Optimal. Leaf size=23 \[ \frac {e^{n \cosh (a c+b c x)}}{b c n} \]
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Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4337, 2194} \[ \frac {e^{n \cosh (a c+b c x)}}{b c n} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 4337
Rubi steps
\begin {align*} \int e^{n \cosh (c (a+b x))} \sinh (a c+b c x) \, dx &=\frac {\operatorname {Subst}\left (\int e^{n x} \, dx,x,\cosh (a c+b c x)\right )}{b c}\\ &=\frac {e^{n \cosh (a c+b c x)}}{b c n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 22, normalized size = 0.96 \[ \frac {e^{n \cosh (c (a+b x))}}{b c n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 35, normalized size = 1.52 \[ \frac {\cosh \left (n \cosh \left (b c x + a c\right )\right ) + \sinh \left (n \cosh \left (b c x + a c\right )\right )}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 38, normalized size = 1.65 \[ \frac {e^{\left (\frac {1}{2} \, n e^{\left (b c x + a c\right )} + \frac {1}{2} \, n e^{\left (-b c x - a c\right )}\right )}}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.83, size = 39, normalized size = 1.70 \[ \frac {\frac {\sinh \left (n \cosh \left (c \left (b x +a \right )\right )\right )}{n}+\frac {\cosh \left (n \cosh \left (c \left (b x +a \right )\right )\right )}{n}}{c b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 22, normalized size = 0.96 \[ \frac {e^{\left (n \cosh \left (b c x + a c\right )\right )}}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 38, normalized size = 1.65 \[ \frac {{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}}\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}}{b\,c\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.45, size = 51, normalized size = 2.22 \[ \begin {cases} 0 & \text {for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \wedge \left (c = 0 \vee n = 0\right ) \\x e^{n \cosh {\left (a c \right )}} \sinh {\left (a c \right )} & \text {for}\: b = 0 \\\frac {\cosh {\left (a c + b c x \right )}}{b c} & \text {for}\: n = 0 \\\frac {e^{n \cosh {\left (a c + b c x \right )}}}{b c n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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