Optimal. Leaf size=63 \[ \frac{1}{8} x \left (8 a^2-b^2\right )+\frac{a b \cosh (2 c+2 d x)}{2 d}+\frac{b^2 \sinh (2 c+2 d x) \cosh (2 c+2 d x)}{16 d} \]
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Rubi [A] time = 0.0354427, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2666, 2644} \[ \frac{1}{8} x \left (8 a^2-b^2\right )+\frac{a b \cosh (2 c+2 d x)}{2 d}+\frac{b^2 \sinh (2 c+2 d x) \cosh (2 c+2 d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2666
Rule 2644
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x) \sinh (c+d x))^2 \, dx &=\int \left (a+\frac{1}{2} b \sinh (2 c+2 d x)\right )^2 \, dx\\ &=\frac{1}{8} \left (8 a^2-b^2\right ) x+\frac{a b \cosh (2 c+2 d x)}{2 d}+\frac{b^2 \cosh (2 c+2 d x) \sinh (2 c+2 d x)}{16 d}\\ \end{align*}
Mathematica [A] time = 0.113222, size = 50, normalized size = 0.79 \[ \frac{4 \left (8 a^2-b^2\right ) (c+d x)+16 a b \cosh (2 (c+d x))+b^2 \sinh (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 68, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{8}}-{\frac{dx}{8}}-{\frac{c}{8}} \right ) +ab \left ( \cosh \left ( dx+c \right ) \right ) ^{2}+{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 = 1.16773, size = 85, normalized size = 1.35 \begin{align*} a^{2} x - \frac{1}{64} \, b^{2}{\left (\frac{8 \,{\left (d x + c\right )}}{d} - \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac{a b \cosh \left (d x + c\right )^{2}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17857, size = 198, normalized size = 3.14 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 4 \, a b \cosh \left (d x + c\right )^{2} + 4 \, a b \sinh \left (d x + c\right )^{2} +{\left (8 \, a^{2} - b^{2}\right )} d x}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.46069, size = 129, normalized size = 2.05 \begin{align*} \begin{cases} a^{2} x + \frac{a b \cosh ^{2}{\left (c + d x \right )}}{d} - \frac{b^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} + \frac{b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac{b^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} + \frac{b^{2} \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh{\left (c \right )} \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 = 1.18512, size = 143, normalized size = 2.27 \begin{align*} \frac{b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 8 \,{\left (8 \, a^{2} - b^{2}\right )}{\left (d x + c\right )} -{\left (48 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 6 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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