Integrand size = 42, antiderivative size = 196 \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {x}{2 \sqrt {b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt {2} a b}+\frac {\text {arctanh}\left (\frac {\sqrt {6-4 \sqrt {2}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt {6+4 \sqrt {2}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b} \]
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Time = 0.41 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.52, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6857, 226, 1418, 425, 537, 418, 1225, 1713, 212, 209} \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{4 \sqrt {2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{4 \sqrt {2} a b}-\frac {x}{2 \sqrt {a^4 x^4+b^4}} \]
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Rule 209
Rule 212
Rule 226
Rule 418
Rule 425
Rule 537
Rule 1225
Rule 1418
Rule 1713
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {b^4+a^4 x^4}}+\frac {2 b^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )}\right ) \, dx \\ & = \left (2 b^8\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx+\int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx \\ & = \frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}+\left (2 b^8\right ) \int \frac {1}{\left (-b^4+a^4 x^4\right ) \left (b^4+a^4 x^4\right )^{3/2}} \, dx \\ & = -\frac {x}{2 \sqrt {b^4+a^4 x^4}}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {\int \frac {3 a^4 b^4-a^8 x^4}{\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 a^4} \\ & = -\frac {x}{2 \sqrt {b^4+a^4 x^4}}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{2} \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx+b^4 \int \frac {1}{\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}} \, dx \\ & = -\frac {x}{2 \sqrt {b^4+a^4 x^4}}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {a^2 x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {a^2 x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx \\ & = -\frac {x}{2 \sqrt {b^4+a^4 x^4}}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 a b \sqrt {b^4+a^4 x^4}}-2 \left (\frac {1}{4} \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx\right )-\frac {1}{4} \int \frac {1-\frac {a^2 x^2}{b^2}}{\left (1+\frac {a^2 x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{4} \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {a^2 x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx \\ & = -\frac {x}{2 \sqrt {b^4+a^4 x^4}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-2 a^2 b^2 x^2} \, dx,x,\frac {x}{\sqrt {b^4+a^4 x^4}}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+2 a^2 b^2 x^2} \, dx,x,\frac {x}{\sqrt {b^4+a^4 x^4}}\right ) \\ & = -\frac {x}{2 \sqrt {b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.55 \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {\frac {4 a b x}{\sqrt {b^4+a^4 x^4}}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{8 a b} \]
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Time = 3.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.67
method | result | size |
elliptic | \(\frac {\left (\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{4 a b}-\frac {\ln \left (a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{8 a b}-\frac {\sqrt {2}\, x}{2 \sqrt {a^{4} x^{4}+b^{4}}}+\frac {\ln \left (-a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{8 a b}\right ) \sqrt {2}}{2}\) | \(131\) |
default | \(-\frac {\sqrt {2}\, \left (\left (\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +\frac {\left (a^{4} x^{4}+b^{4}\right ) \left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}+2 a^{3} b \left (a^{2} x^{2}-a b x +b^{2}\right )}{\left (a x -b \right )^{2}}\right )+\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 a^{3} b \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+2 \ln \left (2\right )\right )}{4}\right ) \sqrt {-a^{2} b^{2}}+\frac {\sqrt {a^{2} b^{2}}\, \left (a^{4} x^{4}+b^{4}\right ) \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 x \,a^{2} b^{2}+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )\right )}{2}\right )}{4 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}\, \left (a^{2} x^{2}-\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right ) \left (a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right )}\) | \(341\) |
pseudoelliptic | \(-\frac {\sqrt {2}\, \left (\left (\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +\frac {\left (a^{4} x^{4}+b^{4}\right ) \left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}+2 a^{3} b \left (a^{2} x^{2}-a b x +b^{2}\right )}{\left (a x -b \right )^{2}}\right )+\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 a^{3} b \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+2 \ln \left (2\right )\right )}{4}\right ) \sqrt {-a^{2} b^{2}}+\frac {\sqrt {a^{2} b^{2}}\, \left (a^{4} x^{4}+b^{4}\right ) \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 x \,a^{2} b^{2}+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )\right )}{2}\right )}{4 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}\, \left (a^{2} x^{2}-\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right ) \left (a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right )}\) | \(341\) |
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Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.81 \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {8 \, \sqrt {a^{4} x^{4} + b^{4}} a b x + 2 \, \sqrt {2} {\left (a^{4} x^{4} + b^{4}\right )} \arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^{4} x^{4} + b^{4}}}\right ) - \sqrt {2} {\left (a^{4} x^{4} + b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4} - 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + b^{4}}\right )}{16 \, {\left (a^{5} b x^{4} + a b^{5}\right )}} \]
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\[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int \frac {a^{8} x^{8} + b^{8}}{\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} + b^{8}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
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\[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} + b^{8}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
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Timed out. \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int -\frac {a^8\,x^8+b^8}{\sqrt {a^4\,x^4+b^4}\,\left (b^8-a^8\,x^8\right )} \,d x \]
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