Optimal. Leaf size=30 \[ \frac{(a+b) \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.0355443, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3676} \[ \frac{(a+b) \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3676
Rubi steps
\begin{align*} \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+(a+b) x^2\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{a \sinh (c+d x)}{d}+\frac{(a+b) \sinh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0155267, size = 44, normalized size = 1.47 \[ \frac{a \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 53, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}\sinh \left ( dx+c \right ) }{3}}-{\frac{\sinh \left ( dx+c \right ) }{3}} \right ) +a \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0968, size = 112, normalized size = 3.73 \begin{align*} \frac{b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3}}{24 \, d} + \frac{1}{24} \, a{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} - \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81256, size = 119, normalized size = 3.97 \begin{align*} \frac{{\left (a + b\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a - b\right )} \sinh \left (d x + c\right )}{12 \, 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 ) \cosh ^{3}{\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 = 1.24649, size = 127, normalized size = 4.23 \begin{align*} -\frac{{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-3 \, d x - 3 \, c\right )} -{\left (a e^{\left (3 \, d x + 12 \, c\right )} + b e^{\left (3 \, d x + 12 \, c\right )} + 9 \, a e^{\left (d x + 10 \, c\right )} - 3 \, b e^{\left (d x + 10 \, c\right )}\right )} e^{\left (-9 \, c\right )}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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