Integrand size = 33, antiderivative size = 133 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \]
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Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {544, 246, 218, 212, 209, 385} \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{6 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{6 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 385
Rule 544
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx-\frac {1}{3} (2 b) \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{3} (2 b) \text {Subst}\left (\int \frac {1}{-b+4 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-2 \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+2 \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 )}{6 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}} \]
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Time = 0.58 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(\frac {-2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right ) \sqrt {2}+\ln \left (\frac {x \,a^{\frac {1}{4}} \sqrt {2}+\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{-x \,a^{\frac {1}{4}} \sqrt {2}+\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) \sqrt {2}-2 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{12 a^{\frac {1}{4}}}\) | \(136\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.78 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} - \frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} - \frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} \]
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\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int \frac {a x^{4} - b}{\sqrt [4]{a x^{4} + b} \left (3 a x^{4} - b\right )}\, dx \]
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\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (3 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (3 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int \frac {b-a\,x^4}{{\left (a\,x^4+b\right )}^{1/4}\,\left (b-3\,a\,x^4\right )} \,d x \]
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