Integrand size = 41, antiderivative size = 150 \[ \int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(150)=300\).
Time = 0.72 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.34, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {6860, 246, 218, 212, 209, 1442, 385, 214, 211} \[ \int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\frac {b \arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {b \arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {b \text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {b \text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \]
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Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 246
Rule 385
Rule 1442
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt [4]{b+a x^4}}+\frac {b}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+b \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {(2 b) \int \frac {1}{\left (a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+4 b}}-\frac {(2 b) \int \frac {1}{\left (a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+4 b}} \\ & = \frac {(2 b) \text {Subst}\left (\int \frac {1}{a-\sqrt {a^2+4 b}-\left (-2 b+a \left (a-\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{a+\sqrt {a^2+4 b}-\left (-2 b+a \left (a+\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b}}+\text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}+\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}+\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}} \\ & = \frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {b \arctan \left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {b \arctan \left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97 \[ \int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \]
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Time = 0.53 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-3 a \right )}\right ) a^{\frac {1}{4}}-4 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+2 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{4 a^{\frac {1}{4}}}\) | \(134\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.56 (sec) , antiderivative size = 5410, normalized size of antiderivative = 36.07 \[ \int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\text {Too large to display} \]
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Not integrable
Time = 129.57 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.23 \[ \int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\int \frac {2 a x^{4} - b + 2 x^{8}}{\sqrt [4]{a x^{4} + b} \left (a x^{4} - b + x^{8}\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.27 \[ \int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} + 2 \, a x^{4} - b}{{\left (x^{8} + a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 1.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.27 \[ \int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} + 2 \, a x^{4} - b}{{\left (x^{8} + a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 6.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.27 \[ \int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\int \frac {2\,x^8+2\,a\,x^4-b}{{\left (a\,x^4+b\right )}^{1/4}\,\left (x^8+a\,x^4-b\right )} \,d x \]
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