Optimal. Leaf size=46 \[ \frac{(a+b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d}-\frac{2 b \cosh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0342356, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3215} \[ \frac{(a+b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d}-\frac{2 b \cosh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3215
Rubi steps
\begin{align*} \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b-2 b x^2+b x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{(a+b) \cosh (c+d x)}{d}-\frac{2 b \cosh ^3(c+d x)}{3 d}+\frac{b \cosh ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.0223954, size = 69, normalized size = 1.5 \[ \frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}+\frac{5 b \cosh (c+d x)}{8 d}-\frac{5 b \cosh (3 (c+d x))}{48 d}+\frac{b \cosh (5 (c+d x))}{80 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 44, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( dx+c \right ) +a\cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05633, size = 131, normalized size = 2.85 \begin{align*} \frac{1}{480} \, b{\left (\frac{3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac{25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{150 \, e^{\left (d x + c\right )}}{d} + \frac{150 \, e^{\left (-d x - c\right )}}{d} - \frac{25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac{a \cosh \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 = 1.66974, size = 250, normalized size = 5.43 \begin{align*} \frac{3 \, b \cosh \left (d x + c\right )^{5} + 15 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - 25 \, b \cosh \left (d x + c\right )^{3} + 15 \,{\left (2 \, b \cosh \left (d x + c\right )^{3} - 5 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 30 \,{\left (8 \, a + 5 \, b\right )} \cosh \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.53825, size = 80, normalized size = 1.74 \begin{align*} \begin{cases} \frac{a \cosh{\left (c + d x \right )}}{d} + \frac{b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{8 b \cosh ^{5}{\left (c + d x \right )}}{15 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right ) \sinh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18581, size = 132, normalized size = 2.87 \begin{align*} \frac{3 \, b e^{\left (5 \, d x + 5 \, c\right )} - 25 \, b e^{\left (3 \, d x + 3 \, c\right )} + 240 \, a e^{\left (d x + c\right )} + 150 \, b e^{\left (d x + c\right )} +{\left (240 \, a e^{\left (4 \, d x + 4 \, c\right )} + 150 \, b e^{\left (4 \, d x + 4 \, c\right )} - 25 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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