Optimal. Leaf size=27 \[ \frac{(a+b) \sinh (c+d x)}{d}-\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]
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Rubi [A] time = 0.0319431, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3676, 388, 203} \[ \frac{(a+b) \sinh (c+d x)}{d}-\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 388
Rule 203
Rubi steps
\begin{align*} \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+(a+b) x^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a+b) \sinh (c+d x)}{d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{b \tan ^{-1}(\sinh (c+d x))}{d}+\frac{(a+b) \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0289552, size = 47, normalized size = 1.74 \[ \frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d}+\frac{b \sinh (c+d x)}{d}-\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 37, normalized size = 1.4 \begin{align*}{\frac{a\sinh \left ( dx+c \right ) }{d}}+{\frac{b\sinh \left ( dx+c \right ) }{d}}-2\,{\frac{b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.60957, size = 74, normalized size = 2.74 \begin{align*} \frac{1}{2} \, b{\left (\frac{4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{a \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86466, size = 296, normalized size = 10.96 \begin{align*} \frac{{\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a + b\right )} \sinh \left (d x + c\right )^{2} - 4 \,{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - a - b}{2 \,{\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\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 ) \cosh{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.265, size = 76, normalized size = 2.81 \begin{align*} -\frac{4 \, b \arctan \left (e^{\left (d x + c\right )}\right ) +{\left (a + b\right )} e^{\left (-d x - c\right )} -{\left (a e^{\left (d x + 4 \, c\right )} + b e^{\left (d x + 4 \, c\right )}\right )} e^{\left (-3 \, c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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