Optimal. Leaf size=18 \[ x (a+b)-\frac{a \coth (c+d x)}{d} \]
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Rubi [A] time = 0.0290586, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3629, 8} \[ x (a+b)-\frac{a \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3629
Rule 8
Rubi steps
\begin{align*} \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=-\frac{a \coth (c+d x)}{d}-\int (-a-b) \, dx\\ &=(a+b) x-\frac{a \coth (c+d x)}{d}\\ \end{align*}
Mathematica [C] time = 0.0259963, size = 32, normalized size = 1.78 \[ b x-\frac{a \coth (c+d x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 28, normalized size = 1.6 \begin{align*}{\frac{a \left ( dx+c-{\rm coth} \left (dx+c\right ) \right ) + \left ( dx+c \right ) b}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04148, size = 42, normalized size = 2.33 \begin{align*} a{\left (x + \frac{c}{d} + \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9345, size = 97, normalized size = 5.39 \begin{align*} -\frac{a \cosh \left (d x + c\right ) -{\left ({\left (a + b\right )} d x + a\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 60.3678, size = 49, normalized size = 2.72 \begin{align*} a \left (\begin{cases} x \coth ^{2}{\left (c \right )} & \text{for}\: d = 0 \\\tilde{\infty } x & \text{for}\: c = \log{\left (- e^{- d x} \right )} \vee c = \log{\left (e^{- d x} \right )} \\x - \frac{1}{d \tanh{\left (c + d x \right )}} & \text{otherwise} \end{cases}\right ) + b \left (\begin{cases} x & \text{for}\: \left |{x}\right | < 1 \\{G_{2, 2}^{1, 1}\left (\begin{matrix} 1 & 2 \\1 & 0 \end{matrix} \middle |{x} \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 2, 1 & \\ & 1, 0 \end{matrix} \middle |{x} \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21697, size = 43, normalized size = 2.39 \begin{align*} \frac{{\left (d x + c\right )}{\left (a + b\right )}}{d} - \frac{2 \, a}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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