Optimal. Leaf size=107 \[ -\frac {4 \tan ^{-1}\left (\frac {a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2-a b x+b^2}\right )}{3 a b}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+2 a b x+b^2}\right )}{3 a b} \]
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Rubi [C] time = 2.66, antiderivative size = 662, normalized size of antiderivative = 6.19, number of steps used = 29, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {6725, 220, 2074, 1725, 1211, 1699, 208, 1248, 725, 206, 6728, 1217, 1707} \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {a^2 x^2+b^2}{\sqrt {2} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {\left (a-\sqrt {3} \sqrt {-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 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {a^4 x^4+b^4}}-\frac {\left (\sqrt {3} \sqrt {-a^2}+a\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 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {a^4 x^4+b^4}}+\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 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \left (\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 x^2+4 b^2\right )}{2 \sqrt {2} \sqrt {\sqrt {3} \sqrt {-a^2}+a} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {\sqrt {3} \sqrt {-a^2}+a} b}-\frac {\left (\sqrt {3} \sqrt {-a^2}+a\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \left (\left (\sqrt {3} \sqrt {-a^2}+a\right )^2 x^2+4 b^2\right )}{2 \sqrt {2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} b} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 220
Rule 725
Rule 1211
Rule 1217
Rule 1248
Rule 1699
Rule 1707
Rule 1725
Rule 2074
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx &=\int \left (\frac {1}{\sqrt {b^4+a^4 x^4}}-\frac {2 b^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx\\ &=-\left (\left (2 b^3\right ) \int \frac {1}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, 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^3\right ) \int \left (\frac {1}{3 b^2 (b+a x) \sqrt {b^4+a^4 x^4}}+\frac {2 b-a x}{3 b^2 \left (b^2-a b x+a^2 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 )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} (2 b) \int \frac {1}{(b+a x) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} (2 b) \int \frac {2 b-a x}{\left (b^2-a b x+a^2 x^2\right ) \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}}-\frac {1}{3} (2 b) \int \left (\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}}+\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx+\frac {1}{3} (2 a b) \int \frac {x}{\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 {1}{\left (b^2-a^2 x^2\right ) \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}}-\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} (a b) \operatorname {Subst}\left (\int \frac {1}{\left (b^2-a^2 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}} \, dx+\frac {1}{3} \left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \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 )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} (a b) \operatorname {Subst}\left (\int \frac {1}{2 a^4 b^4-x^2} \, dx,x,\frac {-a^2 b^4-a^4 b^2 x^2}{\sqrt {b^4+a^4 x^4}}\right )-\frac {1}{3} \left (4 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (4 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} b^2 \operatorname {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-\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \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 (a-\sqrt {3} \sqrt {-a^2}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 a}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 a}-\frac {1}{3} \left (2 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (2 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (4 a \left (a-\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (4 a \left (a+\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=-\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \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 (a-\sqrt {3} \sqrt {-a^2}\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 )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\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 )}{6 a^2 b \sqrt {b^4+a^4 x^4}}+\frac {1}{3} \left (2 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{16 a^8 b^4+a^4 \left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {-4 a^4 b^4-a^4 \left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {b^4+a^4 x^4}}\right )+\frac {1}{3} \left (2 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{16 a^8 b^4+a^4 \left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {-4 a^4 b^4-a^4 \left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {b^4+a^4 x^4}}\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \left (4 b^2+\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {a+\sqrt {3} \sqrt {-a^2}} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a+\sqrt {3} \sqrt {-a^2}} b}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \left (4 b^2+\left (a+\sqrt {3} \sqrt {-a^2}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a-\sqrt {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-\sqrt {3} \sqrt {-a^2}\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 )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\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 )}{6 a^2 b \sqrt {b^4+a^4 x^4}}\\ \end {align*}
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Mathematica [C] time = 3.40, size = 562, normalized size = 5.25 \begin {gather*} \frac {a^3 \left (18 i a \sqrt [4]{a^4} b \sqrt {\frac {a^4 x^4}{b^4}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\frac {3}{2} \left (8 i a \sqrt [4]{a^4} b \sqrt {\frac {a^4 x^4}{b^4}+1} \Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+8 i a \sqrt [4]{a^4} b \sqrt {\frac {a^4 x^4}{b^4}+1} \Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+\sqrt {2} \sqrt {\frac {i a^2}{b^2}} \left ((4+4 i) a b \sqrt [4]{b^4} \sqrt {\frac {a^4 x^4}{b^4}+1} \Pi \left (\frac {i \sqrt {a^4} \sqrt {b^4}}{a^2 b^2};\left .i \sinh ^{-1}\left (\frac {(1+i) \sqrt [4]{a^4} x}{\sqrt {2} \sqrt [4]{b^4}}\right )\right |-1\right )+\sqrt [4]{a^4} \sqrt {a^4 x^4+b^4} \left (\sqrt {1+i \sqrt {3}} \left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\left (\sqrt {3}+i\right ) b^2-2 i a^2 x^2}{\sqrt {2-2 i \sqrt {3}} \sqrt {a^4 x^4+b^4}}\right )+\sqrt {1-i \sqrt {3}} \left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {2 i a^2 x^2+\left (\sqrt {3}-i\right ) b^2}{\sqrt {2+2 i \sqrt {3}} \sqrt {a^4 x^4+b^4}}\right )+2 \tanh ^{-1}\left (\frac {a^2 x^2+b^2}{\sqrt {2} \sqrt {a^4 x^4+b^4}}\right )\right )\right )\right )\right )}{18 \sqrt [4]{a^4} b^5 \left (\frac {i a^2}{b^2}\right )^{5/2} \sqrt {a^4 x^4+b^4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.31, size = 107, normalized size = 1.00 \begin {gather*} -\frac {4 \tan ^{-1}\left (\frac {a b x}{b^2-a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 165, normalized size = 1.54 \begin {gather*} \frac {\sqrt {2} \log \left (-\frac {3 \, a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + 3 \, b^{4} + 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} {\left (a^{2} x^{2} + a b x + b^{2}\right )}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ) - 8 \, \arctan \left (\frac {\sqrt {a^{4} x^{4} + b^{4}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right )}{12 \, a b} \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^{3} x^{3} - b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} + b^{3}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 457, normalized size = 4.27
method | result | size |
default | \(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a +2 b \right ) \left (-\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) a b \left (a \,x^{2}-b \underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {-b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {-b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}}+\frac {2 a \left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{2} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{3 a}-\frac {2 b \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 a^{2} b^{2} x^{2}+2 b^{4}\right ) \sqrt {2}}{4 \sqrt {b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 \sqrt {b^{4}}}+\frac {a \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, -i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b \sqrt {a^{4} x^{4}+b^{4}}}\right )}{3 a}\) | \(457\) |
elliptic | \(-\frac {a^{3} b^{5} \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {-3 a^{4} b^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}+\frac {a \,b^{3} \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}-\frac {2 a^{3} b^{5} \sqrt {2}\, \ln \left (\frac {4 b^{4}+2 a^{2} b^{2} \left (x^{2}-\frac {b^{2}}{a^{2}}\right )+2 \sqrt {2}\, \sqrt {b^{4}}\, \sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right )^{2} a^{4}+2 a^{2} b^{2} \left (x^{2}-\frac {b^{2}}{a^{2}}\right )+2 b^{4}}}{x^{2}-\frac {b^{2}}{a^{2}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {b^{4}}}-\frac {a^{3} b^{5} \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{\left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {-3 a^{4} b^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}-\frac {a \,b^{3} \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{\left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}+\frac {\left (-\frac {\ln \left (a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}+\frac {\ln \left (-a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}\right ) \sqrt {2}}{2}\) | \(1539\) |
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^{3} x^{3} - b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} + b^{3}\right )}}\,{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 {b^3-a^3\,x^3}{\left (a^3\,x^3+b^3\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\left (a x + b\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{2} x^{2} - a b x + b^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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