Optimal. Leaf size=17 \[ \frac {e^{n \cosh (a+b x)}}{b n} \]
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Rubi [A]
time = 0.01, 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 = {4422, 2225}
\begin {gather*} \frac {e^{n \cosh (a+b x)}}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 4422
Rubi steps
\begin {align*} \int e^{n \cosh (a+b x)} \sinh (a+b x) \, dx &=\frac {\text {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 = 0.03, size = 17, normalized size = 1.00 \begin {gather*} \frac {e^{n \cosh (a+b x)}}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 17, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {{\mathrm e}^{n \cosh \left (b x +a \right )}}{b n}\) | \(17\) |
default | \(\frac {{\mathrm e}^{n \cosh \left (b x +a \right )}}{b n}\) | \(17\) |
risch | \(\frac {{\mathrm e}^{\frac {n \left ({\mathrm e}^{b x +a}+{\mathrm e}^{-b x -a}\right )}{2}}}{b n}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 16, normalized size = 0.94 \begin {gather*} \frac {e^{\left (n \cosh \left (b x + a\right )\right )}}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 26, normalized size = 1.53 \begin {gather*} \frac {\cosh \left (n \cosh \left (b x + a\right )\right ) + \sinh \left (n \cosh \left (b x + a\right )\right )}{b n} \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. 36 vs.
\(2 (12) = 24\).
time = 0.13, size = 36, normalized size = 2.12 \begin {gather*} \begin {cases} x \sinh {\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\cosh {\left (a + b x \right )}}{b} & \text {for}\: n = 0 \\x e^{n \cosh {\left (a \right )}} \sinh {\left (a \right )} & \text {for}\: b = 0 \\\frac {e^{n \cosh {\left (a + b x \right )}}}{b n} & \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 = 30, normalized size = 1.76 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \frac {{\mathrm {e}}^{n\,\mathrm {cosh}\left (a+b\,x\right )}}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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