Optimal. Leaf size=35 \[ \frac{\sinh ((m+1) x)}{2 (m+1)}-\frac{\sinh ((1-m) x)}{2 (1-m)} \]
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Rubi [A] time = 0.0319325, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5613, 2637} \[ \frac{\sinh ((m+1) x)}{2 (m+1)}-\frac{\sinh ((1-m) x)}{2 (1-m)} \]
Antiderivative was successfully verified.
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Rule 5613
Rule 2637
Rubi steps
\begin{align*} \int \sinh (x) \sinh (m x) \, dx &=\int \left (-\frac{1}{2} \cosh ((1-m) x)+\frac{1}{2} \cosh ((1+m) x)\right ) \, dx\\ &=-\left (\frac{1}{2} \int \cosh ((1-m) x) \, dx\right )+\frac{1}{2} \int \cosh ((1+m) x) \, dx\\ &=-\frac{\sinh ((1-m) x)}{2 (1-m)}+\frac{\sinh ((1+m) x)}{2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0406219, size = 25, normalized size = 0.71 \[ \frac{m \sinh (x) \cosh (m x)-\cosh (x) \sinh (m x)}{m^2-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 28, normalized size = 0.8 \begin{align*} -{\frac{\sinh \left ( \left ( -1+m \right ) x \right ) }{-2+2\,m}}+{\frac{\sinh \left ( \left ( 1+m \right ) x \right ) }{2+2\,m}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19077, size = 117, normalized size = 3.34 \begin{align*} \frac{m \cosh \left (m x\right ) \sinh \left (x\right ) - \cosh \left (x\right ) \sinh \left (m x\right )}{{\left (m^{2} - 1\right )} \cosh \left (x\right )^{2} -{\left (m^{2} - 1\right )} \sinh \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.09108, size = 78, normalized size = 2.23 \begin{align*} \begin{cases} - \frac{x \sinh ^{2}{\left (x \right )}}{2} + \frac{x \cosh ^{2}{\left (x \right )}}{2} - \frac{\sinh{\left (x \right )} \cosh{\left (x \right )}}{2} & \text{for}\: m = -1 \\\frac{x \sinh ^{2}{\left (x \right )}}{2} - \frac{x \cosh ^{2}{\left (x \right )}}{2} + \frac{\sinh{\left (x \right )} \cosh{\left (x \right )}}{2} & \text{for}\: m = 1 \\\frac{m \sinh{\left (x \right )} \cosh{\left (m x \right )}}{m^{2} - 1} - \frac{\sinh{\left (m x \right )} \cosh{\left (x \right )}}{m^{2} - 1} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15118, size = 80, normalized size = 2.29 \begin{align*} \frac{e^{\left (m x + x\right )}}{4 \,{\left (m + 1\right )}} - \frac{e^{\left (m x - x\right )}}{4 \,{\left (m - 1\right )}} + \frac{e^{\left (-m x + x\right )}}{4 \,{\left (m - 1\right )}} - \frac{e^{\left (-m x - x\right )}}{4 \,{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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