Optimal. Leaf size=25 \[ \frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x) \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5911, 260} \[ \frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 260
Rule 5911
Rubi steps
\begin {align*} \int \coth ^{-1}(a x) \, dx &=x \coth ^{-1}(a x)-a \int \frac {x}{1-a^2 x^2} \, dx\\ &=x \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 1.00 \[ \frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 33, normalized size = 1.32 \[ \frac {a x \log \left (\frac {a x + 1}{a x - 1}\right ) + \log \left (a^{2} x^{2} - 1\right )}{2 \, a} \]
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 \[ \int \operatorname {arcoth}\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 23, normalized size = 0.92 \[ x \,\mathrm {arccoth}\left (a x \right )+\frac {\ln \left (a^{2} x^{2}-1\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 25, normalized size = 1.00 \[ \frac {2 \, a x \operatorname {arcoth}\left (a x\right ) + \log \left (-a^{2} x^{2} + 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 22, normalized size = 0.88 \[ x\,\mathrm {acoth}\left (a\,x\right )+\frac {\ln \left (a^2\,x^2-1\right )}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 27, normalized size = 1.08 \[ \begin {cases} x \operatorname {acoth}{\left (a x \right )} + \frac {\log {\left (a x + 1 \right )}}{a} - \frac {\operatorname {acoth}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\\frac {i \pi x}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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