Integrand size = 35, antiderivative size = 55 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=-\frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(413\) vs. \(2(55)=110\).
Time = 0.39 (sec) , antiderivative size = 413, normalized size of antiderivative = 7.51, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6860, 385, 218, 214, 211} \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\frac {\sqrt [4]{a-\sqrt {a^2+4 b}} \arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\sqrt [4]{\sqrt {a^2+4 b}+a} \arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\sqrt [4]{a-\sqrt {a^2+4 b}} \text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\sqrt [4]{\sqrt {a^2+4 b}+a} \text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}} \]
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a-\sqrt {a^2+4 b}}{\left (a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}+\frac {a+\sqrt {a^2+4 b}}{\left (a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx \\ & = \left (a-\sqrt {a^2+4 b}\right ) \int \frac {1}{\left (a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx+\left (a+\sqrt {a^2+4 b}\right ) \int \frac {1}{\left (a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx \\ & = \left (a-\sqrt {a^2+4 b}\right ) \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 )+\left (a+\sqrt {a^2+4 b}\right ) \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 ) \\ & = \frac {1}{2} \sqrt {a-\sqrt {a^2+4 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 )+\frac {1}{2} \sqrt {a-\sqrt {a^2+4 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 )+\frac {1}{2} \sqrt {a+\sqrt {a^2+4 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 )+\frac {1}{2} \sqrt {a+\sqrt {a^2+4 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 ) \\ & = \frac {\sqrt [4]{a-\sqrt {a^2+4 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 )}{2 \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+4 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 )}{2 \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}+\frac {\sqrt [4]{a-\sqrt {a^2+4 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 )}{2 \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+4 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 )}{2 \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=-\frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Time = 0.97 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{4}\) | \(48\) |
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Timed out. \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 61.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.53 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\int \frac {a x^{4} - 2 b}{\sqrt [4]{a x^{4} + b} \left (a x^{4} - b + x^{8}\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.64 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (x^{8} + a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 1.55 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.64 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (x^{8} + a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )} \, dx=\int -\frac {2\,b-a\,x^4}{{\left (a\,x^4+b\right )}^{1/4}\,\left (x^8+a\,x^4-b\right )} \,d x \]
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