Optimal. Leaf size=21 \[ \frac {1}{2 x^2}-e^a \tanh ^{-1}\left (e^a x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5657, 464, 281,
212} \begin {gather*} \frac {1}{2 x^2}-e^a \tanh ^{-1}\left (e^a x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 281
Rule 464
Rule 5657
Rubi steps
\begin {align*} \int \frac {\coth (a+2 \log (x))}{x^3} \, dx &=\int \frac {\coth (a+2 \log (x))}{x^3} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 27, normalized size = 1.29 \begin {gather*} \frac {1}{2 x^2}-\tanh ^{-1}\left (\frac {\cosh (a)-\sinh (a)}{x^2}\right ) (\cosh (a)+\sinh (a)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 35, normalized size = 1.67
method | result | size |
risch | \(\frac {1}{2 x^{2}}+\frac {{\mathrm e}^{a} \ln \left (-{\mathrm e}^{a} x^{2}+1\right )}{2}-\frac {{\mathrm e}^{a} \ln \left (-{\mathrm e}^{a} x^{2}-1\right )}{2}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 30, normalized size = 1.43 \begin {gather*} -\frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} - e^{a}\right ) + \frac {1}{2 \, x^{2}} \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. 38 vs.
\(2 (17) = 34\).
time = 0.34, size = 38, normalized size = 1.81 \begin {gather*} -\frac {x^{2} e^{a} \log \left (x^{2} e^{a} + 1\right ) - x^{2} e^{a} \log \left (x^{2} e^{a} - 1\right ) - 1}{2 \, x^{2}} \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 {\coth {\left (a + 2 \log {\left (x \right )} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 33, normalized size = 1.57 \begin {gather*} -\frac {1}{2} \, e^{a} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{a} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) + \frac {1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 25, normalized size = 1.19 \begin {gather*} \frac {1}{2\,x^2}-\mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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