Optimal. Leaf size=19 \[ \frac{\sinh ^{n+1}(a+b x)}{b (n+1)} \]
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Rubi [A] time = 0.0225391, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2564, 30} \[ \frac{\sinh ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 30
Rubi steps
\begin{align*} \int \cosh (a+b x) \sinh ^n(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^n \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\sinh ^{1+n}(a+b x)}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0096959, size = 19, normalized size = 1. \[ \frac{\sinh ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 20, normalized size = 1.1 \begin{align*}{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{n+1}}{b \left ( n+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20277, size = 193, normalized size = 10.16 \begin{align*} \frac{\cosh \left (n \log \left (\sinh \left (b x + a\right )\right )\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right ) \sinh \left (n \log \left (\sinh \left (b x + a\right )\right )\right )}{{\left (b n + b\right )} \cosh \left (b x + a\right )^{2} -{\left (b n + b\right )} \sinh \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.66918, size = 49, normalized size = 2.58 \begin{align*} \begin{cases} \frac{x \cosh{\left (a \right )}}{\sinh{\left (a \right )}} & \text{for}\: b = 0 \wedge n = -1 \\x \sinh ^{n}{\left (a \right )} \cosh{\left (a \right )} & \text{for}\: b = 0 \\\frac{\log{\left (\sinh{\left (a + b x \right )} \right )}}{b} & \text{for}\: n = -1 \\\frac{\sinh{\left (a + b x \right )} \sinh ^{n}{\left (a + b x \right )}}{b n + b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (b x + a\right )^{n} \cosh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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