Optimal. Leaf size=334 \[ -4 x-\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )+x \log \left (\frac {1}{x^4}+x^4\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 9, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {2603, 12, 396,
219, 1183, 648, 632, 210, 642} \begin {gather*} x \log \left (x^4+\frac {1}{x^4}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )-4 x-\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 219
Rule 396
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 2603
Rubi steps
\begin {align*} \int \log \left (\frac {1}{x^4}+x^4\right ) \, dx &=x \log \left (\frac {1}{x^4}+x^4\right )-\int \frac {4 \left (-1+x^8\right )}{1+x^8} \, dx\\ &=x \log \left (\frac {1}{x^4}+x^4\right )-4 \int \frac {-1+x^8}{1+x^8} \, dx\\ &=-4 x+x \log \left (\frac {1}{x^4}+x^4\right )+8 \int \frac {1}{1+x^8} \, dx\\ &=-4 x+x \log \left (\frac {1}{x^4}+x^4\right )+\left (2 \sqrt {2}\right ) \int \frac {\sqrt {2}-x^2}{1-\sqrt {2} x^2+x^4} \, dx+\left (2 \sqrt {2}\right ) \int \frac {\sqrt {2}+x^2}{1+\sqrt {2} x^2+x^4} \, dx\\ &=-4 x+x \log \left (\frac {1}{x^4}+x^4\right )+\sqrt {2-\sqrt {2}} \int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx+\sqrt {2-\sqrt {2}} \int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx+\sqrt {2+\sqrt {2}} \int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}-\left (-1+\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx+\sqrt {2+\sqrt {2}} \int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}+\left (-1+\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx\\ &=-4 x+x \log \left (\frac {1}{x^4}+x^4\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx+\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx+\frac {1}{2} \left (2-\sqrt {2}\right ) \int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx+\frac {1}{2} \left (2-\sqrt {2}\right ) \int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx-\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx+\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx+\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx+\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx\\ &=-4 x-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )+x \log \left (\frac {1}{x^4}+x^4\right )+\left (-2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+2 x\right )+\left (-2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+2 x\right )-\left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+2 x\right )-\left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+2 x\right )\\ &=-4 x-\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )+x \log \left (\frac {1}{x^4}+x^4\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.00, size = 30, normalized size = 0.09 \begin {gather*} -4 x+8 x \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-x^8\right )+x \log \left (\frac {1}{x^4}+x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.09, size = 37, normalized size = 0.11 \begin {gather*} -4 x+x \text {Log}\left [\frac {1+x^8}{x^4}\right ]-\text {RootSum}\left [1+\text {\#1}^8\&,\text {Log}\left [x-\text {\#1}\right ] \text {\#1}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.02, size = 36, normalized size = 0.11
method | result | size |
risch | \(x \ln \left (\frac {1}{x^{4}}+x^{4}\right )-4 x +\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{7}}\right )\) | \(34\) |
default | \(x \ln \left (\frac {x^{8}+1}{x^{4}}\right )-4 x +\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{7}}\right )\) | \(36\) |
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*} \text {Failed to integrate} \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. 1058 vs.
\(2 (252) = 504\).
time = 0.39, size = 1058, normalized size = 3.17
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.25, size = 26, normalized size = 0.08 \begin {gather*} x \log {\left (x^{4} + \frac {1}{x^{4}} \right )} - 4 x - \operatorname {RootSum} {\left (t^{8} + 1, \left ( t \mapsto t \log {\left (- t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 344, normalized size = 1.03 \begin {gather*} -\frac {1}{2} \sqrt {\sqrt {2}+2} \ln \left (x^{2}-\sqrt {2+\sqrt {2}} x+1\right )+\sqrt {-\sqrt {2}+2} \arctan \left (\frac {x-\frac {\sqrt {2+\sqrt {2}}}{2}}{\frac {\sqrt {2-\sqrt {2}}}{2}}\right )-\frac {1}{2} \sqrt {-\sqrt {2}+2} \ln \left (x^{2}-\sqrt {2-\sqrt {2}} x+1\right )+\sqrt {\sqrt {2}+2} \arctan \left (\frac {x-\frac {\sqrt {2-\sqrt {2}}}{2}}{\frac {\sqrt {2+\sqrt {2}}}{2}}\right )+\frac {1}{2} \sqrt {-\sqrt {2}+2} \ln \left (x^{2}+\sqrt {2-\sqrt {2}} x+1\right )+\sqrt {\sqrt {2}+2} \arctan \left (\frac {x+\frac {\sqrt {2-\sqrt {2}}}{2}}{\frac {\sqrt {2+\sqrt {2}}}{2}}\right )+\frac {1}{2} \sqrt {\sqrt {2}+2} \ln \left (x^{2}+\sqrt {2+\sqrt {2}} x+1\right )+\sqrt {-\sqrt {2}+2} \arctan \left (\frac {x+\frac {\sqrt {2+\sqrt {2}}}{2}}{\frac {\sqrt {2-\sqrt {2}}}{2}}\right )-4 x+x \ln \left (\frac 1{x^{4}}+x^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 313, normalized size = 0.94 \begin {gather*} x\,\ln \left (\frac {1}{x^4}+x^4\right )-4\,x+\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+2097152\,\sqrt {2}}-\frac {x\,\sqrt {2-\sqrt {2}}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+2097152\,\sqrt {2}}\right )\,\left (\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}-\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2}+2097152\,\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}+\frac {x\,\sqrt {\sqrt {2}+2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2}+2097152\,\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}\right )\,\left (\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}+\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}\right )+\mathrm {atan}\left (-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}}{2}+x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{2}-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i}-\mathrm {atan}\left (x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {2}}{2}-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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