Optimal. Leaf size=48 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{2 \sqrt {2}}-\frac {\sqrt {x^4+1} x}{2 \left (x^2+1\right )^2} \]
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Rubi [A] time = 0.36, antiderivative size = 72, normalized size of antiderivative = 1.50, number of steps used = 20, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {1721, 220, 1224, 1697, 1713, 1196, 1701, 1699, 203, 1211} \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{2 \sqrt {2}}-\frac {\sqrt {x^4+1} x}{2 \left (x^2+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 220
Rule 1196
Rule 1211
Rule 1224
Rule 1697
Rule 1699
Rule 1701
Rule 1713
Rule 1721
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1+x^2\right )^3} \, dx &=\int \left (\frac {1}{\sqrt {1+x^4}}-\frac {4}{\left (1+x^2\right )^3 \sqrt {1+x^4}}+\frac {6}{\left (1+x^2\right )^2 \sqrt {1+x^4}}-\frac {4}{\left (1+x^2\right ) \sqrt {1+x^4}}\right ) \, dx\\ &=-\left (4 \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^4}} \, dx\right )-4 \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx+6 \int \frac {1}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx+\int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=-\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )^2}+\frac {3 x \sqrt {1+x^4}}{2 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {1}{2} \int \frac {-7+4 x^2-x^4}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx-\frac {3}{2} \int \frac {-3+2 x^2+x^4}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx-2 \int \frac {1}{\sqrt {1+x^4}} \, dx-2 \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\\ &=-\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )^2}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{8} \int \frac {16-20 x^2-12 x^4}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx+\frac {3}{2} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {3}{2} \int \frac {-2+2 x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx-2 \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\\ &=-\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )^2}-\frac {3 x \sqrt {1+x^4}}{2 \left (1+x^2\right )}-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{8} \int \frac {4-20 x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx-\frac {3}{2} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+3 \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\\ &=-\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}}-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {3}{2} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx+\int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=-\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}}-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\\ &=-\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{2 \sqrt {2}}-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.25, size = 68, normalized size = 1.42 \begin {gather*} -\frac {\sqrt {x^4+1} x}{2 \left (x^2+1\right )^2}-\frac {1}{2} \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\sqrt [4]{-1} \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 48, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 52, normalized size = 1.08 \begin {gather*} -\frac {\sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) + 2 \, \sqrt {x^{4} + 1} x}{4 \, {\left (x^{4} + 2 \, 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 {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )}^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 53, normalized size = 1.10
method | result | size |
trager | \(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{2}+1\right )^{2}}-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{4}\) | \(53\) |
elliptic | \(\frac {\left (-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{4 \left (\frac {x^{4}+1}{2 x^{2}}+1\right ) x}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(54\) |
risch | \(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{2}+1\right )^{2}}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{\sqrt {x^{4}+1}}\) | \(128\) |
default | \(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{2}+1\right )^{2}}+\frac {i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{\sqrt {x^{4}+1}}\) | \(331\) |
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 {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )}^{3}}\,{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 = -0.02 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {x^4+1}}{{\left (x^2+1\right )}^3} \,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 (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1}}{\left (x^{2} + 1\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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