Optimal. Leaf size=157 \[ -\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {45 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \tan ^{-1}\left (1+\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}} \]
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Rubi [A]
time = 0.07, 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 = {296, 331, 303,
1176, 631, 210, 1179, 642} \begin {gather*} \frac {45 \text {ArcTan}\left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \text {ArcTan}\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 296
Rule 303
Rule 331
Rule 631
Rule 642
Rule 1176
Rule 1179
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 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}+\frac {45 \text {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 = 0.07, size = 134, normalized size = 0.85 \begin {gather*} -\frac {\frac {256 a}{x}+\frac {32 a^5 x^3}{\left (a^4+x^4\right )^2}+\frac {104 a x^3}{a^4+x^4}-90 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )+90 \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} x}{a}\right )+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 )}{256 a^{13}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 124, normalized size = 0.79
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 = 3.73, size = 147, normalized size = 0.94 \begin {gather*} -\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}} \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. 338 vs.
\(2 (127) = 254\).
time = 0.45, size = 338, normalized size = 2.15 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 66, normalized size = 0.42 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 150, normalized size = 0.96 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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