Optimal. Leaf size=37 \[ -\frac {1}{4} i \text {PolyLog}\left (2,-\frac {i}{a x^2}\right )+\frac {1}{4} i \text {PolyLog}\left (2,\frac {i}{a x^2}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4945, 4941,
2438} \begin {gather*} \frac {1}{4} i \text {Li}_2\left (\frac {i}{a x^2}\right )-\frac {1}{4} i \text {Li}_2\left (-\frac {i}{a x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2438
Rule 4941
Rule 4945
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\cot ^{-1}(a x)}{x} \, dx,x,x^2\right )\\ &=\frac {1}{4} i \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{a x}\right )}{x} \, dx,x,x^2\right )-\frac {1}{4} i \text {Subst}\left (\int \frac {\log \left (1+\frac {i}{a x}\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {1}{4} i \text {Li}_2\left (-\frac {i}{a x^2}\right )+\frac {1}{4} i \text {Li}_2\left (\frac {i}{a x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.00 \begin {gather*} -\frac {1}{4} i \text {PolyLog}\left (2,-\frac {i}{a x^2}\right )+\frac {1}{4} i \text {PolyLog}\left (2,\frac {i}{a x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.08, size = 57, normalized size = 1.54
method | result | size |
default | \(\ln \left (x \right ) \mathrm {arccot}\left (a \,x^{2}\right )+\frac {\munderset {\textit {\_R1} =\RootOf \left (a^{2} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}}}{2 a}\) | \(57\) |
risch | \(\frac {\pi \ln \left (x \right )}{2}+\frac {i \ln \left (1+i x \sqrt {-i a}\right ) \ln \left (x \right )}{2}+\frac {i \ln \left (1-i x \sqrt {-i a}\right ) \ln \left (x \right )}{2}+\frac {i \dilog \left (1-i x \sqrt {-i a}\right )}{2}+\frac {i \dilog \left (1+i x \sqrt {-i a}\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (-i a \,x^{2}+1\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (1+i x \sqrt {i a}\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (1-i x \sqrt {i a}\right )}{2}-\frac {i \dilog \left (1+i x \sqrt {i a}\right )}{2}-\frac {i \dilog \left (1-i x \sqrt {i a}\right )}{2}+\frac {i \ln \left (i a \,x^{2}+1\right ) \ln \left (x \right )}{2}\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 68 vs. \(2 (23) = 46\).
time = 0.49, size = 68, normalized size = 1.84 \begin {gather*} \frac {1}{8} \, \pi \log \left (a^{2} x^{4} + 1\right ) - \frac {1}{2} \, \arctan \left (a x^{2}\right ) \log \left (a x^{2}\right ) + \operatorname {arccot}\left (a x^{2}\right ) \log \left (x\right ) + \arctan \left (a x^{2}\right ) \log \left (x\right ) + \frac {1}{4} i \, {\rm Li}_2\left (i \, a x^{2} + 1\right ) - \frac {1}{4} i \, {\rm Li}_2\left (-i \, a x^{2} + 1\right ) \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}{\left (a x^{2} \right )}}{x}\, 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 = -0.03 \begin {gather*} \int \frac {\mathrm {acot}\left (a\,x^2\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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