Optimal. Leaf size=54 \[ -\frac {d e^{c+d x} \cosh (a+b x)}{b^2-d^2}+\frac {b e^{c+d x} \sinh (a+b x)}{b^2-d^2} \]
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Rubi [A]
time = 0.01, 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 = {5583}
\begin {gather*} \frac {b e^{c+d x} \sinh (a+b x)}{b^2-d^2}-\frac {d e^{c+d x} \cosh (a+b x)}{b^2-d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 5583
Rubi steps
\begin {align*} \int e^{c+d x} \cosh (a+b x) \, dx &=-\frac {d e^{c+d x} \cosh (a+b x)}{b^2-d^2}+\frac {b e^{c+d x} \sinh (a+b x)}{b^2-d^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 38, normalized size = 0.70 \begin {gather*} \frac {e^{c+d x} (-d \cosh (a+b x)+b \sinh (a+b x))}{(b-d) (b+d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.92, size = 78, normalized size = 1.44
method | result | size |
risch | \(\frac {\left (b \,{\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 b x +2 a} d -b -d \right ) {\mathrm e}^{-b x +d x -a +c}}{2 \left (b +d \right ) \left (b -d \right )}\) | \(58\) |
default | \(\frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{2 b -2 d}+\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d}-\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{2 \left (b -d \right )}+\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d}\) | \(78\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 97, normalized size = 1.80 \begin {gather*} -\frac {d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) + {\left (d \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{{\left (b^{2} - d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs.
\(2 (42) = 84\).
time = 0.40, size = 184, normalized size = 3.41 \begin {gather*} \begin {cases} x e^{c} \cosh {\left (a \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x e^{c} e^{d x} \sinh {\left (a - d x \right )}}{2} + \frac {x e^{c} e^{d x} \cosh {\left (a - d x \right )}}{2} - \frac {e^{c} e^{d x} \sinh {\left (a - d x \right )}}{2 d} & \text {for}\: b = - d \\- \frac {x e^{c} e^{d x} \sinh {\left (a + d x \right )}}{2} + \frac {x e^{c} e^{d x} \cosh {\left (a + d x \right )}}{2} + \frac {e^{c} e^{d x} \sinh {\left (a + d x \right )}}{d} - \frac {e^{c} e^{d x} \cosh {\left (a + d x \right )}}{2 d} & \text {for}\: b = d \\\frac {b e^{c} e^{d x} \sinh {\left (a + b x \right )}}{b^{2} - d^{2}} - \frac {d e^{c} e^{d x} \cosh {\left (a + b x \right )}}{b^{2} - d^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 40, normalized size = 0.74 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.75, size = 54, normalized size = 1.00 \begin {gather*} -\frac {{\mathrm {e}}^{c-a-b\,x+d\,x}\,\left (b+d-b\,{\mathrm {e}}^{2\,a+2\,b\,x}+d\,{\mathrm {e}}^{2\,a+2\,b\,x}\right )}{2\,\left (b^2-d^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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