Optimal. Leaf size=32 \[ \frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0289156, antiderivative size = 32, 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 = {3013} \[ \frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3013
Rubi steps
\begin{align*} \int \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a-b+b x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0240816, size = 53, normalized size = 1.66 \[ \frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}-\frac{3 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 34, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( b \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \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 = 1.0147, size = 90, normalized size = 2.81 \begin{align*} \frac{1}{24} \, b{\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 )} + \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 [A] time = 1.87512, size = 127, normalized size = 3.97 \begin{align*} \frac{b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \,{\left (4 \, a - 3 \, b\right )} \cosh \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.768842, size = 56, normalized size = 1.75 \begin{align*} \begin{cases} \frac{a \cosh{\left (c + d x \right )}}{d} + \frac{b \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 b \cosh ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\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 = 1.26747, size = 96, normalized size = 3. \begin{align*} \frac{b e^{\left (3 \, d x + 3 \, c\right )} + 12 \, a e^{\left (d x + c\right )} - 9 \, b e^{\left (d x + c\right )} +{\left (12 \, a e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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