Optimal. Leaf size=47 \[ \frac{(a+b) \cosh ^3(c+d x)}{3 d}-\frac{(a+2 b) \cosh (c+d x)}{d}-\frac{b \text{sech}(c+d x)}{d} \]
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Rubi [A] time = 0.0546025, antiderivative size = 47, 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 = {3664, 448} \[ \frac{(a+b) \cosh ^3(c+d x)}{3 d}-\frac{(a+2 b) \cosh (c+d x)}{d}-\frac{b \text{sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 448
Rubi steps
\begin{align*} \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right ) \left (a+b-b x^2\right )}{x^4} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b+\frac{-a-b}{x^4}+\frac{a+2 b}{x^2}\right ) \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{(a+2 b) \cosh (c+d x)}{d}+\frac{(a+b) \cosh ^3(c+d x)}{3 d}-\frac{b \text{sech}(c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0503273, size = 73, normalized size = 1.55 \[ -\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}-\frac{7 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d}-\frac{b \text{sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 73, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) +b \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{3\,\cosh \left ( dx+c \right ) }}+{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3\,\cosh \left ( dx+c \right ) }}-{\frac{8\,\cosh \left ( dx+c \right ) }{3}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15218, size = 184, normalized size = 3.91 \begin{align*} -\frac{1}{24} \, b{\left (\frac{21 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{20 \, e^{\left (-2 \, d x - 2 \, c\right )} + 69 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d{\left (e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \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 [B] time = 1.99928, size = 246, normalized size = 5.23 \begin{align*} \frac{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} +{\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 4 \,{\left (2 \, a + 5 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (a + b\right )} \cosh \left (d x + c\right )^{2} - 4 \, a - 10 \, b\right )} \sinh \left (d x + c\right )^{2} - 9 \, a - 45 \, b}{24 \, 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 ^{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 = 1.30233, size = 162, normalized size = 3.45 \begin{align*} -\frac{{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 21 \, 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 + 24 \, c\right )} + b e^{\left (3 \, d x + 24 \, c\right )} - 9 \, a e^{\left (d x + 22 \, c\right )} - 21 \, b e^{\left (d x + 22 \, c\right )}\right )} e^{\left (-21 \, c\right )} + \frac{48 \, b e^{\left (d x + c\right )}}{e^{\left (2 \, d x + 2 \, c\right )} + 1}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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