Optimal. Leaf size=166 \[ -\frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {4-2 \sqrt {3}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {4+2 \sqrt {3}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 1.52, antiderivative size = 405, normalized size of antiderivative = 2.44, number of steps
used = 17, number of rules used = 9, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6857, 226,
2098, 1225, 1713, 211, 6860, 1231, 1721} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {2 \text {ArcTan}\left (\frac {\sqrt {-a^2} b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {-a^2} b}+\frac {\left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}-\frac {\left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) b \sqrt {a^4 x^4+b^4}}-\frac {\left (\sqrt {3} \sqrt {-a^4}+a^2\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a \left (\sqrt {3} \sqrt {-a^4}+3 a^2\right ) b \sqrt {a^4 x^4+b^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 226
Rule 1225
Rule 1231
Rule 1713
Rule 1721
Rule 2098
Rule 6857
Rule 6860
Rubi steps
\begin {align*} \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx &=\int \left (\frac {1}{\sqrt {b^4+a^4 x^4}}-\frac {2 b^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )}\right ) \, dx\\ &=-\left (\left (2 b^6\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx\right )+\int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}-\left (2 b^6\right ) \int \left (\frac {1}{3 b^4 \left (b^2+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}}+\frac {2 b^2-a^2 x^2}{3 b^4 \sqrt {b^4+a^4 x^4} \left (b^4-a^2 b^2 x^2+a^4 x^4\right )}\right ) \, dx\\ &=\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} \left (2 b^2\right ) \int \frac {1}{\left (b^2+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (2 b^2\right ) \int \frac {2 b^2-a^2 x^2}{\sqrt {b^4+a^4 x^4} \left (b^4-a^2 b^2 x^2+a^4 x^4\right )} \, dx\\ &=\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {b^2-a^2 x^2}{\left (b^2+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (2 b^2\right ) \int \left (\frac {-a^2-\sqrt {3} \sqrt {-a^4}}{\left (-a^2 b^2-\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}}+\frac {-a^2+\sqrt {3} \sqrt {-a^4}}{\left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx\\ &=\frac {\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 )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} b^2 \text {Subst}\left (\int \frac {1}{b^2+2 a^2 b^4 x^2} \, dx,x,\frac {x}{\sqrt {b^4+a^4 x^4}}\right )+\frac {1}{3} \left (2 \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) b^2\right ) \int \frac {1}{\left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx+\frac {1}{3} \left (2 \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) b^2\right ) \int \frac {1}{\left (-a^2 b^2-\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\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 )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (2 \left (a^2-\sqrt {3} \sqrt {-a^4}\right )\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right )}-\frac {\left (2 \left (a^2+\sqrt {3} \sqrt {-a^4}\right )\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right )}+\frac {\left (4 a^2 \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right )}+\frac {\left (4 a^2 \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (-a^2 b^2-\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right )}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {-a^2} b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {-a^2} b}+\frac {\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 )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (a^2-\sqrt {3} \sqrt {-a^4}\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 )}{3 a \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (a^2+\sqrt {3} \sqrt {-a^4}\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 )}{3 a \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (a^2+\sqrt {3} \sqrt {-a^4}\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 {3}{4};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (a^2-\sqrt {3} \sqrt {-a^4}\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 {3}{4};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) b \sqrt {b^4+a^4 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 77, normalized size = 0.46 \begin {gather*} -\frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )+2 \tanh ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.12, size = 441, normalized size = 2.66
method | result | size |
elliptic | \(\frac {\left (\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{3 a b}-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}\right ) \sqrt {2}}{2}\) | \(78\) |
default | \(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{4} \textit {\_Z}^{4}-a^{2} b^{2} \textit {\_Z}^{2}+b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 b^{2}\right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a^{2} x^{2}+b^{2}\right ) a^{2}}{\sqrt {a^{2} b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {a^{2} b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 a^{2} \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right )}{b^{2}}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right )}\right )}{6 a^{2}}-\frac {2 \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{3 \sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}\) | \(441\) |
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 [A]
time = 0.45, size = 101, normalized size = 0.61 \begin {gather*} -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^{4} x^{4} + b^{4}}}\right ) - 2 \, \log \left (\frac {a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4} - 2 \, \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right )}{6 \, a b} \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} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\left (a^{2} x^{2} + b^{2}\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \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 = -0.01 \begin {gather*} \int -\frac {b^6-a^6\,x^6}{\sqrt {a^4\,x^4+b^4}\,\left (a^6\,x^6+b^6\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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