Optimal. Leaf size=16 \[ b x-\frac{a \coth (c+d x)}{d} \]
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Rubi [A] time = 0.0266243, antiderivative size = 16, 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 = {3012, 8} \[ b x-\frac{a \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3012
Rule 8
Rubi steps
\begin{align*} \int \text{csch}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=-\frac{a \coth (c+d x)}{d}+b \int 1 \, dx\\ &=b x-\frac{a \coth (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0195822, size = 16, normalized size = 1. \[ b x-\frac{a \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 22, normalized size = 1.4 \begin{align*}{\frac{-{\rm coth} \left (dx+c\right )a+ \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 = 1.02413, size = 31, normalized size = 1.94 \begin{align*} b x + \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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Fricas [B] time = 1.8503, size = 89, normalized size = 5.56 \begin{align*} -\frac{a \cosh \left (d x + c\right ) -{\left (b 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29289, size = 41, normalized size = 2.56 \begin{align*} \frac{{\left (d x + c\right )} b}{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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