Integrand size = 44, antiderivative size = 297 \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=-\frac {x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {\arctan \left (\frac {2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}{b^4+\sqrt {2} a^2 b^2 x^2-a^4 x^4}\right )}{4\ 2^{3/4} a b}-\frac {\arctan \left (\frac {\frac {b^3}{2 a}+a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{16 a b}+\frac {\text {arctanh}\left (\frac {\frac {b^3}{2 a}-a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{16 a b}+\frac {\text {arctanh}\left (\frac {\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{4\ 2^{3/4} a b} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.96 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.70, number of steps used = 44, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6857, 230, 227, 2098, 1166, 425, 21, 434, 438, 437, 435, 259, 418, 1225, 1713, 209, 212, 1443, 1233, 1232} \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {a^4 x^4-b^4}}-\frac {x \left (a^2 x^2+b^2\right )}{8 b^2 \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {a^4 x^4-b^4}} \]
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Rule 21
Rule 209
Rule 212
Rule 227
Rule 230
Rule 259
Rule 418
Rule 425
Rule 434
Rule 435
Rule 437
Rule 438
Rule 1166
Rule 1225
Rule 1232
Rule 1233
Rule 1443
Rule 1713
Rule 2098
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {-b^4+a^4 x^4}}+\frac {2 b^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )}\right ) \, dx \\ & = \left (2 b^{16}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx+\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx \\ & = \left (2 b^{16}\right ) \int \left (-\frac {1}{8 b^{14} \left (b^2-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{8 b^{14} \left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{4 b^{12} \sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}-\frac {1}{2 b^8 \sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )}\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 {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} b^2 \int \frac {1}{\left (b^2-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} b^2 \int \frac {1}{\left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} b^4 \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx-b^8 \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx \\ & = \frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} \int \frac {1}{\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}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} \left (\sqrt {-a^8} b^4\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4-a^8 x^4\right )} \, dx-\frac {1}{2} \left (\sqrt {-a^8} b^4\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4+a^8 x^4\right )} \, dx-\frac {\left (b^2 \sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2-a^2 x^2} \left (b^2-a^2 x^2\right )^{3/2}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\left (b^2 \sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2+a^2 x^2} \left (b^2+a^2 x^2\right )^{3/2}} \, dx}{4 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}-2 \left (\frac {1}{8} \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx\right )-\frac {1}{8} \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}{8} \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}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {-a^2 b^2+a^4 x^2}{\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}} \, dx}{8 a^2 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {a^2 b^2+a^4 x^2}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{8 a^2 b^2 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{8} \text {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}{8} \text {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 {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {\sqrt {b^2-a^2 x^2}}{\sqrt {-b^2-a^2 x^2}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {-b^2+a^2 x^2}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}-2 \frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}+\frac {3 b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {\sqrt {-b^2-a^2 x^2}}{\sqrt {b^2-a^2 x^2}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {b^2+a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}+\frac {3 b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {-b^2-a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{8 b^2 \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\arcsin \left (\frac {a x}{b}\right )\right |-1\right )}{8 a b \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}+\frac {3 b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (-b^2-a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{8 b^2 \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}-\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 \left (b^2-a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{8 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 15.34 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.26 \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\frac {-\sqrt {-\frac {a^2}{b^2}} x-3 i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-i,i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (i,i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\sqrt [4]{-1},i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\sqrt [4]{-1},i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-(-1)^{3/4},i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left ((-1)^{3/4},i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )}{4 \sqrt {-\frac {a^2}{b^2}} \sqrt {-b^4+a^4 x^4}} \]
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Time = 7.36 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.60
method | result | size |
elliptic | \(\frac {\left (\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{8 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, x}{4 \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )\right )}{32 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(474\) |
pseudoelliptic | \(-\frac {i a^{2} \left (2 \left (-a^{4} x^{4}+b^{4}\right ) \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 a^{8} b^{8}+32 i a^{6} b^{6} \textit {\_Z}^{2}+8 b^{4} \textit {\_Z}^{4} a^{4}-8 i a^{2} b^{2} \textit {\_Z}^{6}+\textit {\_Z}^{8}\right )}{\sum }\frac {\left (8 i a^{6} b^{6}-4 a^{4} b^{4} \textit {\_R}^{2}+2 i a^{2} b^{2} \textit {\_R}^{4}-\textit {\_R}^{6}\right ) \ln \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}+\left (-a^{2} x^{2}-i b^{2}\right ) \operatorname {csgn}\left (a^{2}\right )-\textit {\_R} x}{x}\right )}{\textit {\_R} \left (-8 i a^{6} b^{6}-4 a^{4} b^{4} \textit {\_R}^{2}+6 i a^{2} b^{2} \textit {\_R}^{4}-\textit {\_R}^{6}\right )}\right )+\frac {\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )}{2}+\frac {\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )}{2}+\sqrt {i a^{2} b^{2}}\, \sqrt {2}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+4 \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, x +\ln \left (2\right ) \sqrt {2}\, \left (a x -b \right ) \left (a x +b \right ) \left (a^{2} x^{2}+b^{2}\right ) \left (\sqrt {i a^{2} b^{2}}+\sqrt {-i a^{2} b^{2}}\right )\right ) b^{2}}{8 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right ) \left (a x -b \right ) \left (a x +b \right ) \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (a x +i b \right )}\) | \(784\) |
default | \(\text {Expression too large to display}\) | \(1190\) |
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Result contains complex when optimal does not.
Time = 2.12 (sec) , antiderivative size = 871, normalized size of antiderivative = 2.93 \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\int { \frac {a^{16} x^{16} + b^{16}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
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\[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\int { \frac {a^{16} x^{16} + b^{16}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
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Timed out. \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\int -\frac {a^{16}\,x^{16}+b^{16}}{\sqrt {a^4\,x^4-b^4}\,\left (b^{16}-a^{16}\,x^{16}\right )} \,d x \]
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