Optimal. Leaf size=303 \[ -\frac {x}{8 a^8 b^8 \sqrt {a^4 x^4-b^4}}-\frac {\tan ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}+a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{32 a^9 b^9}+\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}-a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{32 a^9 b^9}-\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2^{3/4} b}+\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}}{x \sqrt {a^4 x^4-b^4}}\right )}{8\ 2^{3/4} a^9 b^9}-\frac {\tan ^{-1}\left (\frac {2^{3/4} a b x \sqrt {a^4 x^4-b^4}}{-a^4 x^4+\sqrt {2} a^2 b^2 x^2+b^4}\right )}{8\ 2^{3/4} a^9 b^9} \]
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Rubi [C] time = 1.25, antiderivative size = 522, normalized size of antiderivative = 1.72, number of steps used = 40, number of rules used = 19, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.528, Rules used = {6725, 1152, 414, 21, 423, 427, 426, 424, 253, 224, 221, 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 )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\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^9 b^7 \sqrt {a^4 x^4-b^4}}-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {a^4 x^4-b^4}}-\frac {x \left (a^2 x^2+b^2\right )}{16 a^8 b^{10} \sqrt {a^4 x^4-b^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\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 )}{8 a^9 b^7 \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 6725
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx &=\int \left (\frac {1}{8 a^8 b^6 \left (-b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{8 a^8 b^6 \left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{4 a^8 b^4 \sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}+\frac {1}{2 a^8 \sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx}{2 a^8}+\frac {\int \frac {1}{\left (-b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^6}-\frac {\int \frac {1}{\left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^6}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{4 a^8 b^4}\\ &=-\frac {\int \frac {1}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}-\frac {\int \frac {1}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4-a^8 x^4\right )} \, dx}{4 \sqrt {-a^8} b^4}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4+a^8 x^4\right )} \, dx}{4 \sqrt {-a^8} b^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} \left (b^2+a^2 x^2\right )^{3/2}} \, dx}{8 a^8 b^6 \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}{\left (-b^2+a^2 x^2\right )^{3/2} \sqrt {b^2+a^2 x^2}} \, dx}{8 a^8 b^6 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-2 \frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{16 a^8 b^8}-\frac {\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}{16 a^8 b^8}-\frac {\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}{16 a^8 b^8}+\frac {\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\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}{16 a^{10} b^{10} \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}{16 a^{10} b^{10} \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\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 )}{16 a^8 b^8}-\frac {\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 )}{16 a^8 b^8}+\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}{16 a^8 b^{10} \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}{16 a^8 b^{10} \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}{16 a^8 b^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}{8 a^8 b^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}{8 a^8 b^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 {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^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 {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \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 )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \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}{16 a^8 b^{10} \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}{8 a^8 b^8 \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}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \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 )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\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}{16 a^8 b^{10} \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}{16 a^8 b^{10} \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 )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \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 )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\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 )}{16 a^9 b^9 \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}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \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}{16 a^8 b^{10} \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}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \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 )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\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^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\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 )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}\\ \end {align*}
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Mathematica [C] time = 1.40, size = 379, normalized size = 1.25 \begin {gather*} \frac {x \left (-\sqrt {-\frac {a^2}{b^2}}\right )+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 )}{8 a^8 b^8 \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 = 2.98, size = 612, normalized size = 2.02 \begin {gather*} \frac {\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 ]}{16 a^8 b^8}-\frac {x}{8 a^8 b^8 \sqrt {a^4 x^4-b^4}}-\frac {\left (\frac {1}{16}-\frac {i}{16}\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^9 b^9}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {-a^4 x^4-(1-i) a^3 b x^3+\left (-a^2 x^2-(1-i) a b x-i b^2\right ) \sqrt {a^4 x^4-b^4}-(1+i) a b^3 x+b^4}{-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^9 b^9} \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 {x^{8}}{{\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 = 0.06, size = 1130, normalized size = 3.73
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 {x^{8}}{{\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 {x^8}{\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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