Optimal. Leaf size=13 \[ x-\frac{\coth (a+b x)}{b} \]
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Rubi [A] time = 0.0081182, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ x-\frac{\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \coth ^2(a+b x) \, dx &=-\frac{\coth (a+b x)}{b}+\int 1 \, dx\\ &=x-\frac{\coth (a+b x)}{b}\\ \end{align*}
Mathematica [C] time = 0.0091073, size = 27, normalized size = 2.08 \[ -\frac{\coth (a+b x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 41, normalized size = 3.2 \begin{align*} -{\frac{{\rm coth} \left (bx+a\right )}{b}}-{\frac{\ln \left ({\rm coth} \left (bx+a\right )-1 \right ) }{2\,b}}+{\frac{\ln \left ({\rm coth} \left (bx+a\right )+1 \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04407, size = 34, normalized size = 2.62 \begin{align*} x + \frac{a}{b} + \frac{2}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.27709, size = 82, normalized size = 6.31 \begin{align*} \frac{{\left (b x + 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \sinh \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.59867, size = 36, normalized size = 2.77 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: a = \log{\left (- e^{- b x} \right )} \vee a = \log{\left (e^{- b x} \right )} \\x \coth ^{2}{\left (a \right )} & \text{for}\: b = 0 \\x - \frac{1}{b \tanh{\left (a + b x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23787, size = 38, normalized size = 2.92 \begin{align*} \frac{b x + a}{b} - \frac{2}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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