Integrand size = 30, antiderivative size = 103 \[ \int \frac {3-x+2 x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{6} \sqrt {21+26 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{6} \sqrt {-21+26 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.47 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.94, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6860, 224, 2160, 2165, 212, 209} \[ \int \frac {3-x+2 x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\sqrt {266+153 \sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {4 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {26-7 \sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {1}{6} \sqrt {21+26 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {x^3+1}}\right )-\frac {1}{6} \sqrt {26 \sqrt {3}-21} \text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right ) \]
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Rule 209
Rule 212
Rule 224
Rule 2160
Rule 2165
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {1+x^3}}+\frac {7-5 x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {1+x^3}} \, dx+\int \frac {7-5 x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx \\ & = \frac {4 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\int \left (\frac {-5+4 \sqrt {3}}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}}+\frac {-5-4 \sqrt {3}}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}}\right ) \, dx \\ & = \frac {4 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\left (-5-4 \sqrt {3}\right ) \int \frac {1}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\left (-5+4 \sqrt {3}\right ) \int \frac {1}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx \\ & = \frac {4 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {1}{576} \left (12-5 \sqrt {3}\right ) \int \frac {96 \left (1+\sqrt {3}\right )+96 x}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\frac {1}{12} \left (-12+5 \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx+\frac {1}{576} \left (12+5 \sqrt {3}\right ) \int \frac {96 \left (1-\sqrt {3}\right )+96 x}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx-\frac {1}{12} \left (12+5 \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx \\ & = -\frac {\sqrt {26-7 \sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {4 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\sqrt {266+153 \sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {1}{6} \left (-12+5 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{1+\left (3-2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1-\sqrt {3}\right ) x}{2-2 \sqrt {3}}}{\sqrt {1+x^3}}\right )-\frac {1}{6} \left (12+5 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{1+\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1+\sqrt {3}\right ) x}{2+2 \sqrt {3}}}{\sqrt {1+x^3}}\right ) \\ & = -\frac {1}{6} \sqrt {21+26 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )-\frac {1}{6} \sqrt {-21+26 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )-\frac {\sqrt {26-7 \sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {4 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\sqrt {266+153 \sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ \end{align*}
Time = 1.58 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \frac {3-x+2 x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{6} \sqrt {21+26 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{6} \sqrt {-21+26 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 1.39 (sec) , antiderivative size = 1501, normalized size of antiderivative = 14.57
method | result | size |
default | \(\text {Expression too large to display}\) | \(1501\) |
elliptic | \(\text {Expression too large to display}\) | \(1706\) |
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Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (79) = 158\).
Time = 0.28 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.91 \[ \int \frac {3-x+2 x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{24} \, \sqrt {26 \, \sqrt {3} - 21} \log \left (\frac {23 \, x^{4} - 46 \, x^{3} + 138 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} - \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 8\right )} - 14 \, x - 10\right )} \sqrt {26 \, \sqrt {3} - 21} + 92 \, \sqrt {3} {\left (x^{3} + 1\right )} + 92 \, x + 92}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {26 \, \sqrt {3} - 21} \log \left (\frac {23 \, x^{4} - 46 \, x^{3} + 138 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} - \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 8\right )} - 14 \, x - 10\right )} \sqrt {26 \, \sqrt {3} - 21} + 92 \, \sqrt {3} {\left (x^{3} + 1\right )} + 92 \, x + 92}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {-26 \, \sqrt {3} - 21} \log \left (\frac {23 \, x^{4} - 46 \, x^{3} + 138 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} + \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 8\right )} - 14 \, x - 10\right )} \sqrt {-26 \, \sqrt {3} - 21} - 92 \, \sqrt {3} {\left (x^{3} + 1\right )} + 92 \, x + 92}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {-26 \, \sqrt {3} - 21} \log \left (\frac {23 \, x^{4} - 46 \, x^{3} + 138 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} + \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 8\right )} - 14 \, x - 10\right )} \sqrt {-26 \, \sqrt {3} - 21} - 92 \, \sqrt {3} {\left (x^{3} + 1\right )} + 92 \, x + 92}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) \]
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\[ \int \frac {3-x+2 x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int \frac {2 x^{2} - x + 3}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{2} + 2 x - 2\right )}\, dx \]
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\[ \int \frac {3-x+2 x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {2 \, x^{2} - x + 3}{\sqrt {x^{3} + 1} {\left (x^{2} + 2 \, x - 2\right )}} \,d x } \]
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\[ \int \frac {3-x+2 x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {2 \, x^{2} - x + 3}{\sqrt {x^{3} + 1} {\left (x^{2} + 2 \, x - 2\right )}} \,d x } \]
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Time = 0.10 (sec) , antiderivative size = 509, normalized size of antiderivative = 4.94 \[ \int \frac {3-x+2 x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {4\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\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 )}{\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 )}}+\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,\sqrt {3}-12\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}}}\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\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 )}{3\,\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 )}}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,\sqrt {3}+12\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}}}\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\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 )}{3\,\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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