Optimal. Leaf size=157 \[ \frac {45 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{64 \sqrt {2} a^{13}}-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {9}{32 a^8 x \left (a^4+x^4\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {290, 325, 297, 1162, 617, 204, 1165, 628} \[ \frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}-\frac {45}{32 a^{12} x}+\frac {45 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{64 \sqrt {2} a^{13}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 325
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx &=\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9 \int \frac {1}{x^2 \left (a^4+x^4\right )^2} \, dx}{8 a^4}\\ &=\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {45 \int \frac {1}{x^2 \left (a^4+x^4\right )} \, dx}{32 a^8}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}-\frac {45 \int \frac {x^2}{a^4+x^4} \, dx}{32 a^{12}}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {45 \int \frac {a^2-x^2}{a^4+x^4} \, dx}{64 a^{12}}-\frac {45 \int \frac {a^2+x^2}{a^4+x^4} \, dx}{64 a^{12}}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}-\frac {45 \int \frac {\sqrt {2} a+2 x}{-a^2-\sqrt {2} a x-x^2} \, dx}{128 \sqrt {2} a^{13}}-\frac {45 \int \frac {\sqrt {2} a-2 x}{-a^2+\sqrt {2} a x-x^2} \, dx}{128 \sqrt {2} a^{13}}-\frac {45 \int \frac {1}{a^2-\sqrt {2} a x+x^2} \, dx}{128 a^{12}}-\frac {45 \int \frac {1}{a^2+\sqrt {2} a x+x^2} \, dx}{128 a^{12}}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}-\frac {45 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}+\frac {45 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {45 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \tan ^{-1}\left (1+\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 134, normalized size = 0.85 \[ -\frac {\frac {104 a x^3}{a^4+x^4}+45 \sqrt {2} \log \left (a^2-\sqrt {2} a x+x^2\right )-45 \sqrt {2} \log \left (a^2+\sqrt {2} a x+x^2\right )+\frac {32 a^5 x^3}{\left (a^4+x^4\right )^2}+\frac {256 a}{x}-90 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )+90 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{256 a^{13}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 107, normalized size = 0.68 \[ \frac {45 \tan ^{-1}\left (\frac {\frac {a}{\sqrt {2}}-\frac {x^2}{\sqrt {2} a}}{x}\right )}{64 \sqrt {2} a^{13}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} a x}{a^2+x^2}\right )}{64 \sqrt {2} a^{13}}+\frac {-32 a^8-81 a^4 x^4-45 x^8}{32 a^{12} x \left (a^4+x^4\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.46, size = 338, normalized size = 2.15 \[ -\frac {256 \, a^{8} + 648 \, a^{4} x^{4} + 360 \, x^{8} - 180 \, \sqrt {2} {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \frac {1}{a^{52}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a^{12} \frac {1}{a^{52}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {\sqrt {2} a^{40} \frac {1}{a^{52}}^{\frac {3}{4}} x + a^{28} \sqrt {\frac {1}{a^{52}}} + x^{2}} a^{12} \frac {1}{a^{52}}^{\frac {1}{4}} - 1\right ) - 180 \, \sqrt {2} {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \frac {1}{a^{52}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a^{12} \frac {1}{a^{52}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {-\sqrt {2} a^{40} \frac {1}{a^{52}}^{\frac {3}{4}} x + a^{28} \sqrt {\frac {1}{a^{52}}} + x^{2}} a^{12} \frac {1}{a^{52}}^{\frac {1}{4}} + 1\right ) - 45 \, \sqrt {2} {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \frac {1}{a^{52}}^{\frac {1}{4}} \log \left (\sqrt {2} a^{40} \frac {1}{a^{52}}^{\frac {3}{4}} x + a^{28} \sqrt {\frac {1}{a^{52}}} + x^{2}\right ) + 45 \, \sqrt {2} {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \frac {1}{a^{52}}^{\frac {1}{4}} \log \left (-\sqrt {2} a^{40} \frac {1}{a^{52}}^{\frac {3}{4}} x + a^{28} \sqrt {\frac {1}{a^{52}}} + x^{2}\right )}{256 \, {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 150, normalized size = 0.96 \[ -\frac {45 \, \sqrt {2} {\left | a \right |} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} + 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{128 \, a^{14}} - \frac {45 \, \sqrt {2} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} - 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{128 \, a^{14}} + \frac {45 \, \sqrt {2} {\left | a \right |} \log \left (\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{256 \, a^{14}} - \frac {45 \, \sqrt {2} {\left | a \right |} \log \left (-\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{256 \, a^{14}} - \frac {17 \, a^{4} x^{3} + 13 \, x^{7}}{32 \, {\left (a^{4} + x^{4}\right )}^{2} a^{12}} - \frac {1}{a^{12} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 75, normalized size = 0.48
method | result | size |
risch | \(\frac {-\frac {45 x^{8}}{32 a^{12}}-\frac {81 x^{4}}{32 a^{8}}-\frac {1}{a^{4}}}{x \left (a^{4}+x^{4}\right )^{2}}+\frac {45 \left (\munderset {\textit {\_R} =\RootOf \left (a^{52} \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4} a^{52}+4\right ) x +\textit {\_R}^{3} a^{40}\right )\right )}{128}\) | \(75\) |
default | \(-\frac {\frac {\frac {17}{32} a^{4} x^{3}+\frac {13}{32} x^{7}}{\left (a^{4}+x^{4}\right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}{x^{2}+\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}-1\right )\right )}{256 \left (a^{4}\right )^{\frac {1}{4}}}}{a^{12}}-\frac {1}{a^{12} x}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 147, normalized size = 0.94 \[ -\frac {32 \, a^{8} + 81 \, a^{4} x^{4} + 45 \, x^{8}}{32 \, {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )}} - \frac {45 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a + 2 \, x\right )}}{2 \, a}\right )}{a} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a - 2 \, x\right )}}{2 \, a}\right )}{a} - \frac {\sqrt {2} \log \left (\sqrt {2} a x + a^{2} + x^{2}\right )}{a} + \frac {\sqrt {2} \log \left (-\sqrt {2} a x + a^{2} + x^{2}\right )}{a}\right )}}{256 \, a^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 76, normalized size = 0.48 \[ \frac {45\,{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{64\,a^{13}}-\frac {45\,{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{64\,a^{13}}-\frac {\frac {1}{a^4}+\frac {81\,x^4}{32\,a^8}+\frac {45\,x^8}{32\,a^{12}}}{a^8\,x+2\,a^4\,x^5+x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 66, normalized size = 0.42 \[ \frac {- 32 a^{8} - 81 a^{4} x^{4} - 45 x^{8}}{32 a^{20} x + 64 a^{16} x^{5} + 32 a^{12} x^{9}} + \frac {\operatorname {RootSum} {\left (268435456 t^{4} + 4100625, \left (t \mapsto t \log {\left (- \frac {2097152 t^{3} a}{91125} + x \right )} \right )\right )}}{a^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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