Optimal. Leaf size=61 \[ \frac{(4 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{1}{8} x (4 a-3 b)+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d} \]
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Rubi [A] time = 0.0428877, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3014, 2635, 8} \[ \frac{(4 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{1}{8} x (4 a-3 b)+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3014
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac{b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{1}{4} (-4 a+3 b) \int \sinh ^2(c+d x) \, dx\\ &=\frac{(4 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{1}{8} (4 a-3 b) \int 1 \, dx\\ &=-\frac{1}{8} (4 a-3 b) x+\frac{(4 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0854699, size = 47, normalized size = 0.77 \[ \frac{-4 (4 a-3 b) (c+d x)+8 (a-b) \sinh (2 (c+d x))+b \sinh (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 66, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( b \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +a \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05928, size = 131, normalized size = 2.15 \begin{align*} \frac{1}{64} \, b{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90866, size = 163, normalized size = 2.67 \begin{align*} \frac{b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} -{\left (4 \, a - 3 \, b\right )} d x +{\left (b \cosh \left (d x + c\right )^{3} + 4 \,{\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.42794, size = 158, normalized size = 2.59 \begin{align*} \begin{cases} \frac{a x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac{a x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{a \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} + \frac{3 b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{5 b \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{3 b \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \sinh ^{2}{\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 = 1.36898, size = 161, normalized size = 2.64 \begin{align*} -\frac{8 \,{\left (d x + c\right )}{\left (4 \, a - 3 \, b\right )} - b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} -{\left (24 \, a e^{\left (4 \, d x + 4 \, c\right )} - 18 \, b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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