Integrand size = 23, antiderivative size = 87 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]
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Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {385, 218, 212, 209} \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]
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Rule 209
Rule 212
Rule 218
Rule 385
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{b-2 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b} \\ & = \frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]
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Time = 1.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right )\right )}{8 a^{\frac {1}{4}} b}\) | \(87\) |
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Result contains complex when optimal does not.
Time = 106.34 (sec) , antiderivative size = 525, normalized size of antiderivative = 6.03 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) - \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) + \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-3 i \, a b x^{4} + i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) - \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {-4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 i \, a b x^{4} - i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) \]
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\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} - b} \left (a x^{4} + b\right )}\, dx \]
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\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int \frac {1}{\left (a\,x^4+b\right )\,{\left (a\,x^4-b\right )}^{1/4}} \,d x \]
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