Integrand size = 42, antiderivative size = 81 \[ \int \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.16 (sec) , antiderivative size = 1639, normalized size of antiderivative = 20.23, number of steps used = 20, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6857, 415, 226, 418, 1231, 1721} \[ \int \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx=-\frac {\left (\sqrt {-a^8}-a^4\right )^{3/2} \arctan \left (\frac {\sqrt [8]{-a^8} \sqrt {\sqrt {-a^8}-a^4} b x}{a^2 \sqrt {b^4+a^4 x^4}}\right ) a^6}{8 \left (-a^8\right )^{13/8} b}-\frac {\left (\sqrt {-a^8}-a^4\right )^{3/2} \arctan \left (\frac {\sqrt {\sqrt {-a^8}-a^4} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{8 \left (-\sqrt {-a^8}\right )^{3/4} b a^4}-\frac {\left (a^4+\sqrt {-a^8}\right )^{3/2} \arctan \left (\frac {\sqrt {a^4+\sqrt {-a^8}} b x}{\sqrt [8]{-a^8} \sqrt {b^4+a^4 x^4}}\right )}{8 \left (-a^8\right )^{3/8} b a^4}+\frac {\left (a^4-\sqrt {-a^8}\right ) \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 b \sqrt {b^4+a^4 x^4} a^5}+\frac {\left (a^4+\sqrt {-a^8}\right ) \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 b \sqrt {b^4+a^4 x^4} a^5}-\frac {\left (a^2-\sqrt [4]{-a^8}\right )^3 \left (a^2+\sqrt [4]{-a^8}\right ) \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 {EllipticPi}\left (\frac {\left (a^2+\sqrt [4]{-a^8}\right )^2}{4 a^2 \sqrt [4]{-a^8}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 \sqrt {-a^8} b \sqrt {b^4+a^4 x^4} a^5}-\frac {\left (a^2-\sqrt [4]{-a^8}\right ) \left (a^4+\sqrt {-a^8}\right ) \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 )}{8 b \sqrt {b^4+a^4 x^4} a^7}-\frac {\sqrt {-a^8} \left (a^2+\sqrt [4]{-a^8}\right ) \left (a^4-\sqrt {-a^8}\right ) \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 )}{8 b \sqrt {b^4+a^4 x^4} a^{11}}+\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2-\sqrt {-\sqrt {-a^8}}\right ) \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 )}{8 b \sqrt {b^4+a^4 x^4} a^{11}}+\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2+\sqrt {-\sqrt {-a^8}}\right ) \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 )}{8 b \sqrt {b^4+a^4 x^4} a^{11}}+\frac {\left (-\sqrt {-a^8}\right )^{5/4} \left (a^4-\sqrt {-a^8}\right )^{3/2} \arctan \left (\frac {\sqrt {a^4-\sqrt {-a^8}} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{8 b a^{12}}+\frac {\sqrt {-a^8} \left (a^2+\sqrt [4]{-a^8}\right )^2 \left (a^4-\sqrt {-a^8}\right ) \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 {EllipticPi}\left (\frac {a^6 \left (a^2-\sqrt [4]{-a^8}\right )^2}{4 \left (-a^8\right )^{5/4}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 b \sqrt {b^4+a^4 x^4} a^{13}}-\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2+\sqrt {-\sqrt {-a^8}}\right )^2 \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 {EllipticPi}\left (-\frac {\left (a^2-\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 b \sqrt {b^4+a^4 x^4} a^{13}}-\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2-\sqrt {-\sqrt {-a^8}}\right )^2 \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 {EllipticPi}\left (\frac {\left (a^2+\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 b \sqrt {b^4+a^4 x^4} a^{13}} \]
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Rule 226
Rule 415
Rule 418
Rule 1231
Rule 1721
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {-a^8} \left (a^4 b^4-\sqrt {-a^8} b^4\right ) \sqrt {b^4+a^4 x^4}}{2 a^8 b^4 \left (b^4-\sqrt {-a^8} x^4\right )}+\frac {\sqrt {-a^8} \left (a^4 b^4+\sqrt {-a^8} b^4\right ) \sqrt {b^4+a^4 x^4}}{2 a^8 b^4 \left (b^4+\sqrt {-a^8} x^4\right )}\right ) \, dx \\ & = -\frac {\left (a^4+\sqrt {-a^8}\right ) \int \frac {\sqrt {b^4+a^4 x^4}}{b^4-\sqrt {-a^8} x^4} \, dx}{2 a^4}+\frac {\left (\sqrt {-a^8} \left (a^4 b^4+\sqrt {-a^8} b^4\right )\right ) \int \frac {\sqrt {b^4+a^4 x^4}}{b^4+\sqrt {-a^8} x^4} \, dx}{2 a^8 b^4} \\ & = \frac {1}{2} \left (1+\frac {a^4}{\sqrt {-a^8}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx+\frac {\left (a^4+\sqrt {-a^8}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 a^4}-b^4 \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^4-\sqrt {-a^8} x^4\right )} \, dx-b^4 \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^4+\sqrt {-a^8} x^4\right )} \, dx \\ & = \frac {\left (1+\frac {a^4}{\sqrt {-a^8}}\right ) \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 {\left (a^4+\sqrt {-a^8}\right ) \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^5 b \sqrt {b^4+a^4 x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx \\ & = \frac {\left (1+\frac {a^4}{\sqrt {-a^8}}\right ) \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 {\left (a^4+\sqrt {-a^8}\right ) \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^5 b \sqrt {b^4+a^4 x^4}}-\frac {a^2 \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^2-\sqrt [4]{-a^8}\right )}+\frac {\sqrt [4]{-a^8} \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^2-\sqrt [4]{-a^8}\right )}-\frac {a^2 \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^2+\sqrt [4]{-a^8}\right )}-\frac {\sqrt [4]{-a^8} \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^2+\sqrt [4]{-a^8}\right )}-\frac {\left (a^2 \left (a^2-\sqrt {-\sqrt {-a^8}}\right )\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^4+\sqrt {-a^8}\right )}-\frac {\left (a^2 \left (a^2+\sqrt {-\sqrt {-a^8}}\right )\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^4+\sqrt {-a^8}\right )}+\frac {\left (\sqrt {-\sqrt {-a^8}} \left (a^2+\sqrt {-\sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^4+\sqrt {-a^8}\right )}-\frac {\left (\sqrt {-a^8}+a^2 \sqrt {-\sqrt {-a^8}}\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^4+\sqrt {-a^8}\right )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \]
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Time = 6.54 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.41
method | result | size |
pseudoelliptic | \(-\frac {\left (-2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 2^{\frac {3}{4}}}{2 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}{2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}\right )\right ) 2^{\frac {3}{4}}}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\) | \(114\) |
default | \(\frac {2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right )-\ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\) | \(144\) |
elliptic | \(\frac {2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right )-\ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\) | \(144\) |
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Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 623, normalized size of antiderivative = 7.69 \[ \int \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx=-\frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} + a^{4} x^{5} + b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 4 \, a^{4} b^{4} x^{4} + b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) + \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} + a^{4} x^{5} + b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 4 \, a^{4} b^{4} x^{4} + b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) + \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (i \, a^{8} b^{4} x^{6} + i \, a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} - a^{4} x^{5} - b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-i \, a^{8} x^{8} - 4 i \, a^{4} b^{4} x^{4} - i \, b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) - \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (-i \, a^{8} b^{4} x^{6} - i \, a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} - a^{4} x^{5} - b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (i \, a^{8} x^{8} + 4 i \, a^{4} b^{4} x^{4} + i \, b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) \]
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\[ \int \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \sqrt {a^{4} x^{4} + b^{4}}}{a^{8} x^{8} + b^{8}}\, dx \]
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\[ \int \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx=\int { \frac {\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{4} x^{4} - b^{4}\right )}}{a^{8} x^{8} + b^{8}} \,d x } \]
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\[ \int \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx=\int { \frac {\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{4} x^{4} - b^{4}\right )}}{a^{8} x^{8} + b^{8}} \,d x } \]
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Timed out. \[ \int \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx=\int -\frac {\sqrt {a^4\,x^4+b^4}\,\left (b^4-a^4\,x^4\right )}{a^8\,x^8+b^8} \,d x \]
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