Optimal. Leaf size=44 \[ \frac{(a+b) \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{1}{2} x (a+3 b)+\frac{b \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.0507793, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3663, 455, 388, 206} \[ \frac{(a+b) \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{1}{2} x (a+3 b)+\frac{b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 455
Rule 388
Rule 206
Rubi steps
\begin{align*} \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b) \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{a+b+2 b x^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac{(a+b) \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b \tanh (c+d x)}{d}-\frac{(a+3 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac{1}{2} (a+3 b) x+\frac{(a+b) \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b \tanh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.189214, size = 41, normalized size = 0.93 \[ \frac{-2 (a+3 b) (c+d x)+(a+b) \sinh (2 (c+d x))+4 b \tanh (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 66, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) +b \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{2\,\cosh \left ( dx+c \right ) }}-{\frac{3\,dx}{2}}-{\frac{3\,c}{2}}+{\frac{3\,\tanh \left ( dx+c \right ) }{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12602, size = 136, normalized size = 3.09 \begin{align*} -\frac{1}{8} \, a{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{8} \, b{\left (\frac{12 \,{\left (d x + c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95945, size = 193, normalized size = 4.39 \begin{align*} \frac{{\left (a + b\right )} \sinh \left (d x + c\right )^{3} - 4 \,{\left ({\left (a + 3 \, b\right )} d x + 2 \, b\right )} \cosh \left (d x + c\right ) +{\left (3 \,{\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a + 9 \, b\right )} \sinh \left (d x + c\right )}{8 \, d \cosh \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 ) \sinh ^{2}{\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 = 1.2702, size = 147, normalized size = 3.34 \begin{align*} -\frac{4 \,{\left (a + 3 \, b\right )} d x -{\left (a e^{\left (2 \, d x + 8 \, c\right )} + b e^{\left (2 \, d x + 8 \, c\right )}\right )} e^{\left (-6 \, c\right )} - \frac{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 14 \, b e^{\left (2 \, d x + 2 \, c\right )} - a - b\right )} e^{\left (-2 \, c\right )}}{e^{\left (2 \, d x\right )} + e^{\left (4 \, d x + 2 \, c\right )}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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