Optimal. Leaf size=196 \[ -\frac {x}{2 \sqrt {a^4 x^4+b^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{2 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {6-4 \sqrt {2}} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{4 \sqrt {2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {6+4 \sqrt {2}} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{4 \sqrt {2} a b} \]
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Rubi [A] time = 0.64, antiderivative size = 101, normalized size of antiderivative = 0.52, number of steps used = 16, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6725, 220, 1404, 414, 523, 409, 1211, 1699, 206, 203} \begin {gather*} -\frac {x}{2 \sqrt {a^4 x^4+b^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{4 \sqrt {2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{4 \sqrt {2} a b} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 220
Rule 409
Rule 414
Rule 523
Rule 1211
Rule 1404
Rule 1699
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 \left (\frac {1}{\sqrt {b^4+a^4 x^4}}+\frac {2 b^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )}\right ) \, dx\\ &=\left (2 b^8\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx+\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^8\right ) \int \frac {1}{\left (-b^4+a^4 x^4\right ) \left (b^4+a^4 x^4\right )^{3/2}} \, dx\\ &=-\frac {x}{2 \sqrt {b^4+a^4 x^4}}+\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 {\int \frac {3 a^4 b^4-a^8 x^4}{\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 a^4}\\ &=-\frac {x}{2 \sqrt {b^4+a^4 x^4}}+\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}{2} \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx+b^4 \int \frac {1}{\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=-\frac {x}{2 \sqrt {b^4+a^4 x^4}}+\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 )}{4 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {a^2 x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {a^2 x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=-\frac {x}{2 \sqrt {b^4+a^4 x^4}}+\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 )}{4 a b \sqrt {b^4+a^4 x^4}}-2 \left (\frac {1}{4} \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx\right )-\frac {1}{4} \int \frac {1-\frac {a^2 x^2}{b^2}}{\left (1+\frac {a^2 x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{4} \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {a^2 x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=-\frac {x}{2 \sqrt {b^4+a^4 x^4}}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-2 a^2 b^2 x^2} \, dx,x,\frac {x}{\sqrt {b^4+a^4 x^4}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+2 a^2 b^2 x^2} \, dx,x,\frac {x}{\sqrt {b^4+a^4 x^4}}\right )\\ &=-\frac {x}{2 \sqrt {b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b}\\ \end {align*}
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Mathematica [C] time = 0.58, size = 199, normalized size = 1.02 \begin {gather*} \frac {x \left (-\frac {5 \left (a^4 b^4 x^4+b^8\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-\frac {a^4 x^4}{b^4},\frac {a^4 x^4}{b^4}\right )}{\left (b^4-a^4 x^4\right ) \left (5 b^4 F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-\frac {a^4 x^4}{b^4},\frac {a^4 x^4}{b^4}\right )+2 a^4 x^4 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};-\frac {a^4 x^4}{b^4},\frac {a^4 x^4}{b^4}\right )+F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\frac {a^4 x^4}{b^4},\frac {a^4 x^4}{b^4}\right )\right )\right )}-1\right )}{2 \sqrt {a^4 x^4+b^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.85, size = 114, normalized size = 0.58 \begin {gather*} -\frac {x}{2 \sqrt {b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt {2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 159, normalized size = 0.81 \begin {gather*} -\frac {8 \, \sqrt {a^{4} x^{4} + b^{4}} a b x + 2 \, \sqrt {2} {\left (a^{4} x^{4} + b^{4}\right )} \arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^{4} x^{4} + b^{4}}}\right ) - \sqrt {2} {\left (a^{4} x^{4} + b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4} - 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + b^{4}}\right )}{16 \, {\left (a^{5} b x^{4} + a b^{5}\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 [A] time = 0.20, size = 131, normalized size = 0.67
method | result | size |
elliptic | \(\frac {\left (-\frac {\sqrt {2}\, x}{2 \sqrt {a^{4} x^{4}+b^{4}}}+\frac {\ln \left (-a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{8 a b}-\frac {\ln \left (a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{8 a b}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{4 a b}\right ) \sqrt {2}}{2}\) | \(131\) |
default | \(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 a^{2} b^{2} x^{2}+2 b^{4}\right ) \sqrt {2}}{4 \sqrt {b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 \sqrt {b^{4}}}+\frac {a \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, -i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 a}+\frac {b \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 a^{2} b^{2} x^{2}+2 b^{4}\right ) \sqrt {2}}{4 \sqrt {b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 \sqrt {b^{4}}}-\frac {a \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, -i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 a}-b^{4} \left (\frac {x}{2 b^{4} \sqrt {\left (x^{4}+\frac {b^{4}}{a^{4}}\right ) a^{4}}}+\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{2 b^{4} \sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )-\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{2 \sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}\) | \(595\) |
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 {a^8\,x^8+b^8}{\sqrt {a^4\,x^4+b^4}\,\left (b^8-a^8\,x^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 {a^{8} x^{8} + b^{8}}{\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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