt {3}}{\sqrt {7}}\right )}{\sqrt {7}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}-2 x}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Rule 212
Rule 632
Rule 1175
Rubi steps \begin{align*} \text {integral}& = -\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 \\ & = \text {Subst}\left (\int \frac {1}{7-x^2} \, dx,x,-\sqrt {3}+2 x\right )+\text {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*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87 \[ \int \frac {1-x^2}{1-5 x^2+x^4} \, dx=\frac {-\log \left (-1+\sqrt {7} x-x^2\right )+\log \left (1+\sqrt {7} x+x^2\right )}{2 \sqrt {7}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {\sqrt {7}\, \ln \left (x^{2}+x \sqrt {7}+1\right )}{14}-\frac {\sqrt {7}\, \ln \left (x^{2}-x \sqrt {7}+1\right )}{14}\) | \(35\) |
default | \(\frac {2 \left (3+\sqrt {21}\right ) \sqrt {21}\, \operatorname {arctanh}\left (\frac {4 x}{2 \sqrt {7}+2 \sqrt {3}}\right )}{21 \left (2 \sqrt {7}+2 \sqrt {3}\right )}+\frac {2 \left (-3+\sqrt {21}\right ) \sqrt {21}\, \operatorname {arctanh}\left (\frac {4 x}{2 \sqrt {7}-2 \sqrt {3}}\right )}{21 \left (2 \sqrt {7}-2 \sqrt {3}\right )}\) | \(82\) |
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1-x^2}{1-5 x^2+x^4} \, dx=\frac {1}{14} \, \sqrt {7} \log \left (\frac {x^{4} + 9 \, x^{2} + 2 \, \sqrt {7} {\left (x^{3} + x\right )} + 1}{x^{4} - 5 \, x^{2} + 1}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1-x^2}{1-5 x^2+x^4} \, dx=- \frac {\sqrt {7} \log {\left (x^{2} - \sqrt {7} x + 1 \right )}}{14} + \frac {\sqrt {7} \log {\left (x^{2} + \sqrt {7} x + 1 \right )}}{14} \]
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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 = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1-x^2}{1-5 x^2+x^4} \, dx=-\frac {1}{14} \, \sqrt {7} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {7} + \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {7} + \frac {2}{x} \right |}}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.39 \[ \int \frac {1-x^2}{1-5 x^2+x^4} \, dx=\frac {\sqrt {7}\,\mathrm {atanh}\left (\frac {\sqrt {7}\,x}{x^2+1}\right )}{7} \]
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