Integrand size = 16, antiderivative size = 36 \[ \int \frac {x}{a+b x^2+c x^4} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1121, 632, 212} \[ \int \frac {x}{a+b x^2+c x^4} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Rule 212
Rule 632
Rule 1121
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right ) \\ & = -\text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \frac {x}{a+b x^2+c x^4} \, dx=\frac {\arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}} \]
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Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\) | \(36\) |
risch | \(-\frac {\ln \left (\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}-2 a \right )}{2 \sqrt {-4 a c +b^{2}}}+\frac {\ln \left (\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}+2 a \right )}{2 \sqrt {-4 a c +b^{2}}}\) | \(70\) |
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none
Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.58 \[ \int \frac {x}{a+b x^2+c x^4} \, dx=\left [\frac {\log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c - {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{2 \, \sqrt {b^{2} - 4 \, a c}}, -\frac {\sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{b^{2} - 4 \, a c}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (34) = 68\).
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.64 \[ \int \frac {x}{a+b x^2+c x^4} \, dx=- \frac {\sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x^{2} + \frac {- 4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )}}{2} + \frac {\sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x^{2} + \frac {4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )}}{2} \]
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Exception generated. \[ \int \frac {x}{a+b x^2+c x^4} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.65 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {x}{a+b x^2+c x^4} \, dx=\frac {\arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \frac {x}{a+b x^2+c x^4} \, dx=\frac {\mathrm {atan}\left (\frac {2\,a\,c\,x^2+a\,b}{a\,\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}} \]
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