Optimal. Leaf size=27 \[ \frac{\sinh (a+2 b x+c)}{4 b}-\frac{1}{2} x \cosh (a-c) \]
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Rubi [A] time = 0.0266107, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5613, 2637} \[ \frac{\sinh (a+2 b x+c)}{4 b}-\frac{1}{2} x \cosh (a-c) \]
Antiderivative was successfully verified.
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Rule 5613
Rule 2637
Rubi steps
\begin{align*} \int \sinh (a+b x) \sinh (c+b x) \, dx &=\int \left (-\frac{1}{2} \cosh (a-c)+\frac{1}{2} \cosh (a+c+2 b x)\right ) \, dx\\ &=-\frac{1}{2} x \cosh (a-c)+\frac{1}{2} \int \cosh (a+c+2 b x) \, dx\\ &=-\frac{1}{2} x \cosh (a-c)+\frac{\sinh (a+c+2 b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.033118, size = 26, normalized size = 0.96 \[ \frac{\sinh (a+2 b x+c)-2 b x \cosh (a-c)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 24, normalized size = 0.9 \begin{align*} -{\frac{x\cosh \left ( a-c \right ) }{2}}+{\frac{\sinh \left ( 2\,bx+a+c \right ) }{4\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1784, size = 78, normalized size = 2.89 \begin{align*} -\frac{{\left (b x + a\right )}{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{4 \, b} + \frac{e^{\left (2 \, b x + a + c\right )}}{8 \, b} - \frac{e^{\left (-2 \, b x - a - c\right )}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9213, size = 232, normalized size = 8.59 \begin{align*} -\frac{2 \, b x \cosh \left (-a + c\right ) - 2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (b x + c\right ) + \cosh \left (b x + c\right )^{2} \sinh \left (-a + c\right ) + \sinh \left (b x + c\right )^{2} \sinh \left (-a + c\right )}{4 \,{\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.09259, size = 58, normalized size = 2.15 \begin{align*} \begin{cases} \frac{x \sinh{\left (a + b x \right )} \sinh{\left (b x + c \right )}}{2} - \frac{x \cosh{\left (a + b x \right )} \cosh{\left (b x + c \right )}}{2} + \frac{\sinh{\left (a + b x \right )} \cosh{\left (b x + c \right )}}{2 b} & \text{for}\: b \neq 0 \\x \sinh{\left (a \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 = 1.18586, size = 96, normalized size = 3.56 \begin{align*} -\frac{2 \, b x{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} -{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (2 \, b x + 2 \, c\right )} - 1\right )} e^{\left (-2 \, b x - a - c\right )} - e^{\left (2 \, b x + a + c\right )}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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