Integrand size = 23, antiderivative size = 29 \[ \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx=-2 \text {arctanh}\left (\frac {x}{1-2 x+x^2+\sqrt {1-x^2+x^4}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1755, 12, 1261, 738, 212, 1712} \[ \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {x^4-x^2+1}}\right )-\text {arctanh}\left (\frac {x^2+1}{2 \sqrt {x^4-x^2+1}}\right ) \]
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Rule 12
Rule 212
Rule 738
Rule 1261
Rule 1712
Rule 1755
Rubi steps \begin{align*} \text {integral}& = \int -\frac {2 x}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx+\int \frac {-1-x^2}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx \\ & = -\left (2 \int \frac {x}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx\right )-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1-x^2+x^4}}\right ) \\ & = -\text {arctanh}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right )-\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1-x+x^2}} \, dx,x,x^2\right ) \\ & = -\text {arctanh}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right )+2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1-x^2}{\sqrt {1-x^2+x^4}}\right ) \\ & = -\text {arctanh}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right )-\text {arctanh}\left (\frac {1+x^2}{2 \sqrt {1-x^2+x^4}}\right ) \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx=-2 \text {arctanh}\left (\frac {x}{1-2 x+x^2+\sqrt {1-x^2+x^4}}\right ) \]
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Time = 1.89 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\operatorname {arctanh}\left (\frac {2 x^{2}-3 x +2}{\sqrt {x^{4}-x^{2}+1}}\right )\) | \(27\) |
pseudoelliptic | \(-\operatorname {arctanh}\left (\frac {2 x^{2}-3 x +2}{\sqrt {x^{4}-x^{2}+1}}\right )\) | \(27\) |
trager | \(\ln \left (-\frac {-2 x^{2}+\sqrt {x^{4}-x^{2}+1}+3 x -2}{\left (x -1\right )^{2}}\right )\) | \(31\) |
elliptic | \(-\operatorname {arctanh}\left (\frac {x^{2}+1}{2 \sqrt {\left (x^{2}-1\right )^{2}+x^{2}}}\right )-\operatorname {arctanh}\left (\frac {\sqrt {x^{4}-x^{2}+1}}{x}\right )\) | \(44\) |
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx=\log \left (\frac {2 \, x^{2} - 3 \, x - \sqrt {x^{4} - x^{2} + 1} + 2}{x^{2} - 2 \, x + 1}\right ) \]
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\[ \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx=\int \frac {x + 1}{\left (x - 1\right ) \sqrt {x^{4} - x^{2} + 1}}\, dx \]
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\[ \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx=\int { \frac {x + 1}{\sqrt {x^{4} - x^{2} + 1} {\left (x - 1\right )}} \,d x } \]
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\[ \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx=\int { \frac {x + 1}{\sqrt {x^{4} - x^{2} + 1} {\left (x - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx=\int \frac {x+1}{\left (x-1\right )\,\sqrt {x^4-x^2+1}} \,d x \]
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