Optimal. Leaf size=41 \[ \frac{e^x \sinh (a+b x)}{1-b^2}-\frac{b e^x \cosh (a+b x)}{1-b^2} \]
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Rubi [A] time = 0.0132719, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5474} \[ \frac{e^x \sinh (a+b x)}{1-b^2}-\frac{b e^x \cosh (a+b x)}{1-b^2} \]
Antiderivative was successfully verified.
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Rule 5474
Rubi steps
\begin{align*} \int e^x \sinh (a+b x) \, dx &=-\frac{b e^x \cosh (a+b x)}{1-b^2}+\frac{e^x \sinh (a+b x)}{1-b^2}\\ \end{align*}
Mathematica [A] time = 0.0479623, size = 28, normalized size = 0.68 \[ \frac{e^x (b \cosh (a+b x)-\sinh (a+b x))}{b^2-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 62, normalized size = 1.5 \begin{align*} -{\frac{\sinh \left ( \left ( b-1 \right ) x+a \right ) }{2\,b-2}}+{\frac{\sinh \left ( \left ( 1+b \right ) x+a \right ) }{2+2\,b}}+{\frac{\cosh \left ( \left ( b-1 \right ) x+a \right ) }{2\,b-2}}+{\frac{\cosh \left ( \left ( 1+b \right ) x+a \right ) }{2+2\,b}} \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 = 1.74592, size = 134, normalized size = 3.27 \begin{align*} \frac{b \cosh \left (b x + a\right ) \cosh \left (x\right ) + b \cosh \left (b x + a\right ) \sinh \left (x\right ) -{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \sinh \left (b x + a\right )}{b^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.842607, size = 99, normalized size = 2.41 \begin{align*} \begin{cases} \frac{x e^{x} \sinh{\left (a - x \right )}}{2} + \frac{x e^{x} \cosh{\left (a - x \right )}}{2} + \frac{e^{x} \sinh{\left (a - x \right )}}{2} & \text{for}\: b = -1 \\\frac{x e^{x} \sinh{\left (a + x \right )}}{2} - \frac{x e^{x} \cosh{\left (a + x \right )}}{2} + \frac{e^{x} \cosh{\left (a + x \right )}}{2} & \text{for}\: b = 1 \\\frac{b e^{x} \cosh{\left (a + b x \right )}}{b^{2} - 1} - \frac{e^{x} \sinh{\left (a + b x \right )}}{b^{2} - 1} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16365, size = 43, normalized size = 1.05 \begin{align*} \frac{e^{\left (b x + a + x\right )}}{2 \,{\left (b + 1\right )}} + \frac{e^{\left (-b x - a + x\right )}}{2 \,{\left (b - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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