Optimal. Leaf size=81 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{2 \sqrt [4]{2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{2 \sqrt [4]{2} a b} \]
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Rubi [C] time = 2.74, antiderivative size = 1639, normalized size of antiderivative = 20.23, number of steps used = 21, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1586, 6725, 406, 220, 409, 1217, 1707}
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Warning: Unable to verify antiderivative.
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Rule 220
Rule 406
Rule 409
Rule 1217
Rule 1586
Rule 1707
Rule 6725
Rubi steps
\begin {align*} \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx &=\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 \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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 )}\\ &=-\frac {\sqrt [4]{-\sqrt {-a^8}} \tan ^{-1}\left (\frac {\sqrt {a^4-\sqrt {-a^8}} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {a^4-\sqrt {-a^8}} b}-\frac {a^2 \tan ^{-1}\left (\frac {\sqrt [8]{-a^8} \sqrt {-a^4+\sqrt {-a^8}} b x}{a^2 \sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt [8]{-a^8} \sqrt {-a^4+\sqrt {-a^8}} b}-\frac {\sqrt [4]{-\sqrt {-a^8}} \tan ^{-1}\left (\frac {\sqrt {-a^4+\sqrt {-a^8}} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {-a^4+\sqrt {-a^8}} b}-\frac {\sqrt [8]{-a^8} \tan ^{-1}\left (\frac {\sqrt {a^4+\sqrt {-a^8}} b x}{\sqrt [8]{-a^8} \sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {a^4+\sqrt {-a^8}} b}+\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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 a b \sqrt {b^4+a^4 x^4}}-\frac {a \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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \left (a^2-\sqrt [4]{-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {a \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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \left (a^2+\sqrt [4]{-a^8}\right ) 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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 a^5 b \sqrt {b^4+a^4 x^4}}-\frac {a \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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \left (a^4+\sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {a \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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \left (a^4+\sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt [4]{-a^8} \left (a^4+a^2 \sqrt [4]{-a^8}+\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}} \Pi \left (\frac {a^6 \left (a^2-\sqrt [4]{-a^8}\right )^2}{4 \left (-a^8\right )^{5/4}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{8 a^7 b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt [4]{-a^8} \left (a^4+\sqrt {-a^8}+\frac {\left (-a^8\right )^{5/4}}{a^6}\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}} \Pi \left (\frac {\left (a^2+\sqrt [4]{-a^8}\right )^2}{4 a^2 \sqrt [4]{-a^8}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{8 a^7 b \sqrt {b^4+a^4 x^4}}+\frac {\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}} \Pi \left (-\frac {\left (a^2-\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{8 a \left (a^4+\sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^4-\sqrt {-a^8}-2 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}} \Pi \left (\frac {\left (a^2+\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{8 a \left (a^4+\sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.56, size = 201, normalized size = 2.48 \begin {gather*} -\frac {i \sqrt {\frac {a^4 x^4}{b^4}+1} \left (2 F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-(-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left ((-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )\right )}{2 \sqrt {\frac {i a^2}{b^2}} \sqrt {a^4 x^4+b^4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.53, size = 81, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 4.86, size = 500, normalized size = 6.17 \begin {gather*} -\frac {1}{2} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (2 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{4} b^{4} x^{3} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{5} + a^{4} b^{8} x\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{4} x^{4} + b^{4}} + {\left (\left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{12} b^{4} x^{8} + 4 \, a^{8} b^{8} x^{4} + a^{4} b^{12}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}\right )} \sqrt {\sqrt {\frac {1}{2}} \sqrt {\frac {1}{a^{4} b^{4}}}}\right )}}{a^{8} x^{8} + b^{8}}\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} \, \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 ) \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^{8} x^{8} - b^{8}}{{\left (a^{8} x^{8} + b^{8}\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 [B] time = 0.16, size = 168, normalized size = 2.07
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right )}{2 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, \ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\) | \(168\) |
default | \(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b^{8} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{8} \textit {\_Z}^{8}+b^{8}\right )}{\sum }\frac {-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-a^{4} \underline {\hspace {1.25 ex}}\alpha ^{6}+b^{4} x^{2}\right ) a^{4}}{b^{4} \sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+b^{4}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{7} a^{8} \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}}{b^{6}}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}+b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{7}}\right )}{8 a^{8}}\) | \(287\) |
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^{8} x^{8} - b^{8}}{{\left (a^{8} x^{8} + b^{8}\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 = -0.01 \begin {gather*} \int -\frac {b^8-a^8\,x^8}{\sqrt {a^4\,x^4+b^4}\,\left (a^8\,x^8+b^8\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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