Optimal. Leaf size=26 \[ \frac {(a \sinh (c+d x)+a \cosh (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \[ \frac {(a \sinh (c+d x)+a \cosh (c+d x))^2}{2 d} \]
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))^2 \, dx &=\frac {(a \cosh (c+d x)+a \sinh (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 25, normalized size = 0.96 \[ \frac {a^2 (\sinh (c+d x)+\cosh (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 43, normalized size = 1.65 \[ \frac {a^{2} \cosh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right ) - d \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 17, normalized size = 0.65 \[ \frac {a^{2} e^{\left (2 \, d x + 2 \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 24, normalized size = 0.92 \[ \frac {a^{2} \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )^{2}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 88, normalized size = 3.38 \[ \frac {1}{8} \, a^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac {a^{2} \cosh \left (d x + c\right )^{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 17, normalized size = 0.65 \[ \frac {a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 44, normalized size = 1.69 \[ \begin {cases} \frac {a^{2} \sinh ^{2}{\left (c + d x \right )}}{d} + \frac {a^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sinh {\relax (c )} + a \cosh {\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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