Integrand size = 38, antiderivative size = 21 \[ \int \frac {-1+2 x^4}{\left (1+2 x^2+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+3 x^2+2 x^4}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2137, 212} \[ \int \frac {-1+2 x^4}{\left (1+2 x^2+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {2 x^4+3 x^2+1}}\right ) \]
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Rule 212
Rule 2137
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1+3 x^2+2 x^4}}\right ) \\ & = -\text {arctanh}\left (\frac {x}{\sqrt {1+3 x^2+2 x^4}}\right ) \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^4}{\left (1+2 x^2+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+3 x^2+2 x^4}}\right ) \]
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Time = 3.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05
method | result | size |
elliptic | \(-\operatorname {arctanh}\left (\frac {\sqrt {2 x^{4}+3 x^{2}+1}}{x}\right )\) | \(22\) |
trager | \(\frac {\ln \left (-\frac {-2 x^{4}+2 x \sqrt {2 x^{4}+3 x^{2}+1}-4 x^{2}-1}{2 x^{4}+2 x^{2}+1}\right )}{2}\) | \(49\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}+2 x \sqrt {2}-3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}-2 x \sqrt {2}+3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )}{2}\) | \(92\) |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}+2 x \sqrt {2}-3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}-2 x \sqrt {2}+3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )}{2}\) | \(92\) |
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24 \[ \int \frac {-1+2 x^4}{\left (1+2 x^2+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}} \, dx=\frac {1}{2} \, \log \left (\frac {2 \, x^{4} + 4 \, x^{2} - 2 \, \sqrt {2 \, x^{4} + 3 \, x^{2} + 1} x + 1}{2 \, x^{4} + 2 \, x^{2} + 1}\right ) \]
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\[ \int \frac {-1+2 x^4}{\left (1+2 x^2+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}} \, dx=\int \frac {2 x^{4} - 1}{\sqrt {\left (x^{2} + 1\right ) \left (2 x^{2} + 1\right )} \left (2 x^{4} + 2 x^{2} + 1\right )}\, dx \]
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\[ \int \frac {-1+2 x^4}{\left (1+2 x^2+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}} \, dx=\int { \frac {2 \, x^{4} - 1}{\sqrt {2 \, x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}} \,d x } \]
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\[ \int \frac {-1+2 x^4}{\left (1+2 x^2+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}} \, dx=\int { \frac {2 \, x^{4} - 1}{\sqrt {2 \, x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {-1+2 x^4}{\left (1+2 x^2+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}} \, dx=\int \frac {2\,x^4-1}{\left (2\,x^4+2\,x^2+1\right )\,\sqrt {2\,x^4+3\,x^2+1}} \,d x \]
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