Optimal. Leaf size=54 \[ \frac {b e^{a+b x} \sinh (c+d x)}{b^2-d^2}-\frac {d e^{a+b x} \cosh (c+d x)}{b^2-d^2} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5474} \[ \frac {b e^{a+b x} \sinh (c+d x)}{b^2-d^2}-\frac {d e^{a+b x} \cosh (c+d x)}{b^2-d^2} \]
Antiderivative was successfully verified.
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Rule 5474
Rubi steps
\begin {align*} \int e^{a+b x} \sinh (c+d x) \, dx &=-\frac {d e^{a+b x} \cosh (c+d x)}{b^2-d^2}+\frac {b e^{a+b x} \sinh (c+d x)}{b^2-d^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 38, normalized size = 0.70 \[ \frac {e^{a+b x} (b \sinh (c+d x)-d \cosh (c+d x))}{(b-d) (b+d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 67, normalized size = 1.24 \[ -\frac {d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + d \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - {\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{b^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 40, normalized size = 0.74 \[ \frac {e^{\left (b x + d x + a + c\right )}}{2 \, {\left (b + d\right )}} - \frac {e^{\left (b x - d x + a - c\right )}}{2 \, {\left (b - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 78, normalized size = 1.44 \[ -\frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{2 \left (b -d \right )}+\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d}-\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{2 \left (b -d \right )}+\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d} \]
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 = 0.11, size = 54, normalized size = 1.00 \[ -\frac {{\mathrm {e}}^{a-c+b\,x-d\,x}\,\left (b+d-b\,{\mathrm {e}}^{2\,c+2\,d\,x}+d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{2\,\left (b^2-d^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.31, size = 201, normalized size = 3.72 \[ \begin {cases} x e^{a} \sinh {\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x e^{a} e^{- d x} \sinh {\left (c + d x \right )}}{2} + \frac {x e^{a} e^{- d x} \cosh {\left (c + d x \right )}}{2} + \frac {e^{a} e^{- d x} \sinh {\left (c + d x \right )}}{2 d} + \frac {e^{a} e^{- d x} \cosh {\left (c + d x \right )}}{d} & \text {for}\: b = - d \\\frac {x e^{a} e^{d x} \sinh {\left (c + d x \right )}}{2} - \frac {x e^{a} e^{d x} \cosh {\left (c + d x \right )}}{2} - \frac {e^{a} e^{d x} \sinh {\left (c + d x \right )}}{2 d} + \frac {e^{a} e^{d x} \cosh {\left (c + d x \right )}}{d} & \text {for}\: b = d \\\frac {b e^{a} e^{b x} \sinh {\left (c + d x \right )}}{b^{2} - d^{2}} - \frac {d e^{a} e^{b x} \cosh {\left (c + d x \right )}}{b^{2} - d^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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