Optimal. Leaf size=113 \[ -\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^3 \text {ArcTan}(a x)+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {1}{3} i a^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4947, 5039,
331, 209, 5045, 4989, 2497} \begin {gather*} -\frac {1}{3} a^3 \text {ArcTan}(a x)+\frac {1}{3} i a^3 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2+\frac {2}{3} a^3 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac {a^2}{3 x}-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {a \cot ^{-1}(a x)}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 331
Rule 2497
Rule 4947
Rule 4989
Rule 5039
Rule 5045
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx &=-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\cot ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\cot ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 i a^3\right ) \int \frac {\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=-\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{3} a^4 \int \frac {1}{1+a^2 x^2} \, dx+\frac {1}{3} \left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^3 \tan ^{-1}(a x)+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {1}{3} i a^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 96, normalized size = 0.85 \begin {gather*} \frac {-a^2 x^2+\left (-1-i a^3 x^3\right ) \cot ^{-1}(a x)^2+a x \cot ^{-1}(a x) \left (1+a^2 x^2+2 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )-i a^3 x^3 \text {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 253 vs. \(2 (95 ) = 190\).
time = 0.28, size = 254, normalized size = 2.25
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\mathrm {arccot}\left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {\mathrm {arccot}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \ln \left (a x \right ) \mathrm {arccot}\left (a x \right )}{3}-\frac {\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}-\frac {i \dilog \left (i a x +1\right )}{3}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3}-\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}-\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}+\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{6}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{3}+\frac {i \ln \left (a x +i\right )^{2}}{12}+\frac {i \dilog \left (-i a x +1\right )}{3}-\frac {\arctan \left (a x \right )}{3}-\frac {1}{3 a x}+\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}-\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}+\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{6}-\frac {i \ln \left (a x -i\right )^{2}}{12}\right )\) | \(254\) |
default | \(a^{3} \left (-\frac {\mathrm {arccot}\left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {\mathrm {arccot}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \ln \left (a x \right ) \mathrm {arccot}\left (a x \right )}{3}-\frac {\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}-\frac {i \dilog \left (i a x +1\right )}{3}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3}-\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}-\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}+\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{6}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{3}+\frac {i \ln \left (a x +i\right )^{2}}{12}+\frac {i \dilog \left (-i a x +1\right )}{3}-\frac {\arctan \left (a x \right )}{3}-\frac {1}{3 a x}+\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}-\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}+\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{6}-\frac {i \ln \left (a x -i\right )^{2}}{12}\right )\) | \(254\) |
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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