Integrand size = 42, antiderivative size = 333 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {b^4+a^4 x^4}}+\frac {1}{6} \text {RootSum}\left [16 a^8 b^8-32 a^6 b^6 \text {$\#$1}^2-24 a^4 b^4 \text {$\#$1}^4-8 a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right )-4 a^4 b^4 \log (x) \text {$\#$1}^2+4 a^4 b^4 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a^2 b^2 \log (x) \text {$\#$1}^4+2 a^2 b^2 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+\log (x) \text {$\#$1}^6-\log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{8 a^6 b^6 \text {$\#$1}+12 a^4 b^4 \text {$\#$1}^3+6 a^2 b^2 \text {$\#$1}^5-\text {$\#$1}^7}\&\right ] \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(2431\) vs. \(2(333)=666\).
Time = 7.81 (sec) , antiderivative size = 2431, normalized size of antiderivative = 7.30, number of steps used = 25, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6857, 226, 2098, 425, 537, 418, 1231, 1721} \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {2 \sqrt [4]{2} \arctan \left (\frac {\sqrt {3 a^4-\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4-\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right ) a^{12}}{\sqrt {3} \sqrt {-a^8} \left (a^4-\sqrt {3} \sqrt {-a^8}\right )^{3/4} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right )^{3/2} b}+\frac {2 \sqrt [4]{2} \arctan \left (\frac {\sqrt {\sqrt {3} \sqrt {-a^8}-3 a^4} b x}{\sqrt [4]{2} \sqrt [4]{a^4-\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right ) a^{12}}{\sqrt {3} \sqrt {-a^8} \left (a^4-\sqrt {3} \sqrt {-a^8}\right )^{3/4} \left (\sqrt {3} \sqrt {-a^8}-3 a^4\right )^{3/2} b}-\frac {2 \sqrt [4]{2} \arctan \left (\frac {\sqrt {-3 a^4-\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4+\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right ) a^{12}}{\sqrt {3} \sqrt {-a^8} \left (-3 a^4-\sqrt {3} \sqrt {-a^8}\right )^{3/2} \left (a^4+\sqrt {3} \sqrt {-a^8}\right )^{3/4} b}+\frac {2 \sqrt [4]{2} \arctan \left (\frac {\sqrt {3 a^4+\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4+\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right ) a^{12}}{\sqrt {3} \sqrt {-a^8} \left (a^4+\sqrt {3} \sqrt {-a^8}\right )^{3/4} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right )^{3/2} b}+\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}}\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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right ) a^7}{\sqrt {3} \sqrt {-a^8} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\left (\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}}+1\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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right ) a^7}{\sqrt {3} \sqrt {-a^8} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}}\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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right ) a^7}{\sqrt {3} \sqrt {-a^8} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}}+1\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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right ) a^7}{\sqrt {3} \sqrt {-a^8} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {2 x a^4}{\sqrt {3} \left (\sqrt {3} a^4-3 \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 x a^4}{\sqrt {3} \left (\sqrt {3} a^4+3 \sqrt {-a^8}\right ) \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right ) a^3}{\sqrt {3} \left (\sqrt {3} a^4-3 \sqrt {-a^8}\right ) b \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right ) a^3}{\sqrt {3} \left (\sqrt {3} a^4+3 \sqrt {-a^8}\right ) b \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 b \sqrt {b^4+a^4 x^4} a}-\frac {\left (\sqrt {2} a^2+\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )^2 \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}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2} a^2-\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 \sqrt {3} \left (\sqrt {3} a^4+3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4} a}-\frac {\left (\sqrt {2} a^2-\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )^2 \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}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} a^2+\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 \sqrt {3} \left (\sqrt {3} a^4+3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4} a}-\frac {\left (\sqrt {2} a^2+\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )^2 \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}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2} a^2-\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 \sqrt {3} \left (\sqrt {3} a^4-3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4} a}-\frac {\left (\sqrt {2} a^2-\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )^2 \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}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} a^2+\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 \sqrt {3} \left (\sqrt {3} a^4-3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4} a} \]
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Rule 226
Rule 418
Rule 425
Rule 537
Rule 1231
Rule 1721
Rule 2098
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {b^4+a^4 x^4}}-\frac {2 b^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )}\right ) \, dx \\ & = -\left (\left (2 b^{12}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\left (2 b^{12}\right ) \int \left (-\frac {2 a^8}{\sqrt {3} \sqrt {-a^8} b^4 \left (b^4+a^4 x^4\right )^{3/2} \left (a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4-2 a^8 x^4\right )}-\frac {2 a^8}{\sqrt {3} \sqrt {-a^8} b^4 \left (b^4+a^4 x^4\right )^{3/2} \left (-a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4+2 a^8 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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (4 \sqrt {-a^8} b^8\right ) \int \frac {1}{\left (b^4+a^4 x^4\right )^{3/2} \left (a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4-2 a^8 x^4\right )} \, dx}{\sqrt {3}}-\frac {\left (4 \sqrt {-a^8} b^8\right ) \int \frac {1}{\left (b^4+a^4 x^4\right )^{3/2} \left (-a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4+2 a^8 x^4\right )} \, dx}{\sqrt {3}} \\ & = \frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (2 \sqrt {-a^8}\right ) \int \frac {a^4 \left (5 a^4-\sqrt {3} \sqrt {-a^8}\right ) b^4-2 a^{12} x^4}{\sqrt {b^4+a^4 x^4} \left (-a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4+2 a^8 x^4\right )} \, dx}{\sqrt {3} a^4 \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right )}+\frac {\left (2 \sqrt {-a^8}\right ) \int \frac {-a^4 \left (5 a^4+\sqrt {3} \sqrt {-a^8}\right ) b^4+2 a^{12} x^4}{\sqrt {b^4+a^4 x^4} \left (a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4-2 a^8 x^4\right )} \, dx}{\sqrt {3} a^4 \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right )} \\ & = \frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {\left (2 \sqrt {-a^8}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (2 \sqrt {-a^8}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (8 a^4 \sqrt {-a^8} b^4\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (-a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4+2 a^8 x^4\right )} \, dx}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (8 a^4 \sqrt {-a^8} b^4\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4-2 a^8 x^4\right )} \, dx}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right )} \\ & = \frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt {-a^8} \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{\sqrt {3} a \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt {-a^8} \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{\sqrt {3} a \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} \int \frac {1}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx \\ & = \frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt {-a^8} \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{\sqrt {3} a \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt {-a^8} \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{\sqrt {3} a \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (1-\frac {2 a^4}{a^4-\sqrt {3} \sqrt {-a^8}}\right )}-\frac {\left (1+\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (1-\frac {2 a^4}{a^4-\sqrt {3} \sqrt {-a^8}}\right )}-\frac {\left (a^2 \left (2 a^2-\sqrt {2} \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (a^4+\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (a^2 \left (2 a^2+\sqrt {2} \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (a^4+\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (1-\frac {2 a^4}{a^4+\sqrt {3} \sqrt {-a^8}}\right )}-\frac {\left (1+\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (1-\frac {2 a^4}{a^4+\sqrt {3} \sqrt {-a^8}}\right )}-\frac {\left (a^2 \left (2 a^2-\sqrt {2} \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (a^4-\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (a^2 \left (2 a^2+\sqrt {2} \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (a^4-\sqrt {3} \sqrt {-a^8}\right )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 1.06 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.99 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {b^4+a^4 x^4}}+\frac {1}{6} \text {RootSum}\left [16 a^8 b^8-32 a^6 b^6 \text {$\#$1}^2-24 a^4 b^4 \text {$\#$1}^4-8 a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right )+4 a^4 b^4 \log (x) \text {$\#$1}^2-4 a^4 b^4 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2+2 a^2 b^2 \log (x) \text {$\#$1}^4-2 a^2 b^2 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-\log (x) \text {$\#$1}^6+\log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{-8 a^6 b^6 \text {$\#$1}-12 a^4 b^4 \text {$\#$1}^3-6 a^2 b^2 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
[In]
[Out]
Time = 3.82 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.52
method | result | size |
elliptic | \(\frac {\left (-\frac {\sqrt {2}\, x}{3 \sqrt {a^{4} x^{4}+b^{4}}}+\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}\right )-\ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}{2}}\right )\right )}{6 \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\) | \(172\) |
pseudoelliptic | \(-\frac {6 \sqrt {a^{4} x^{4}+b^{4}}\, \left (a^{4} b^{4}\right )^{\frac {1}{4}} x +3^{\frac {3}{4}} \left (a^{4} x^{4}+b^{4}\right ) \left (-2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 3^{\frac {3}{4}}}{3 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}{-x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}\right )\right )}{18 \left (a^{4} b^{4}\right )^{\frac {1}{4}} \left (a^{2} x^{2}-\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right ) \left (a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right )}\) | \(203\) |
default | \(\frac {\left (2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 3^{\frac {3}{4}}}{3 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) a^{4} x^{4}-\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}{-x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}\right ) a^{4} x^{4}+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 3^{\frac {3}{4}}}{3 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) b^{4}-\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}{-x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}\right ) b^{4}-2 \sqrt {a^{4} x^{4}+b^{4}}\, 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x \right ) 3^{\frac {3}{4}}}{18 \left (a^{4} b^{4}\right )^{\frac {1}{4}} \left (a^{2} x^{2}-\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right ) \left (a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right )}\) | \(309\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.77 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.24 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{4} x^{4} + b^{4}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {6 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (3 \, \sqrt {\frac {1}{3}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} + a^{4} x^{5} + b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 5 \, a^{4} b^{4} x^{4} + b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}\right )}}\right ) - \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{4} x^{4} + b^{4}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {6 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (3 \, \sqrt {\frac {1}{3}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} + a^{4} x^{5} + b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 5 \, a^{4} b^{4} x^{4} + b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}\right )}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, a^{4} x^{4} - i \, b^{4}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {6 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (i \, a^{8} b^{4} x^{6} + i \, a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (3 \, \sqrt {\frac {1}{3}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} - a^{4} x^{5} - b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, a^{8} x^{8} - 5 i \, a^{4} b^{4} x^{4} - i \, b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}\right )}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (i \, a^{4} x^{4} + i \, b^{4}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {6 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (-i \, a^{8} b^{4} x^{6} - i \, a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (3 \, \sqrt {\frac {1}{3}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} - a^{4} x^{5} - b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (i \, a^{8} x^{8} + 5 i \, a^{4} b^{4} x^{4} + i \, b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}\right )}}\right ) + 4 \, \sqrt {a^{4} x^{4} + b^{4}} x}{12 \, {\left (a^{4} x^{4} + b^{4}\right )}} \]
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Not integrable
Time = 127.47 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.32 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}{\left (a^{4} x^{4} + b^{4}\right )^{\frac {3}{2}} \left (a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}\right )}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.13 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=\int { \frac {a^{12} x^{12} - b^{12}}{{\left (a^{12} x^{12} + b^{12}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.13 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=\int { \frac {a^{12} x^{12} - b^{12}}{{\left (a^{12} x^{12} + b^{12}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
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Not integrable
Time = 7.68 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.13 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=\int -\frac {b^{12}-a^{12}\,x^{12}}{\sqrt {a^4\,x^4+b^4}\,\left (a^{12}\,x^{12}+b^{12}\right )} \,d x \]
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