Optimal. Leaf size=313 \[ -\frac {x}{2 \sqrt {a^4 x^4-b^4}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {a^4 x^4-b^4}}{\sqrt {3-2 \sqrt {2}} a b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3-2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3-2 \sqrt {2}} a}}{x}\right )}{a b}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {a^4 x^4-b^4}}{\sqrt {3+2 \sqrt {2}} a b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3+2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3+2 \sqrt {2}} a}}{x}\right )}{a b}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2}\right )}{a b} \]
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Rubi [A] time = 0.71, antiderivative size = 135, normalized size of antiderivative = 0.43, number of steps used = 20, number of rules used = 11, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6725, 224, 221, 1404, 414, 523, 409, 1211, 1699, 203, 206} \begin {gather*} -\frac {x}{2 \sqrt {a^4 x^4-b^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{4 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{4 \sqrt {2} \sqrt [4]{-a^4} b} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 221
Rule 224
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\\ &=\left (2 b^8\right ) \int \frac {1}{\left (-b^4+a^4 x^4\right )^{3/2} \left (b^4+a^4 x^4\right )} \, dx+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{\sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {\int \frac {-3 a^4 b^4-a^8 x^4}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{2 a^4}\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \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}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \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}}-2 \left (\frac {1}{4} \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx\right )-\frac {1}{4} \int \frac {1-\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} \int \frac {1+\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \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 {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-2 \sqrt {-a^4} 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 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )-2 \frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{4 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{4 \sqrt {2} \sqrt [4]{-a^4} b}\\ \end {align*}
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Mathematica [C] time = 0.63, size = 201, normalized size = 0.64 \begin {gather*} \frac {x \left (-\frac {5 \left (b^8-a^4 b^4 x^4\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 (a^4 x^4+b^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 = 1.09, size = 247, normalized size = 0.79 \begin {gather*} -\frac {x}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a b}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tanh ^{-1}\left (\frac {b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-a^4 x^4+\left (-i b^2-(1-i) a b x-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left (b^2+(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right )}{a b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 162, normalized size = 0.52 \begin {gather*} -\frac {4 \, \sqrt {a^{4} x^{4} - b^{4}} a b x - 2 \, {\left (a^{4} x^{4} - b^{4}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) - {\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 {a^{4} x^{4} - b^{4}} a b x}{a^{4} x^{4} + b^{4}}\right )}{8 \, {\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 = 0.43, size = 249, normalized size = 0.80
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{16 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, x}{2 \sqrt {a^{4} x^{4}-b^{4}}}\right ) \sqrt {2}}{2}\) | \(249\) |
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 {b \left (\frac {a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}\right )}}+\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 )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{4}+\frac {b \left (-\frac {a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}\right )}}-\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 )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{4}-\frac {b^{4} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}+b^{4}\right )}{\sum }\frac {-\frac {\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) a^{4}}{\sqrt {-2 b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {-b^{4}}}+\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{4} \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} a^{2}}{b^{2}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{16 a^{4}}-\frac {b^{2} \left (-\frac {\left (a^{4} x^{2}-a^{2} b^{2}\right ) x}{2 b^{4} a^{2} \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) \left (a^{4} x^{2}-a^{2} b^{2}\right )}}+\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 )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{2}\) | \(981\) |
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 = -0.00 \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}}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \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 )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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