Optimal. Leaf size=50 \[ \frac{1}{2} x \left (2 a^2+b^2\right )+\frac{2 a b \sinh (c+d x)}{d}+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d} \]
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Rubi [A] time = 0.0172046, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2644} \[ \frac{1}{2} x \left (2 a^2+b^2\right )+\frac{2 a b \sinh (c+d x)}{d}+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2644
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x))^2 \, dx &=\frac{1}{2} \left (2 a^2+b^2\right ) x+\frac{2 a b \sinh (c+d x)}{d}+\frac{b^2 \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0728621, size = 46, normalized size = 0.92 \[ \frac{2 \left (2 a^2+b^2\right ) (c+d x)+8 a b \sinh (c+d x)+b^2 \sinh (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 51, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,ab\sinh \left ( dx+c \right ) +{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04553, size = 74, normalized size = 1.48 \begin{align*} \frac{1}{8} \, b^{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 )} + a^{2} x + \frac{2 \, a b \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07314, size = 96, normalized size = 1.92 \begin{align*} \frac{{\left (2 \, a^{2} + b^{2}\right )} d x +{\left (b^{2} \cosh \left (d x + c\right ) + 4 \, a b\right )} \sinh \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.307524, size = 78, normalized size = 1.56 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b \sinh{\left (c + d x \right )}}{d} - \frac{b^{2} x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac{b^{2} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{b^{2} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \cosh{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1479, size = 95, normalized size = 1.9 \begin{align*} \frac{b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b e^{\left (d x + c\right )} + 4 \,{\left (2 \, a^{2} + b^{2}\right )}{\left (d x + c\right )} -{\left (8 \, a b e^{\left (d x + c\right )} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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