Optimal. Leaf size=25 \[ \frac{a \log (\sinh (c+d x))}{d}+\frac{b \log (\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.0412611, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3625, 3475} \[ \frac{a \log (\sinh (c+d x))}{d}+\frac{b \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3625
Rule 3475
Rubi steps
\begin{align*} \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=a \int \coth (c+d x) \, dx+b \int \tanh (c+d x) \, dx\\ &=\frac{b \log (\cosh (c+d x))}{d}+\frac{a \log (\sinh (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0396557, size = 33, normalized size = 1.32 \[ \frac{a (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{d}+\frac{b \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 26, normalized size = 1. \begin{align*}{\frac{b\ln \left ( \cosh \left ( dx+c \right ) \right ) }{d}}+{\frac{a\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10261, size = 47, normalized size = 1.88 \begin{align*} \frac{b \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{d} + \frac{a \log \left (\sinh \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02863, size = 178, normalized size = 7.12 \begin{align*} -\frac{{\left (a + b\right )} d x - b \log \left (\frac{2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - a \log \left (\frac{2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{d} \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 ) \coth{\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 = 1.17575, size = 66, normalized size = 2.64 \begin{align*} -\frac{{\left (d x + c\right )}{\left (a + b\right )}}{d} + \frac{b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} + \frac{a \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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