Optimal. Leaf size=17 \[ \frac {e^{n \cosh (a+b x)}}{b n} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4337, 2194} \[ \frac {e^{n \cosh (a+b x)}}{b n} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 4337
Rubi steps
\begin {align*} \int e^{n \cosh (a+b x)} \sinh (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int e^{n x} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {e^{n \cosh (a+b x)}}{b n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 17, normalized size = 1.00 \[ \frac {e^{n \cosh (a+b x)}}{b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 26, normalized size = 1.53 \[ \frac {\cosh \left (n \cosh \left (b x + a\right )\right ) + \sinh \left (n \cosh \left (b x + a\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 30, normalized size = 1.76 \[ \frac {e^{\left (\frac {1}{2} \, n e^{\left (b x + a\right )} + \frac {1}{2} \, n e^{\left (-b x - a\right )}\right )}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 17, normalized size = 1.00 \[ \frac {{\mathrm e}^{n \cosh \left (b x +a \right )}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 16, normalized size = 0.94 \[ \frac {e^{\left (n \cosh \left (b x + a\right )\right )}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 16, normalized size = 0.94 \[ \frac {{\mathrm {e}}^{n\,\mathrm {cosh}\left (a+b\,x\right )}}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 36, normalized size = 2.12 \[ \begin {cases} x \sinh {\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\x e^{n \cosh {\relax (a )}} \sinh {\relax (a )} & \text {for}\: b = 0 \\\frac {\cosh {\left (a + b x \right )}}{b} & \text {for}\: n = 0 \\\frac {e^{n \cosh {\left (a + b x \right )}}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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