Optimal. Leaf size=297 \[ -\frac {x}{4 \sqrt {a^4 x^4-b^4}}-\frac {\tan ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}+a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{16 a b}+\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}-a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{16 a b}+\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2^{3/4} b}+\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}}{x \sqrt {a^4 x^4-b^4}}\right )}{4\ 2^{3/4} a b}+\frac {\tan ^{-1}\left (\frac {2^{3/4} a b x \sqrt {a^4 x^4-b^4}}{-a^4 x^4+\sqrt {2} a^2 b^2 x^2+b^4}\right )}{4\ 2^{3/4} a b} \]
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Rubi [C] time = 1.71, antiderivative size = 506, normalized size of antiderivative = 1.70, number of steps used = 44, number of rules used = 20, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6725, 224, 221, 2073, 1152, 414, 21, 423, 427, 426, 424, 253, 409, 1211, 1699, 203, 206, 1429, 1219, 1218} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {a^4 x^4-b^4}}-\frac {x \left (a^2 x^2+b^2\right )}{8 b^2 \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {a^4 x^4-b^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 203
Rule 206
Rule 221
Rule 224
Rule 253
Rule 409
Rule 414
Rule 423
Rule 424
Rule 426
Rule 427
Rule 1152
Rule 1211
Rule 1218
Rule 1219
Rule 1429
Rule 1699
Rule 2073
Rule 6725
Rubi steps
\begin {align*} \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx &=\int \left (\frac {1}{\sqrt {-b^4+a^4 x^4}}+\frac {2 b^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )}\right ) \, dx\\ &=\left (2 b^{16}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx+\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx\\ &=\left (2 b^{16}\right ) \int \left (-\frac {1}{8 b^{14} \left (b^2-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{8 b^{14} \left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{4 b^{12} \sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}-\frac {1}{2 b^8 \sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )}\right ) \, dx+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{\sqrt {-b^4+a^4 x^4}}\\ &=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} b^2 \int \frac {1}{\left (b^2-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} b^2 \int \frac {1}{\left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} b^4 \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx-b^8 \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx\\ &=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} \int \frac {1}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} \left (\sqrt {-a^8} b^4\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4-a^8 x^4\right )} \, dx-\frac {1}{2} \left (\sqrt {-a^8} b^4\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4+a^8 x^4\right )} \, dx-\frac {\left (b^2 \sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2-a^2 x^2} \left (b^2-a^2 x^2\right )^{3/2}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\left (b^2 \sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2+a^2 x^2} \left (b^2+a^2 x^2\right )^{3/2}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-2 \left (\frac {1}{8} \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx\right )-\frac {1}{8} \int \frac {1-\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{8} \int \frac {1+\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} \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}{4} \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}{4} \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}{4} \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 (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {-a^2 b^2+a^4 x^2}{\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}} \, dx}{8 a^2 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {a^2 b^2+a^4 x^2}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{8 a^2 b^2 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {\sqrt {b^2-a^2 x^2}}{\sqrt {-b^2-a^2 x^2}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {-b^2+a^2 x^2}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}-2 \frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}+\frac {3 b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {\sqrt {-b^2-a^2 x^2}}{\sqrt {b^2-a^2 x^2}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {b^2+a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}+\frac {3 b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {-b^2-a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{8 b^2 \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a b \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}+\frac {3 b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (-b^2-a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{8 b^2 \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}\\ \end {align*}
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Mathematica [C] time = 1.32, size = 373, normalized size = 1.26 \begin {gather*} \frac {x \left (-\sqrt {-\frac {a^2}{b^2}}\right )-3 i \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (i;\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-(-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left ((-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )}{4 \sqrt {-\frac {a^2}{b^2}} \sqrt {a^4 x^4-b^4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 4.62, size = 624, normalized size = 2.10 \begin {gather*} -\frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-8 i \text {$\#$1}^6 a^2 b^2+8 \text {$\#$1}^4 a^4 b^4+32 i \text {$\#$1}^2 a^6 b^6+16 a^8 b^8\&,\frac {i \text {$\#$1}^6 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2\right )-i \text {$\#$1}^6 \log (x)-2 \text {$\#$1}^4 a^2 b^2 \log (x)+2 \text {$\#$1}^4 a^2 b^2 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2\right )-4 i \text {$\#$1}^2 a^4 b^4 \log (x)+4 i \text {$\#$1}^2 a^4 b^4 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2\right )+8 a^6 b^6 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2\right )-8 a^6 b^6 \log (x)}{-i \text {$\#$1}^7-6 \text {$\#$1}^5 a^2 b^2-4 i \text {$\#$1}^3 a^4 b^4+8 \text {$\#$1} a^6 b^6}\&\right ]-\frac {x}{4 \sqrt {a^4 x^4-b^4}}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2}\right )}{a b}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) \log \left (-i a^4 x^4-(1-i) a^3 b x^3+\left (-i a^2 x^2-(1-i) a b x+b^2\right ) \sqrt {a^4 x^4-b^4}-(1+i) a b^3 x+i b^4\right )}{a b}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) \log \left (a^5 b x^4+(1+i) a^4 b^2 x^3-(1-i) a^2 b^4 x+\left (a^3 b x^2+(1+i) a^2 b^2 x+i a b^3\right ) \sqrt {a^4 x^4-b^4}-a b^5\right )}{a b} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{16} x^{16} + b^{16}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 1190, normalized size = 4.01
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{16} x^{16} + b^{16}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {a^{16}\,x^{16}+b^{16}}{\sqrt {a^4\,x^4-b^4}\,\left (b^{16}-a^{16}\,x^{16}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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