Optimal. Leaf size=26 \[ \frac{(a \sinh (c+d x)+a \cosh (c+d x))^n}{d n} \]
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Rubi [A] time = 0.0151521, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3071} \[ \frac{(a \sinh (c+d x)+a \cosh (c+d x))^n}{d n} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin{align*} \int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx &=\frac{(a \cosh (c+d x)+a \sinh (c+d x))^n}{d n}\\ \end{align*}
Mathematica [A] time = 0.0741302, size = 24, normalized size = 0.92 \[ \frac{(a (\sinh (c+d x)+\cosh (c+d x)))^n}{d n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a\cosh \left ( dx+c \right ) +a\sinh \left ( dx+c \right ) \right ) ^{n}}{dn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02406, size = 24, normalized size = 0.92 \begin{align*} \frac{a^{n} e^{\left ({\left (d x + c\right )} n\right )}}{d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0289, size = 93, normalized size = 3.58 \begin{align*} \frac{\cosh \left (d n x + c n + n \log \left (a\right )\right ) + \sinh \left (d n x + c n + n \log \left (a\right )\right )}{d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.283659, size = 36, normalized size = 1.38 \begin{align*} \begin{cases} x & \text{for}\: d = 0 \wedge n = 0 \\x \left (a \sinh{\left (c \right )} + a \cosh{\left (c \right )}\right )^{n} & \text{for}\: d = 0 \\x & \text{for}\: n = 0 \\\frac{\left (a \sinh{\left (c + d x \right )} + a \cosh{\left (c + d x \right )}\right )^{n}}{d n} & \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{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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