Optimal. Leaf size=41 \[ \frac {e^x \cosh (a+b x)}{1-b^2}-\frac {b e^x \sinh (a+b x)}{1-b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5475} \[ \frac {e^x \cosh (a+b x)}{1-b^2}-\frac {b e^x \sinh (a+b x)}{1-b^2} \]
Antiderivative was successfully verified.
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Rule 5475
Rubi steps
\begin {align*} \int e^x \cosh (a+b x) \, dx &=\frac {e^x \cosh (a+b x)}{1-b^2}-\frac {b e^x \sinh (a+b x)}{1-b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 28, normalized size = 0.68 \[ \frac {e^x (b \sinh (a+b x)-\cosh (a+b x))}{b^2-1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 45, normalized size = 1.10 \[ -\frac {\cosh \left (b x + a\right ) \cosh \relax (x) - {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right ) \sinh \relax (x)}{b^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 32, normalized size = 0.78 \[ \frac {e^{\left (b x + a + x\right )}}{2 \, {\left (b + 1\right )}} - \frac {e^{\left (-b x - a + x\right )}}{2 \, {\left (b - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 62, normalized size = 1.51 \[ \frac {\sinh \left (\left (b -1\right ) x +a \right )}{2 b -2}+\frac {\sinh \left (\left (1+b \right ) x +a \right )}{2+2 b}-\frac {\cosh \left (\left (b -1\right ) x +a \right )}{2 \left (b -1\right )}+\frac {\cosh \left (\left (1+b \right ) x +a \right )}{2+2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 45, normalized size = 1.10 \[ -\frac {{\mathrm {e}}^{x-a-b\,x}\,\left (b+{\mathrm {e}}^{2\,a+2\,b\,x}-b\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}{2\,\left (b^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 99, normalized size = 2.41 \[ \begin {cases} \frac {x e^{x} \sinh {\left (a - x \right )}}{2} + \frac {x e^{x} \cosh {\left (a - x \right )}}{2} - \frac {e^{x} \sinh {\left (a - x \right )}}{2} & \text {for}\: b = -1 \\- \frac {x e^{x} \sinh {\left (a + x \right )}}{2} + \frac {x e^{x} \cosh {\left (a + x \right )}}{2} + \frac {e^{x} \sinh {\left (a + x \right )}}{2} & \text {for}\: b = 1 \\\frac {b e^{x} \sinh {\left (a + b x \right )}}{b^{2} - 1} - \frac {e^{x} \cosh {\left (a + b x \right )}}{b^{2} - 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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