Optimal. Leaf size=68 \[ \frac{\sinh (2 a+x (2 b-d)-c)}{4 (2 b-d)}+\frac{\sinh (2 a+x (2 b+d)+c)}{4 (2 b+d)}-\frac{\sinh (c+d x)}{2 d} \]
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Rubi [A] time = 0.0536795, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5618, 2637} \[ \frac{\sinh (2 a+x (2 b-d)-c)}{4 (2 b-d)}+\frac{\sinh (2 a+x (2 b+d)+c)}{4 (2 b+d)}-\frac{\sinh (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 5618
Rule 2637
Rubi steps
\begin{align*} \int \cosh (c+d x) \sinh ^2(a+b x) \, dx &=\int \left (\frac{1}{4} \cosh (2 a-c+(2 b-d) x)-\frac{1}{2} \cosh (c+d x)+\frac{1}{4} \cosh (2 a+c+(2 b+d) x)\right ) \, dx\\ &=\frac{1}{4} \int \cosh (2 a-c+(2 b-d) x) \, dx+\frac{1}{4} \int \cosh (2 a+c+(2 b+d) x) \, dx-\frac{1}{2} \int \cosh (c+d x) \, dx\\ &=\frac{\sinh (2 a-c+(2 b-d) x)}{4 (2 b-d)}-\frac{\sinh (c+d x)}{2 d}+\frac{\sinh (2 a+c+(2 b+d) x)}{4 (2 b+d)}\\ \end{align*}
Mathematica [A] time = 0.720775, size = 74, normalized size = 1.09 \[ \frac{1}{4} \left (\frac{\sinh (2 a+2 b x-c-d x)}{2 b-d}+\frac{\sinh (2 a+2 b x+c+d x)}{2 b+d}-\frac{2 \sinh (c) \cosh (d x)}{d}-\frac{2 \cosh (c) \sinh (d x)}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 63, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{8\,b-4\,d}}-{\frac{\sinh \left ( dx+c \right ) }{2\,d}}+{\frac{\sinh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{8\,b+4\,d}} \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 = 1.83368, size = 266, normalized size = 3.91 \begin{align*} \frac{4 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) -{\left (d^{2} \cosh \left (b x + a\right )^{2} + d^{2} \sinh \left (b x + a\right )^{2} + 4 \, b^{2} - d^{2}\right )} \sinh \left (d x + c\right )}{2 \,{\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2} -{\left (4 \, b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.19713, size = 405, normalized size = 5.96 \begin{align*} \begin{cases} x \sinh ^{2}{\left (a \right )} \cosh{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \sinh ^{2}{\left (a - \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{4} + \frac{x \sinh{\left (a - \frac{d x}{2} \right )} \sinh{\left (c + d x \right )} \cosh{\left (a - \frac{d x}{2} \right )}}{2} + \frac{x \cosh ^{2}{\left (a - \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{4} + \frac{\sinh ^{2}{\left (a - \frac{d x}{2} \right )} \sinh{\left (c + d x \right )}}{d} + \frac{\sinh{\left (a - \frac{d x}{2} \right )} \cosh{\left (a - \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: b = - \frac{d}{2} \\\frac{x \sinh ^{2}{\left (a + \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{4} - \frac{x \sinh{\left (a + \frac{d x}{2} \right )} \sinh{\left (c + d x \right )} \cosh{\left (a + \frac{d x}{2} \right )}}{2} + \frac{x \cosh ^{2}{\left (a + \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{4} + \frac{3 \sinh ^{2}{\left (a + \frac{d x}{2} \right )} \sinh{\left (c + d x \right )}}{4 d} - \frac{\sinh{\left (c + d x \right )} \cosh ^{2}{\left (a + \frac{d x}{2} \right )}}{4 d} & \text{for}\: b = \frac{d}{2} \\\left (\frac{x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac{x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac{\sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b}\right ) \cosh{\left (c \right )} & \text{for}\: d = 0 \\\frac{2 b^{2} \sinh ^{2}{\left (a + b x \right )} \sinh{\left (c + d x \right )}}{4 b^{2} d - d^{3}} - \frac{2 b^{2} \sinh{\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b^{2} d - d^{3}} + \frac{2 b d \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )} \cosh{\left (c + d x \right )}}{4 b^{2} d - d^{3}} - \frac{d^{2} \sinh ^{2}{\left (a + b x \right )} \sinh{\left (c + d x \right )}}{4 b^{2} d - d^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16759, size = 167, normalized size = 2.46 \begin{align*} \frac{e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \,{\left (2 \, b + d\right )}} + \frac{e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{8 \,{\left (2 \, b - d\right )}} - \frac{e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \,{\left (2 \, b - d\right )}} - \frac{e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{8 \,{\left (2 \, b + d\right )}} - \frac{e^{\left (d x + c\right )}}{4 \, d} + \frac{e^{\left (-d x - c\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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