Integrand size = 27, antiderivative size = 18 \[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=-2 \text {arctanh}\left (\frac {1-x}{\sqrt {-1+x^3}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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}& = -\left (2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {1-x}{\sqrt {-1+x^3}}\right )\right ) \\ & = -2 \tanh ^{-1}\left (\frac {1-x}{\sqrt {-1+x^3}}\right ) \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=2 \text {arctanh}\left (\frac {\sqrt {-1+x^3}}{1+x+x^2}\right ) \]
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Time = 1.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44
method | result | size |
trager | \(\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-1}+2 x}{x^{2}+2}\right )\) | \(26\) |
default | \(\text {Expression too large to display}\) | \(1656\) |
elliptic | \(\text {Expression too large to display}\) | \(1865\) |
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=\log \left (\frac {x^{2} + 2 \, x + 2 \, \sqrt {x^{3} - 1}}{x^{2} + 2}\right ) \]
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\[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=- \int \left (- \frac {2 x}{x^{2} \sqrt {x^{3} - 1} + 2 \sqrt {x^{3} - 1}}\right )\, dx - \int \frac {x^{2}}{x^{2} \sqrt {x^{3} - 1} + 2 \sqrt {x^{3} - 1}}\, dx - \int \left (- \frac {2}{x^{2} \sqrt {x^{3} - 1} + 2 \sqrt {x^{3} - 1}}\right )\, dx \]
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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 = 0.12 (sec) , antiderivative size = 276, normalized size of antiderivative = 15.33 \[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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