Optimal. Leaf size=24 \[ \frac{b \tanh (c+d x)}{d}-\frac{a \coth (c+d x)}{d} \]
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Rubi [A] time = 0.0327757, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3663, 14} \[ \frac{b \tanh (c+d x)}{d}-\frac{a \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 14
Rubi steps
\begin{align*} \int \text{csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b x^2}{x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (b+\frac{a}{x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{a \coth (c+d x)}{d}+\frac{b \tanh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0197248, size = 24, normalized size = 1. \[ \frac{b \tanh (c+d x)}{d}-\frac{a \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 23, normalized size = 1. \begin{align*}{\frac{-{\rm coth} \left (dx+c\right )a+b\tanh \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14149, size = 53, normalized size = 2.21 \begin{align*} \frac{2 \, b}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} + \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 = 1.8861, size = 238, normalized size = 9.92 \begin{align*} -\frac{4 \,{\left (a \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )}}{d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right ) +{\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \operatorname{csch}^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26439, size = 61, normalized size = 2.54 \begin{align*} -\frac{2 \,{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b\right )}}{d{\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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