Integrand size = 30, antiderivative size = 116 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a+a^2-a b-\text {$\#$1}^4-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-2 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(447\) vs. \(2(116)=232\).
Time = 0.55 (sec) , antiderivative size = 447, normalized size of antiderivative = 3.85, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1442, 385, 218, 214, 211} \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\frac {a^{3/4} \arctan \left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}-\frac {a^{3/4} \arctan \left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}+\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}-\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}} \]
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 1442
Rubi steps \begin{align*} \text {integral}& = \frac {(2 a) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-1-\sqrt {1+4 a b}+2 a x^4\right )} \, dx}{\sqrt {1+4 a b}}-\frac {(2 a) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-1+\sqrt {1+4 a b}+2 a x^4\right )} \, dx}{\sqrt {1+4 a b}} \\ & = \frac {(2 a) \text {Subst}\left (\int \frac {1}{-1-\sqrt {1+4 a b}-\left (2 a b+a \left (-1-\sqrt {1+4 a b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b}}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{-1+\sqrt {1+4 a b}-\left (2 a b+a \left (-1+\sqrt {1+4 a b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b}} \\ & = \frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1-\sqrt {1+4 a b}}-\sqrt {a} \sqrt {1-2 b-\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1-\sqrt {1+4 a b}}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1-\sqrt {1+4 a b}}+\sqrt {a} \sqrt {1-2 b-\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1-\sqrt {1+4 a b}}}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {1+4 a b}}-\sqrt {a} \sqrt {1-2 b+\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1+\sqrt {1+4 a b}}}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {1+4 a b}}+\sqrt {a} \sqrt {1-2 b+\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1+\sqrt {1+4 a b}}} \\ & = \frac {a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b-\sqrt {1+4 a b}} x}{\sqrt [4]{1-\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1-\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b-\sqrt {1+4 a b}}}-\frac {a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b+\sqrt {1+4 a b}} x}{\sqrt [4]{1+\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1+\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b+\sqrt {1+4 a b}}}+\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b-\sqrt {1+4 a b}} x}{\sqrt [4]{1-\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1-\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b-\sqrt {1+4 a b}}}-\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b+\sqrt {1+4 a b}} x}{\sqrt [4]{1+\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1+\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b+\sqrt {1+4 a b}}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a+a^2-a b-\text {$\#$1}^4-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-2 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \]
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Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-2 a -1\right ) \textit {\_Z}^{4}+a^{2}-a b +a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-2 a -1\right )}}{4 b}\) | \(75\) |
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Timed out. \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 7.91 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} - b} \left (a x^{8} - b - x^{4}\right )}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} - x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 1.89 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} - x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 5.89 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\int \frac {1}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-a\,x^8+x^4+b\right )} \,d x \]
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