Optimal. Leaf size=67 \[ \frac {4 x}{\sqrt [4]{x^2-1}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^2-1}}{x}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.39, antiderivative size = 65, normalized size of antiderivative = 0.97, number of steps used = 20, number of rules used = 10, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1688, 6725, 199, 230, 305, 220, 1196, 288, 403, 398} \begin {gather*} \frac {4 x}{\sqrt [4]{x^2-1}}-\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 220
Rule 230
Rule 288
Rule 305
Rule 398
Rule 403
Rule 1196
Rule 1688
Rule 6725
Rubi steps
\begin {align*} \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx &=\int \frac {3-3 x^2+2 x^4}{\left (-2+x^2\right ) \left (-1+x^2\right )^{5/4}} \, dx\\ &=\int \left (\frac {1}{\left (-1+x^2\right )^{5/4}}+\frac {2 x^2}{\left (-1+x^2\right )^{5/4}}+\frac {5}{\left (-2+x^2\right ) \left (-1+x^2\right )^{5/4}}\right ) \, dx\\ &=2 \int \frac {x^2}{\left (-1+x^2\right )^{5/4}} \, dx+5 \int \frac {1}{\left (-2+x^2\right ) \left (-1+x^2\right )^{5/4}} \, dx+\int \frac {1}{\left (-1+x^2\right )^{5/4}} \, dx\\ &=-\frac {6 x}{\sqrt [4]{-1+x^2}}+4 \int \frac {1}{\sqrt [4]{-1+x^2}} \, dx-5 \int \frac {1}{\left (-1+x^2\right )^{5/4}} \, dx+5 \int \frac {1}{\left (-2+x^2\right ) \sqrt [4]{-1+x^2}} \, dx+\int \frac {1}{\sqrt [4]{-1+x^2}} \, dx\\ &=\frac {4 x}{\sqrt [4]{-1+x^2}}-\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-5 \int \frac {1}{\sqrt [4]{-1+x^2}} \, dx+\frac {\left (2 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}+\frac {\left (8 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}\\ &=\frac {4 x}{\sqrt [4]{-1+x^2}}-\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}+\frac {\left (2 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}-\frac {\left (2 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}+\frac {\left (8 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}-\frac {\left (8 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}-\frac {\left (10 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}\\ &=\frac {4 x}{\sqrt [4]{-1+x^2}}+\frac {10 x \sqrt [4]{-1+x^2}}{1+\sqrt {-1+x^2}}-\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {10 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+x^2}\right )^2}} \left (1+\sqrt {-1+x^2}\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{-1+x^2}\right )|\frac {1}{2}\right )}{x}+\frac {5 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+x^2}\right )^2}} \left (1+\sqrt {-1+x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+x^2}\right )|\frac {1}{2}\right )}{x}-\frac {\left (10 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}+\frac {\left (10 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}\\ &=\frac {4 x}{\sqrt [4]{-1+x^2}}-\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.25, size = 119, normalized size = 1.78 \begin {gather*} \frac {2 x \left (\frac {15 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};x^2,\frac {x^2}{2}\right )}{\left (x^2-2\right ) \left (x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};x^2,\frac {x^2}{2}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};x^2,\frac {x^2}{2}\right )\right )+6 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};x^2,\frac {x^2}{2}\right )\right )}+2\right )}{\sqrt [4]{x^2-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.44, size = 67, normalized size = 1.00 \begin {gather*} \frac {4 x}{\sqrt [4]{-1+x^2}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{x}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 120, normalized size = 1.79 \begin {gather*} \frac {10 \, \sqrt {2} {\left (x^{2} - 1\right )} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{x}\right ) + 5 \, \sqrt {2} {\left (x^{2} - 1\right )} \log \left (-\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{2} - 1} x^{2} - 4 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} x + 4 \, x^{2} - 4}{x^{4} - 4 \, x^{2} + 4}\right ) + 32 \, {\left (x^{2} - 1\right )}^{\frac {3}{4}} x}{8 \, {\left (x^{2} - 1\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 {2 \, x^{4} - 3 \, x^{2} + 3}{{\left (x^{4} - 3 \, x^{2} + 2\right )} {\left (x^{2} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.57, size = 127, normalized size = 1.90
method | result | size |
trager | \(\frac {4 x}{\left (x^{2}-1\right )^{\frac {1}{4}}}-\frac {5 \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+2\right )+x \sqrt {x^{2}-1}-\RootOf \left (\textit {\_Z}^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{4}-\frac {5 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-2\right )+x \sqrt {x^{2}-1}+\RootOf \left (\textit {\_Z}^{2}-2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}+x}{x^{2}-2}\right )}{4}\) | \(127\) |
risch | \(\frac {4 x}{\left (x^{2}-1\right )^{\frac {1}{4}}}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+2\right )-x \sqrt {x^{2}-1}-\RootOf \left (\textit {\_Z}^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}+x}{x^{2}-2}\right )}{4}+\frac {5 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-2\right )-x \sqrt {x^{2}-1}+\RootOf \left (\textit {\_Z}^{2}-2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{4}\) | \(131\) |
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 {2 \, x^{4} - 3 \, x^{2} + 3}{{\left (x^{4} - 3 \, x^{2} + 2\right )} {\left (x^{2} - 1\right )}^{\frac {1}{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 {2\,x^4-3\,x^2+3}{{\left (x^2-1\right )}^{1/4}\,\left (x^4-3\,x^2+2\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 {2 x^{4} - 3 x^{2} + 3}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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