Optimal. Leaf size=83 \[ -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}+\frac {5 \tan ^{-1}\left (\frac {x^2}{a^2}\right )}{256 a^6} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 294, 205,
209} \begin {gather*} -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 \text {ArcTan}\left (\frac {x^2}{a^2}\right )}{256 a^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 209
Rule 281
Rule 294
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^6}{\left (a^4+x^2\right )^5} \, dx,x,x^2\right )\\ &=-\frac {x^{10}}{16 \left (a^4+x^4\right )^4}+\frac {5}{16} \text {Subst}\left (\int \frac {x^4}{\left (a^4+x^2\right )^4} \, dx,x,x^2\right )\\ &=-\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}+\frac {5}{32} \text {Subst}\left (\int \frac {x^2}{\left (a^4+x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5}{128} \text {Subst}\left (\int \frac {1}{\left (a^4+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}+\frac {5 \text {Subst}\left (\int \frac {1}{a^4+x^2} \, dx,x,x^2\right )}{256 a^4}\\ &=-\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}+\frac {5 \tan ^{-1}\left (\frac {x^2}{a^2}\right )}{256 a^6}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 62, normalized size = 0.75 \begin {gather*} \frac {-\frac {a^2 x^2 \left (15 a^{12}+55 a^8 x^4+73 a^4 x^8-15 x^{12}\right )}{\left (a^4+x^4\right )^4}+15 \tan ^{-1}\left (\frac {x^2}{a^2}\right )}{768 a^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 56, normalized size = 0.67
method | result | size |
risch | \(\frac {-\frac {5 a^{8} x^{2}}{256}-\frac {55 a^{4} x^{6}}{768}-\frac {73 x^{10}}{768}+\frac {5 x^{14}}{256 a^{4}}}{\left (a^{4}+x^{4}\right )^{4}}+\frac {5 \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 a^{6}}\) | \(55\) |
default | \(\frac {\frac {5 x^{14}}{128 a^{4}}-\frac {73 x^{10}}{384}-\frac {55 a^{4} x^{6}}{384}-\frac {5 a^{8} x^{2}}{128}}{2 \left (a^{4}+x^{4}\right )^{4}}+\frac {5 \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 a^{6}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.37, size = 83, normalized size = 1.00 \begin {gather*} -\frac {15 \, a^{12} x^{2} + 55 \, a^{8} x^{6} + 73 \, a^{4} x^{10} - 15 \, x^{14}}{768 \, {\left (a^{20} + 4 \, a^{16} x^{4} + 6 \, a^{12} x^{8} + 4 \, a^{8} x^{12} + a^{4} x^{16}\right )}} + \frac {5 \, \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 \, a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 113, normalized size = 1.36 \begin {gather*} -\frac {15 \, a^{14} x^{2} + 55 \, a^{10} x^{6} + 73 \, a^{6} x^{10} - 15 \, a^{2} x^{14} - 15 \, {\left (a^{16} + 4 \, a^{12} x^{4} + 6 \, a^{8} x^{8} + 4 \, a^{4} x^{12} + x^{16}\right )} \arctan \left (\frac {x^{2}}{a^{2}}\right )}{768 \, {\left (a^{22} + 4 \, a^{18} x^{4} + 6 \, a^{14} x^{8} + 4 \, a^{10} x^{12} + a^{6} x^{16}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.32, size = 102, normalized size = 1.23 \begin {gather*} \frac {- 15 a^{12} x^{2} - 55 a^{8} x^{6} - 73 a^{4} x^{10} + 15 x^{14}}{768 a^{20} + 3072 a^{16} x^{4} + 4608 a^{12} x^{8} + 3072 a^{8} x^{12} + 768 a^{4} x^{16}} + \frac {- \frac {5 i \log {\left (- i a^{2} + x^{2} \right )}}{512} + \frac {5 i \log {\left (i a^{2} + x^{2} \right )}}{512}}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.37, size = 58, normalized size = 0.70 \begin {gather*} \frac {5 \, \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 \, a^{6}} - \frac {15 \, a^{12} x^{2} + 55 \, a^{8} x^{6} + 73 \, a^{4} x^{10} - 15 \, x^{14}}{768 \, {\left (a^{4} + x^{4}\right )}^{4} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 79, normalized size = 0.95 \begin {gather*} \frac {5\,\mathrm {atan}\left (\frac {x^2}{a^2}\right )}{256\,a^6}-\frac {\frac {73\,x^{10}}{768}+\frac {55\,a^4\,x^6}{768}+\frac {5\,a^8\,x^2}{256}-\frac {5\,x^{14}}{256\,a^4}}{a^{16}+4\,a^{12}\,x^4+6\,a^8\,x^8+4\,a^4\,x^{12}+x^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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