Optimal. Leaf size=60 \[ \frac{a \cosh (c+d x)}{d}+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b x}{8} \]
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Rubi [A] time = 0.0684057, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3220, 2638, 2635, 8} \[ \frac{a \cosh (c+d x)}{d}+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b x}{8} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 2638
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=-\left (i \int \left (i a \sinh (c+d x)+i b \sinh ^4(c+d x)\right ) \, dx\right )\\ &=a \int \sinh (c+d x) \, dx+b \int \sinh ^4(c+d x) \, dx\\ &=\frac{a \cosh (c+d x)}{d}+\frac{b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{1}{4} (3 b) \int \sinh ^2(c+d x) \, dx\\ &=\frac{a \cosh (c+d x)}{d}-\frac{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} (3 b) \int 1 \, dx\\ &=\frac{3 b x}{8}+\frac{a \cosh (c+d x)}{d}-\frac{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 = 0.125911, size = 45, normalized size = 0.75 \[ \frac{32 a \cosh (c+d x)+b (12 (c+d x)-8 \sinh (2 (c+d x))+\sinh (4 (c+d x)))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 50, normalized size = 0.8 \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\cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15149, size = 100, normalized size = 1.67 \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{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 = 1.80175, size = 171, normalized size = 2.85 \begin{align*} \frac{b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 3 \, b d x + 8 \, a \cosh \left (d x + c\right ) +{\left (b \cosh \left (d x + c\right )^{3} - 4 \, b \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 = 1.24827, size = 121, normalized size = 2.02 \begin{align*} \begin{cases} \frac{a \cosh{\left (c + d x \right )}}{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 ^{3}{\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 [A] time = 1.17231, size = 113, normalized size = 1.88 \begin{align*} \frac{24 \,{\left (d x + c\right )} b + b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a e^{\left (d x + c\right )} +{\left (32 \, a e^{\left (3 \, d x + 3 \, 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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