Integrand size = 28, antiderivative size = 33 \[ \int \frac {-2+2 x+x^2}{\left (-1-3 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {1+x^3}}{1-x+x^2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2170, 212} \[ \int \frac {-2+2 x+x^2}{\left (-1-3 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} (x+1)}{\sqrt {x^3+1}}\right ) \]
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Rule 212
Rule 2170
Rubi steps \begin{align*} \text {integral}& = -\left (4 \text {Subst}\left (\int \frac {1}{2-4 x^2} \, dx,x,\frac {1+x}{\sqrt {1+x^3}}\right )\right ) \\ & = -\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} (1+x)}{\sqrt {1+x^3}}\right ) \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-2+2 x+x^2}{\left (-1-3 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {1+x^3}}{1-x+x^2}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-4 \sqrt {x^{3}+1}}{x^{2}-3 x -1}\right )}{2}\) | \(58\) |
default | \(\text {Expression too large to display}\) | \(1625\) |
elliptic | \(\text {Expression too large to display}\) | \(1830\) |
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {-2+2 x+x^2}{\left (-1-3 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 10 \, x^{3} - 4 \, \sqrt {2} \sqrt {x^{3} + 1} {\left (x^{2} + x + 3\right )} + 7 \, x^{2} + 6 \, x + 17}{x^{4} - 6 \, x^{3} + 7 \, x^{2} + 6 \, x + 1}\right ) \]
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\[ \int \frac {-2+2 x+x^2}{\left (-1-3 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int \frac {x^{2} + 2 x - 2}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{2} - 3 x - 1\right )}\, dx \]
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\[ \int \frac {-2+2 x+x^2}{\left (-1-3 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (x^{2} - 3 \, x - 1\right )}} \,d x } \]
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\[ \int \frac {-2+2 x+x^2}{\left (-1-3 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (x^{2} - 3 \, x - 1\right )}} \,d x } \]
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Time = 5.42 (sec) , antiderivative size = 272, normalized size of antiderivative = 8.24 \[ \int \frac {-2+2 x+x^2}{\left (-1-3 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {\sqrt {13}}{2}+\frac {5}{2}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {\sqrt {13}}{2}-\frac {5}{2}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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