Integrand size = 20, antiderivative size = 121 \[ \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{\sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.55 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.90, number of steps used = 17, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {2081, 6865, 6857, 251, 1452, 440, 476, 502, 2174, 206, 31, 648, 631, 210, 642} \[ \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx=-\frac {6 x \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^4+x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x^2+1} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{x^4+x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}+\frac {3 x \sqrt [3]{x^2+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^4+x^2}}-\frac {x^{2/3} \sqrt [3]{x^2+1} \log \left (\left (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{4 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}-\frac {x^{2/3} \sqrt [3]{x^2+1} \log \left (\frac {2^{2/3} \left (x^{2/3}+1\right )^2}{\left (x^2+1\right )^{2/3}}-\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{2 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}+\frac {x^{2/3} \sqrt [3]{x^2+1} \log \left (\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{\sqrt [3]{2} \sqrt [3]{x^4+x^2}}+\frac {3 x^{2/3} \sqrt [3]{x^2+1} \log \left (x^{2/3}-2^{2/3} \sqrt [3]{x^2+1}+1\right )}{4 \sqrt [3]{2} \sqrt [3]{x^4+x^2}} \]
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Rule 31
Rule 206
Rule 210
Rule 251
Rule 440
Rule 476
Rule 502
Rule 631
Rule 642
Rule 648
Rule 1452
Rule 2081
Rule 2174
Rule 6857
Rule 6865
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1+x}{(-1+x) x^{2/3} \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^6}}+\frac {2}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = -\frac {6 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x^2+x^4}} \\ & = -\frac {6 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+2 x^3} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{\sqrt [3]{x^2+x^4}} \\ & = -\frac {6 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {3 x^{2/3} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{\sqrt [3]{x^2+x^4}} \\ & = -\frac {6 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{\sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {3 x^{2/3} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}} \\ & = -\frac {6 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}-\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (1+\frac {2^{2/3} \left (1+x^{2/3}\right )^2}{\left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{\sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {3 x^{2/3} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{\sqrt [3]{2} \sqrt [3]{x^2+x^4}} \\ & = -\frac {6 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x^2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{x^2+x^4}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}-\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (1+\frac {2^{2/3} \left (1+x^{2/3}\right )^2}{\left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {x^{2/3} \sqrt [3]{1+x^2} \log \left (1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{\sqrt [3]{2} \sqrt [3]{x^2+x^4}}+\frac {3 x^{2/3} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2+x^4}} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21 \[ \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx=-\frac {x^{2/3} \sqrt [3]{1+x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )+\log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^4}} \]
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Time = 7.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right )+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{4}\) | \(105\) |
trager | \(\text {Expression too large to display}\) | \(1463\) |
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Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (94) = 188\).
Time = 2.44 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.54 \[ \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{4} + 2 \, x^{3} - 6 \, x^{2} + 2 \, x + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{5} + 14 \, x^{4} + 6 \, x^{3} + 14 \, x^{2} + x\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{7} + 30 \, x^{6} + 51 \, x^{5} + 52 \, x^{4} + 51 \, x^{3} + 30 \, x^{2} + x\right )}}{3 \, {\left (x^{7} - 6 \, x^{6} - 93 \, x^{5} - 20 \, x^{4} - 93 \, x^{3} - 6 \, x^{2} + x\right )}}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + 4 \, x + 1\right )} + 4^{\frac {2}{3}} {\left (x^{5} + 14 \, x^{4} + 6 \, x^{3} + 14 \, x^{2} + x\right )} + 24 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} + x\right )}}{x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + x}\right ) + \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} - 6 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - 2 \, x^{2} + x}\right ) \]
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\[ \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx=\int \frac {x + 1}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right )}\, dx \]
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\[ \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx=\int { \frac {x + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x - 1\right )}} \,d x } \]
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\[ \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx=\int { \frac {x + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx=\int \frac {x+1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x-1\right )} \,d x \]
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