Integrand size = 25, antiderivative size = 21 \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=-2 \text {arctanh}\left (\frac {\sqrt {-1+x^3}}{1+x+x^2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2171, 212} \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=2 \text {arctanh}\left (\frac {1-x}{\sqrt {x^3-1}}\right ) \]
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Rule 212
Rule 2171
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {1-x}{\sqrt {-1+x^3}}\right ) \\ & = 2 \text {arctanh}\left (\frac {1-x}{\sqrt {-1+x^3}}\right ) \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=-2 \text {arctanh}\left (\frac {\sqrt {-1+x^3}}{1+x+x^2}\right ) \]
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Time = 2.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33
method | result | size |
trager | \(-\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-1}+2 x}{x^{2}+2}\right )\) | \(28\) |
default | \(\text {Expression too large to display}\) | \(1656\) |
elliptic | \(\text {Expression too large to display}\) | \(1865\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=\log \left (\frac {x^{2} + 2 \, x - 2 \, \sqrt {x^{3} - 1}}{x^{2} + 2}\right ) \]
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\[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=\int \frac {x^{2} - 2 x - 2}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 2\right )}\, dx \]
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\[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=\int { \frac {x^{2} - 2 \, x - 2}{\sqrt {x^{3} - 1} {\left (x^{2} + 2\right )}} \,d x } \]
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\[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=\int { \frac {x^{2} - 2 \, x - 2}{\sqrt {x^{3} - 1} {\left (x^{2} + 2\right )}} \,d x } \]
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Time = 0.11 (sec) , antiderivative size = 275, normalized size of antiderivative = 13.10 \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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