Optimal. Leaf size=11 \[ \frac {\log (\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3475} \[ \frac {\log (\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin {align*} \int \coth (a+b x) \, dx &=\frac {\log (\sinh (a+b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.73 \[ \frac {\log (\tanh (a+b x))+\log (\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 37, normalized size = 3.36 \[ -\frac {b x - \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 25, normalized size = 2.27 \[ -\frac {b x + a - \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 30, normalized size = 2.73 \[ -\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2 b}-\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 11, normalized size = 1.00 \[ \frac {\log \left (\sinh \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 21, normalized size = 1.91 \[ \frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 27, normalized size = 2.45 \[ \begin {cases} x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} + \frac {\log {\left (\tanh {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\x \coth {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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