Optimal. Leaf size=229 \[ -\frac {\tan ^{-1}\left (\frac {-\frac {a^4 x^4}{\sqrt {2} \sqrt [4]{2 a^4 b^4-c}}+\frac {x^2 \sqrt [4]{2 a^4 b^4-c}}{\sqrt {2}}+\frac {b^4}{\sqrt {2} \sqrt [4]{2 a^4 b^4-c}}}{x \sqrt {a^4 x^4-b^4}}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{2 a^4 b^4-c} \sqrt {a^4 x^4-b^4}}{x^2 \sqrt {2 a^4 b^4-c}+a^4 x^4-b^4}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}} \]
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Rubi [C] time = 1.22, antiderivative size = 508, normalized size of antiderivative = 2.22, number of steps used = 19, number of rules used = 8, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1586, 6728, 406, 224, 221, 409, 1219, 1218} \begin {gather*} \frac {b \left (1-\frac {2 a^4 b^4+c}{\sqrt {c^2-4 a^8 b^8}}\right ) \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 {b \left (\frac {2 a^4 b^4+c}{\sqrt {c^2-4 a^8 b^8}}+1\right ) \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 {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {2} a^2 b^2}{\sqrt {c-\sqrt {c^2-4 a^8 b^8}}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {2} a^2 b^2}{\sqrt {c-\sqrt {c^2-4 a^8 b^8}}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {2} a^2 b^2}{\sqrt {c+\sqrt {c^2-4 a^8 b^8}}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {2} a^2 b^2}{\sqrt {c+\sqrt {c^2-4 a^8 b^8}}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 406
Rule 409
Rule 1218
Rule 1219
Rule 1586
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8-c x^4+a^8 x^8\right )} \, dx &=\int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8-c x^4+a^8 x^8} \, dx\\ &=\int \left (\frac {\left (a^4+\frac {a^4 \left (2 a^4 b^4+c\right )}{\sqrt {-4 a^8 b^8+c^2}}\right ) \sqrt {-b^4+a^4 x^4}}{-c-\sqrt {-4 a^8 b^8+c^2}+2 a^8 x^4}+\frac {\left (a^4-\frac {a^4 \left (2 a^4 b^4+c\right )}{\sqrt {-4 a^8 b^8+c^2}}\right ) \sqrt {-b^4+a^4 x^4}}{-c+\sqrt {-4 a^8 b^8+c^2}+2 a^8 x^4}\right ) \, dx\\ &=\left (a^4 \left (1-\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right )\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{-c+\sqrt {-4 a^8 b^8+c^2}+2 a^8 x^4} \, dx+\left (a^4 \left (1+\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right )\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{-c-\sqrt {-4 a^8 b^8+c^2}+2 a^8 x^4} \, dx\\ &=\frac {1}{2} \left (1-\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{2} \left (1+\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx+\frac {\left (4 a^8 b^8-c \left (c-\sqrt {-4 a^8 b^8+c^2}\right )\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (-c+\sqrt {-4 a^8 b^8+c^2}+2 a^8 x^4\right )} \, dx}{\sqrt {-4 a^8 b^8+c^2}}-\frac {\left (4 a^8 b^8-c \left (c+\sqrt {-4 a^8 b^8+c^2}\right )\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (-c-\sqrt {-4 a^8 b^8+c^2}+2 a^8 x^4\right )} \, dx}{\sqrt {-4 a^8 b^8+c^2}}\\ &=-\left (\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {c-\sqrt {-4 a^8 b^8+c^2}}}\right ) \sqrt {-b^4+a^4 x^4}} \, dx\right )-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {c-\sqrt {-4 a^8 b^8+c^2}}}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {c+\sqrt {-4 a^8 b^8+c^2}}}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {c+\sqrt {-4 a^8 b^8+c^2}}}\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {\left (\left (1-\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (1+\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}\\ &=\frac {b \left (1-\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right ) \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 \left (1+\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right ) \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 {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {c-\sqrt {-4 a^8 b^8+c^2}}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {c-\sqrt {-4 a^8 b^8+c^2}}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {c+\sqrt {-4 a^8 b^8+c^2}}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {c+\sqrt {-4 a^8 b^8+c^2}}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}\\ &=\frac {b \left (1-\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right ) \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 \left (1+\frac {2 a^4 b^4+c}{\sqrt {-4 a^8 b^8+c^2}}\right ) \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 {\sqrt {2} a^2 b^2}{\sqrt {c-\sqrt {-4 a^8 b^8+c^2}}};\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 {\sqrt {2} a^2 b^2}{\sqrt {c-\sqrt {-4 a^8 b^8+c^2}}};\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 {\sqrt {2} a^2 b^2}{\sqrt {c+\sqrt {-4 a^8 b^8+c^2}}};\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 {\sqrt {2} a^2 b^2}{\sqrt {c+\sqrt {-4 a^8 b^8+c^2}}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}\\ \end {align*}
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Mathematica [C] time = 1.31, size = 326, normalized size = 1.42 \begin {gather*} -\frac {i \sqrt {1-\frac {a^4 x^4}{b^4}} \left (2 F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-\frac {\sqrt {2} b^2}{a^2 \sqrt {\frac {c-\sqrt {c^2-4 a^8 b^8}}{a^8}}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2} b^2}{a^2 \sqrt {\frac {c-\sqrt {c^2-4 a^8 b^8}}{a^8}}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-\frac {\sqrt {2} b^2}{a^2 \sqrt {\frac {c+\sqrt {c^2-4 a^8 b^8}}{a^8}}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2} b^2}{a^2 \sqrt {\frac {c+\sqrt {c^2-4 a^8 b^8}}{a^8}}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )\right )}{2 \sqrt {-\frac {a^2}{b^2}} \sqrt {a^4 x^4-b^4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.17, size = 228, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2 a^4 b^4-c} x \sqrt {-b^4+a^4 x^4}}{b^4+\sqrt {2 a^4 b^4-c} x^2-a^4 x^4}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b^4}{\sqrt {2} \sqrt [4]{2 a^4 b^4-c}}+\frac {\sqrt [4]{2 a^4 b^4-c} x^2}{\sqrt {2}}+\frac {a^4 x^4}{\sqrt {2} \sqrt [4]{2 a^4 b^4-c}}}{x \sqrt {-b^4+a^4 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 23.91, size = 746, normalized size = 3.26 \begin {gather*} \frac {1}{2} \, \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, \sqrt {a^{4} x^{4} - b^{4}} {\left ({\left (2 \, a^{4} b^{4} - c\right )} x^{3} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}} + {\left ({\left (2 \, a^{8} b^{4} - a^{4} c\right )} x^{5} - {\left (2 \, a^{4} b^{8} - b^{4} c\right )} x\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {3}{4}}\right )} - {\left ({\left (2 \, a^{4} b^{12} - b^{8} c + {\left (2 \, a^{12} b^{4} - a^{8} c\right )} x^{8} - {\left (8 \, a^{8} b^{8} - 6 \, a^{4} b^{4} c + c^{2}\right )} x^{4}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {3}{4}} + 2 \, {\left ({\left (2 \, a^{8} b^{4} - a^{4} c\right )} x^{6} - {\left (2 \, a^{4} b^{8} - b^{4} c\right )} x^{2}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}}}{a^{8} x^{8} + b^{8} - c x^{4}}\right ) + \frac {1}{8} \, \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}} \log \left (\frac {2 \, {\left ({\left (2 \, a^{8} b^{4} - a^{4} c\right )} x^{6} - {\left (2 \, a^{4} b^{8} - b^{4} c\right )} x^{2}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {3}{4}} + 2 \, {\left (a^{4} x^{5} - b^{4} x - {\left (2 \, a^{4} b^{4} - c\right )} x^{3} \sqrt {-\frac {1}{2 \, a^{4} b^{4} - c}}\right )} \sqrt {a^{4} x^{4} - b^{4}} - {\left (a^{8} x^{8} + b^{8} - {\left (4 \, a^{4} b^{4} - c\right )} x^{4}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8} - c x^{4}\right )}}\right ) - \frac {1}{8} \, \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, {\left ({\left (2 \, a^{8} b^{4} - a^{4} c\right )} x^{6} - {\left (2 \, a^{4} b^{8} - b^{4} c\right )} x^{2}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {3}{4}} - 2 \, {\left (a^{4} x^{5} - b^{4} x - {\left (2 \, a^{4} b^{4} - c\right )} x^{3} \sqrt {-\frac {1}{2 \, a^{4} b^{4} - c}}\right )} \sqrt {a^{4} x^{4} - b^{4}} - {\left (a^{8} x^{8} + b^{8} - {\left (4 \, a^{4} b^{4} - c\right )} x^{4}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8} - c x^{4}\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} - c x^{4}\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 [A] time = 0.18, size = 281, normalized size = 1.23
method | result | size |
elliptic | \(\frac {\left (\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2 a^{4} b^{4}-c}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2 a^{4} b^{4}-c}}{2}}\right )}{4 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} x}+1\right )}{2 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} x}-1\right )}{2 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(281\) |
default | \(\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{8} \textit {\_Z}^{8}+b^{8}-c \,\textit {\_Z}^{4}\right )}{\sum }\frac {\left (-2 b^{8}+\underline {\hspace {1.25 ex}}\alpha ^{4} c \right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6} a^{8}+a^{4} b^{4} x^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} c \right )}{b^{4} \sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} \left (\underline {\hspace {1.25 ex}}\alpha ^{4} a^{8}-c \right ) \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{4} a^{8}-c \right )}{a^{2} b^{6}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \left (2 \underline {\hspace {1.25 ex}}\alpha ^{4} a^{8}-c \right )}\right )}{8}\) | \(331\) |
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} - c x^{4}\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 {b^8-a^8\,x^8}{\sqrt {a^4\,x^4-b^4}\,\left (a^8\,x^8+b^8-c\,x^4\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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