Optimal. Leaf size=109 \[ -\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a} \]
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Rubi [A]
time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {303, 1176, 631,
210, 1179, 642} \begin {gather*} \frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{a}+1\right )}{2 \sqrt {2} a} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^2}{a^4+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4} \, dx\right )+\frac {1}{2} \int \frac {a^2+x^2}{a^4+x^4} \, dx\\ &=\frac {1}{4} \int \frac {1}{a^2-\sqrt {2} a x+x^2} \, dx+\frac {1}{4} \int \frac {1}{a^2+\sqrt {2} a x+x^2} \, dx+\frac {\int \frac {\sqrt {2} a+2 x}{-a^2-\sqrt {2} a x-x^2} \, dx}{4 \sqrt {2} a}+\frac {\int \frac {\sqrt {2} a-2 x}{-a^2+\sqrt {2} a x-x^2} \, dx}{4 \sqrt {2} a}\\ &=\frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}\\ &=-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 79, normalized size = 0.72 \begin {gather*} \frac {-2 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )+2 \tan ^{-1}\left (1+\frac {\sqrt {2} x}{a}\right )+\log \left (a^2-\sqrt {2} a x+x^2\right )-\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 85, normalized size = 0.78
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+a^{4}\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}}\right )}{4}\) | \(24\) |
default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}{x^{2}+\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a^{4}\right )^{\frac {1}{4}}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.43, size = 98, normalized size = 0.90 \begin {gather*} \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a + 2 \, x\right )}}{2 \, a}\right )}{4 \, a} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a - 2 \, x\right )}}{2 \, a}\right )}{4 \, a} - \frac {\sqrt {2} \log \left (\sqrt {2} a x + a^{2} + x^{2}\right )}{8 \, a} + \frac {\sqrt {2} \log \left (-\sqrt {2} a x + a^{2} + x^{2}\right )}{8 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 199 vs.
\(2 (85) = 170\).
time = 0.42, size = 199, normalized size = 1.83 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {\sqrt {2} a^{4} \frac {1}{a^{4}}^{\frac {3}{4}} x + a^{4} \sqrt {\frac {1}{a^{4}}} + x^{2}} \frac {1}{a^{4}}^{\frac {1}{4}} - 1\right ) - \frac {1}{2} \, \sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {-\sqrt {2} a^{4} \frac {1}{a^{4}}^{\frac {3}{4}} x + a^{4} \sqrt {\frac {1}{a^{4}}} + x^{2}} \frac {1}{a^{4}}^{\frac {1}{4}} + 1\right ) - \frac {1}{8} \, \sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (\sqrt {2} a^{4} \frac {1}{a^{4}}^{\frac {3}{4}} x + a^{4} \sqrt {\frac {1}{a^{4}}} + x^{2}\right ) + \frac {1}{8} \, \sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (-\sqrt {2} a^{4} \frac {1}{a^{4}}^{\frac {3}{4}} x + a^{4} \sqrt {\frac {1}{a^{4}}} + x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.17 \begin {gather*} \frac {\operatorname {RootSum} {\left (256 t^{4} + 1, \left ( t \mapsto t \log {\left (64 t^{3} a + x \right )} \right )\right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 114, normalized size = 1.05 \begin {gather*} \frac {\sqrt {2} {\left | a \right |} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} + 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{4 \, a^{2}} + \frac {\sqrt {2} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} - 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} {\left | a \right |} \log \left (\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{8 \, a^{2}} + \frac {\sqrt {2} {\left | a \right |} \log \left (-\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{8 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 33, normalized size = 0.30 \begin {gather*} \frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )-{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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