Integrand size = 36, antiderivative size = 319 \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a^5 b^5}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \text {arctanh}\left (\frac {\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3-2 \sqrt {2}} a}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3-2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-b^4+a^4 x^4}}{\sqrt {3-2 \sqrt {2}} a b}}{x}\right )}{a^5 b^5}+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \text {arctanh}\left (\frac {\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3+2 \sqrt {2}} a}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3+2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-b^4+a^4 x^4}}{\sqrt {3+2 \sqrt {2}} a b}}{x}\right )}{a^5 b^5} \]
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Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.38, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1493, 482, 21, 414} \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {x}{4 a^4 b^4 \sqrt {a^4 x^4-b^4}}+\frac {\arctan \left (\frac {a x \left (b^2-a^2 x^2\right )}{b \sqrt {a^4 x^4-b^4}}\right )}{8 a^5 b^5}+\frac {\text {arctanh}\left (\frac {a x \left (a^2 x^2+b^2\right )}{b \sqrt {a^4 x^4-b^4}}\right )}{8 a^5 b^5} \]
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Rule 21
Rule 414
Rule 482
Rule 1493
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4}{\left (-b^4+a^4 x^4\right )^{3/2} \left (b^4+a^4 x^4\right )} \, dx \\ & = -\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}+\frac {\int \frac {b^4-a^4 x^4}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{4 a^4 b^4} \\ & = -\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}-\frac {\int \frac {\sqrt {-b^4+a^4 x^4}}{b^4+a^4 x^4} \, dx}{4 a^4 b^4} \\ & = -\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}+\frac {\arctan \left (\frac {a x \left (b^2-a^2 x^2\right )}{b \sqrt {-b^4+a^4 x^4}}\right )}{8 a^5 b^5}+\frac {\text {arctanh}\left (\frac {a x \left (b^2+a^2 x^2\right )}{b \sqrt {-b^4+a^4 x^4}}\right )}{8 a^5 b^5} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\frac {-\frac {4 a b x}{\sqrt {-b^4+a^4 x^4}}+(2-2 i) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )+(1+i) \arctan \left (\frac {i b^4+(1-i) a b^3 x-(1+i) a^3 b x^3-i a^4 x^4+\left (b^2-(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left (b^2+(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right )}{16 a^5 b^5} \]
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Time = 5.00 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.75
method | result | size |
elliptic | \(\frac {\left (-\frac {\sqrt {2}\, x}{4 a^{4} b^{4} \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )\right )}{32 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(238\) |
default | \(\frac {i \left (\left (-8 \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\, x +\left (\ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )+2 \ln \left (2\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right )\right ) \sqrt {-i a^{2} b^{2}}+2 \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \left (\ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+\ln \left (2\right )\right ) \sqrt {2}\, \left (a x +b \right ) \sqrt {i a^{2} b^{2}}\right )}{16 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, a^{2} \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (a x -b \right ) \left (a x +i b \right ) \left (a x +b \right ) b^{2} \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right )}\) | \(508\) |
pseudoelliptic | \(\frac {i \left (\left (-8 \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\, x +\left (\ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )+2 \ln \left (2\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right )\right ) \sqrt {-i a^{2} b^{2}}+2 \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \left (\ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+\ln \left (2\right )\right ) \sqrt {2}\, \left (a x +b \right ) \sqrt {i a^{2} b^{2}}\right )}{16 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, a^{2} \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (a x -b \right ) \left (a x +i b \right ) \left (a x +b \right ) b^{2} \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right )}\) | \(508\) |
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Time = 0.71 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.52 \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {4 \, \sqrt {a^{4} x^{4} - b^{4}} a b x + 2 \, {\left (a^{4} x^{4} - b^{4}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) - {\left (a^{4} x^{4} - b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} - b^{4} + 2 \, \sqrt {a^{4} x^{4} - b^{4}} a b x}{a^{4} x^{4} + b^{4}}\right )}{16 \, {\left (a^{9} b^{5} x^{4} - a^{5} b^{9}\right )}} \]
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\[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int \frac {x^{4}}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )}\, dx \]
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\[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
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\[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\int \frac {x^4}{\sqrt {a^4\,x^4-b^4}\,\left (b^8-a^8\,x^8\right )} \,d x \]
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