Optimal. Leaf size=303 \[ -\frac {x}{8 a^8 b^8 \sqrt {a^4 x^4-b^4}}-\frac {\tan ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}+a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{32 a^9 b^9}+\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}-a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{32 a^9 b^9}-\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2^{3/4} b}+\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}}{x \sqrt {a^4 x^4-b^4}}\right )}{8\ 2^{3/4} a^9 b^9}-\frac {\tan ^{-1}\left (\frac {2^{3/4} a b x \sqrt {a^4 x^4-b^4}}{-a^4 x^4+\sqrt {2} a^2 b^2 x^2+b^4}\right )}{8\ 2^{3/4} a^9 b^9} \]
________________________________________________________________________________________
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.
[In]
[Out]
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*}
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [C] time = 2.78, size = 612, normalized size = 2.02 \begin {gather*} -\frac {x}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a^9 b^9}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-a^4 x^4+\left (-i b^2-(1-i) a b x-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 )}{a^9 b^9}+\frac {\text {RootSum}\left [16 a^8 b^8+32 i a^6 b^6 \text {$\#$1}^2+8 a^4 b^4 \text {$\#$1}^4-8 i a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )-4 i a^4 b^4 \log (x) \text {$\#$1}^2+4 i a^4 b^4 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a^2 b^2 \log (x) \text {$\#$1}^4+2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-i \log (x) \text {$\#$1}^6+i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{8 a^6 b^6 \text {$\#$1}-4 i a^4 b^4 \text {$\#$1}^3-6 a^2 b^2 \text {$\#$1}^5-i \text {$\#$1}^7}\&\right ]}{16 a^8 b^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.26, size = 568, normalized size = 1.87
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{64 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{32 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{32 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, x}{8 a^{8} b^{8} \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )}{16 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )}{8 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{8 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\) | \(568\) |
default | \(-\frac {\frac {a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}\right )}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{16 b^{7} a^{8}}+\frac {-\frac {a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}\right )}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{16 b^{7} a^{8}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8} a^{8}+b^{8}\right )}{\sum }\frac {-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6} a^{4}+b^{4} x^{2}\right ) a^{4}}{b^{4} \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4}-b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4}-b^{4}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{7} a^{8} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}}{b^{6}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{7}}}{32 a^{16}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}+b^{4}\right )}{\sum }\frac {-\frac {\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) a^{4}}{\sqrt {-2 b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {-b^{4}}}+\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{4} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}}{b^{2}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{64 a^{12} b^{4}}-\frac {-\frac {\left (a^{4} x^{2}-a^{2} b^{2}\right ) x}{2 b^{4} a^{2} \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) \left (a^{4} x^{2}-a^{2} b^{2}\right )}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{8 a^{8} b^{6}}\) | \(1130\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________