Integrand size = 27, antiderivative size = 96 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=-\frac {\text {RootSum}\left [b+2 a \text {$\#$1}^4-2 \text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{8 b} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(441\) vs. \(2(96)=192\).
Time = 0.16 (sec) , antiderivative size = 441, normalized size of antiderivative = 4.59, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1442, 385, 218, 214, 211} \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+2 b} \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}-\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+2 b} \left (\sqrt {a^2+2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+2 b} \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}-\frac {\text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+2 b} \left (\sqrt {a^2+2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}} \]
[In]
[Out]
Rule 211
Rule 214
Rule 218
Rule 385
Rule 1442
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\left (-2 a-2 \sqrt {a^2+2 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+2 b}}-\frac {\int \frac {1}{\left (-2 a+2 \sqrt {a^2+2 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+2 b}} \\ & = \frac {\text {Subst}\left (\int \frac {1}{-2 a-2 \sqrt {a^2+2 b}-\left (-2 b+a \left (-2 a-2 \sqrt {a^2+2 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+2 b}}-\frac {\text {Subst}\left (\int \frac {1}{-2 a+2 \sqrt {a^2+2 b}-\left (-2 b+a \left (-2 a+2 \sqrt {a^2+2 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+2 b}} \\ & = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+2 b}}-\sqrt {a^2+b-a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \sqrt {a-\sqrt {a^2+2 b}}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+2 b}}+\sqrt {a^2+b-a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \sqrt {a-\sqrt {a^2+2 b}}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+2 b}}-\sqrt {a^2+b+a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \sqrt {a+\sqrt {a^2+2 b}}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+2 b}}+\sqrt {a^2+b+a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \sqrt {a+\sqrt {a^2+2 b}}} \\ & = \frac {\arctan \left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+2 b}} x}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{a^2+b-a \sqrt {a^2+2 b}}}-\frac {\arctan \left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+2 b}} x}{\sqrt [4]{a+\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \left (a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{a^2+b+a \sqrt {a^2+2 b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+2 b}} x}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{a^2+b-a \sqrt {a^2+2 b}}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+2 b}} x}{\sqrt [4]{a+\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \left (a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{a^2+b+a \sqrt {a^2+2 b}}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=-\frac {\text {RootSum}\left [b+2 a \text {$\#$1}^4-2 \text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{8 b} \]
[In]
[Out]
Time = 0.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-2 \textit {\_Z}^{4} a -b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-a \right )}}{8 b}\) | \(66\) |
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 6.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} + b} \left (- 2 a x^{4} - 2 b + x^{8}\right )}\, dx \]
[In]
[Out]
Not integrable
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 2 \, a x^{4} - 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.92 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 2 \, a x^{4} - 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=-\int \frac {1}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+2\,a\,x^4+2\,b\right )} \,d x \]
[In]
[Out]