Optimal. Leaf size=229 \[ -\frac {\text {ArcTan}\left (\frac {\frac {b^4}{\sqrt {2} \sqrt [4]{2 a^4 b^4-c}}+\frac {\sqrt [4]{2 a^4 b^4-c} x^2}{\sqrt {2}}-\frac {a^4 x^4}{\sqrt {2} \sqrt [4]{2 a^4 b^4-c}}}{x \sqrt {-b^4+a^4 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2 a^4 b^4-c} x \sqrt {-b^4+a^4 x^4}}{-b^4+\sqrt {2 a^4 b^4-c} x^2+a^4 x^4}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.96, antiderivative size = 508, normalized size of antiderivative = 2.22, number of steps
used = 18, number of rules used = 7, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {6860, 415,
230, 227, 418, 1233, 1232} \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 .\text {ArcSin}\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 .\text {ArcSin}\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 .\text {ArcSin}\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 .\text {ArcSin}\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 .\text {ArcSin}\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 .\text {ArcSin}\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 227
Rule 230
Rule 415
Rule 418
Rule 1232
Rule 1233
Rule 6860
Rubi steps
\begin {align*} \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 [A]
time = 0.72, size = 171, normalized size = 0.75 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{2 a^4 b^4-c} x \sqrt {-2 b^4+2 a^4 x^4}}{b^4+\sqrt {2 a^4 b^4-c} x^2-a^4 x^4}\right )-\tanh ^{-1}\left (\frac {-b^4+\sqrt {2 a^4 b^4-c} x^2+a^4 x^4}{\sqrt [4]{2 a^4 b^4-c} x \sqrt {-2 b^4+2 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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Maple [A]
time = 0.11, size = 281, normalized size = 1.23
method | result | size |
default | \(\frac {\left (\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2 a^{4} b^{4}-c}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2 a^{4} b^{4}-c}}{2}}\right )}{4 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} x}+1\right )}{2 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} x}-1\right )}{2 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(281\) |
elliptic | \(\frac {\left (\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2 a^{4} b^{4}-c}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2 a^{4} b^{4}-c}}{2}}\right )}{4 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} x}+1\right )}{2 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} x}-1\right )}{2 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(281\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 746 vs.
\(2 (197) = 394\).
time = 11.94, 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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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 {\left (a^4\,x^4+b^4\right )\,\sqrt {a^4\,x^4-b^4}}{a^8\,x^8+b^8-c\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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