Optimal. Leaf size=41 \[ \frac {a}{12 x^3}-\frac {a^3}{4 x}-\frac {\cot ^{-1}(a x)}{4 x^4}-\frac {1}{4} a^4 \text {ArcTan}(a x) \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4947, 331, 209}
\begin {gather*} -\frac {1}{4} a^4 \text {ArcTan}(a x)-\frac {a^3}{4 x}-\frac {\cot ^{-1}(a x)}{4 x^4}+\frac {a}{12 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 331
Rule 4947
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)}{x^5} \, dx &=-\frac {\cot ^{-1}(a x)}{4 x^4}-\frac {1}{4} a \int \frac {1}{x^4 \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a}{12 x^3}-\frac {\cot ^{-1}(a x)}{4 x^4}+\frac {1}{4} a^3 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a}{12 x^3}-\frac {a^3}{4 x}-\frac {\cot ^{-1}(a x)}{4 x^4}-\frac {1}{4} a^5 \int \frac {1}{1+a^2 x^2} \, dx\\ &=\frac {a}{12 x^3}-\frac {a^3}{4 x}-\frac {\cot ^{-1}(a x)}{4 x^4}-\frac {1}{4} a^4 \tan ^{-1}(a x)\\ \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 = 36, normalized size = 0.88 \begin {gather*} -\frac {\cot ^{-1}(a x)}{4 x^4}+\frac {a \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-a^2 x^2\right )}{12 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 40, normalized size = 0.98
method | result | size |
derivativedivides | \(a^{4} \left (-\frac {\mathrm {arccot}\left (a x \right )}{4 a^{4} x^{4}}+\frac {1}{12 a^{3} x^{3}}-\frac {1}{4 a x}-\frac {\arctan \left (a x \right )}{4}\right )\) | \(40\) |
default | \(a^{4} \left (-\frac {\mathrm {arccot}\left (a x \right )}{4 a^{4} x^{4}}+\frac {1}{12 a^{3} x^{3}}-\frac {1}{4 a x}-\frac {\arctan \left (a x \right )}{4}\right )\) | \(40\) |
risch | \(-\frac {i \ln \left (i a x +1\right )}{8 x^{4}}-\frac {3 i a^{4} \ln \left (-a x -i\right ) x^{4}-3 i a^{4} \ln \left (-a x +i\right ) x^{4}+6 a^{3} x^{3}-3 i \ln \left (-i a x +1\right )-2 a x +3 \pi }{24 x^{4}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 37, normalized size = 0.90 \begin {gather*} -\frac {1}{12} \, {\left (3 \, a^{3} \arctan \left (a x\right ) + \frac {3 \, a^{2} x^{2} - 1}{x^{3}}\right )} a - \frac {\operatorname {arccot}\left (a x\right )}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.28, size = 33, normalized size = 0.80 \begin {gather*} -\frac {3 \, a^{3} x^{3} - a x - 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 32, normalized size = 0.78 \begin {gather*} \frac {a^{4} \operatorname {acot}{\left (a x \right )}}{4} - \frac {a^{3}}{4 x} + \frac {a}{12 x^{3}} - \frac {\operatorname {acot}{\left (a x \right )}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 51, normalized size = 1.24 \begin {gather*} -\frac {1}{12} \, {\left (a^{3} {\left (\frac {3}{a x} - \frac {1}{a^{3} x^{3}} - 3 \, \arctan \left (\frac {1}{a x}\right )\right )} + \frac {3 \, \arctan \left (\frac {1}{a x}\right )}{a x^{4}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 47, normalized size = 1.15 \begin {gather*} \left \{\begin {array}{cl} -\frac {\pi }{8\,x^4} & \text {\ if\ \ }a=0\\ -\frac {a^4\,\mathrm {atan}\left (a\,x\right )}{4}-\frac {\frac {\mathrm {acot}\left (a\,x\right )}{4}-\frac {a\,x}{12}+\frac {a^3\,x^3}{4}}{x^4} & \text {\ if\ \ }a\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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