Integrand size = 27, antiderivative size = 211 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=-2 \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {3}{2} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]+\frac {1}{2} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
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Result contains complex when optimal does not.
Time = 1.62 (sec) , antiderivative size = 1271, normalized size of antiderivative = 6.02, number of steps used = 35, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2081, 6860, 920, 65, 338, 304, 209, 212, 6857, 95, 211, 214} \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}} \]
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Rule 65
Rule 95
Rule 209
Rule 211
Rule 212
Rule 214
Rule 304
Rule 338
Rule 920
Rule 2081
Rule 6857
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x} \left (-1+x^2\right )}{1+x^2+x^4} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int \left (\frac {\left (1+i \sqrt {3}\right ) x^{3/4} \sqrt [4]{1+x}}{1-i \sqrt {3}+2 x^2}+\frac {\left (1-i \sqrt {3}\right ) x^{3/4} \sqrt [4]{1+x}}{1+i \sqrt {3}+2 x^2}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x}}{1+i \sqrt {3}+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x}}{1-i \sqrt {3}+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {-1-i \sqrt {3}+2 x}{\sqrt [4]{x} (1+x)^{3/4} \left (1+i \sqrt {3}+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {-1+i \sqrt {3}+2 x}{\sqrt [4]{x} (1+x)^{3/4} \left (1-i \sqrt {3}+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \left (\frac {\sqrt {2} \left (-1-i \sqrt {3}\right )+\left (-1-i \sqrt {3}\right )^{3/2}}{2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}-\sqrt {2} x\right )}+\frac {-\sqrt {2} \left (-1-i \sqrt {3}\right )+\left (-1-i \sqrt {3}\right )^{3/2}}{2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \left (\frac {\sqrt {2} \left (-1+i \sqrt {3}\right )+\left (-1+i \sqrt {3}\right )^{3/2}}{2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}-\sqrt {2} x\right )}+\frac {-\sqrt {2} \left (-1+i \sqrt {3}\right )+\left (-1+i \sqrt {3}\right )^{3/2}}{2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}-\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}+\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}-\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}+\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1-i \sqrt {3}}-\left (\sqrt {2}+\sqrt {-1-i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1-i \sqrt {3}}-\left (-\sqrt {2}+\sqrt {-1-i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+i \sqrt {3}}-\left (\sqrt {2}+\sqrt {-1+i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+i \sqrt {3}}-\left (-\sqrt {2}+\sqrt {-1+i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (16 \left (-\arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )+\text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )\right )-3 \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (1+x)\right )^{3/4}} \]
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Time = 32.94 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(2 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )+\ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+x}{x}\right )-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )}{2}-\ln \left (\frac {-x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\left (\textit {\_R}^{4}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-3\right )}\right )}{2}\) | \(160\) |
trager | \(\text {Expression too large to display}\) | \(4123\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.30 (sec) , antiderivative size = 1070, normalized size of antiderivative = 5.07 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\text {Too large to display} \]
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Not integrable
Time = 1.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{4} + x^{2} + 1} \,d x } \]
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Exception generated. \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x^2-1\right )}{x^4+x^2+1} \,d x \]
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