Optimal. Leaf size=57 \[ \frac{1}{2} x \left (3 a^2-c^2\right )+\frac{3}{2} a^2 \sinh (x)+\frac{3}{2} a c \cosh (x)+\frac{1}{2} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x)) \]
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Rubi [A] time = 0.0345935, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3120, 2637, 2638} \[ \frac{1}{2} x \left (3 a^2-c^2\right )+\frac{3}{2} a^2 \sinh (x)+\frac{3}{2} a c \cosh (x)+\frac{1}{2} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3120
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (a+a \cosh (x)+c \sinh (x))^2 \, dx &=\frac{1}{2} (c \cosh (x)+a \sinh (x)) (a+a \cosh (x)+c \sinh (x))+\frac{1}{2} \int \left (3 a^2-c^2+3 a^2 \cosh (x)+3 a c \sinh (x)\right ) \, dx\\ &=\frac{1}{2} \left (3 a^2-c^2\right ) x+\frac{1}{2} (c \cosh (x)+a \sinh (x)) (a+a \cosh (x)+c \sinh (x))+\frac{1}{2} \left (3 a^2\right ) \int \cosh (x) \, dx+\frac{1}{2} (3 a c) \int \sinh (x) \, dx\\ &=\frac{1}{2} \left (3 a^2-c^2\right ) x+\frac{3}{2} a c \cosh (x)+\frac{3}{2} a^2 \sinh (x)+\frac{1}{2} (c \cosh (x)+a \sinh (x)) (a+a \cosh (x)+c \sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0697247, size = 55, normalized size = 0.96 \[ \frac{1}{2} x \left (3 a^2-c^2\right )+\frac{1}{4} \left (a^2+c^2\right ) \sinh (2 x)+2 a^2 \sinh (x)+2 a c \cosh (x)+\frac{1}{2} a c \cosh (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 55, normalized size = 1. \begin{align*}{a}^{2}x+2\,{a}^{2}\sinh \left ( x \right ) +2\,ac\cosh \left ( x \right ) +{a}^{2} \left ({\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}}+{\frac{x}{2}} \right ) +ac \left ( \cosh \left ( x \right ) \right ) ^{2}+{c}^{2} \left ({\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}}-{\frac{x}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98096, size = 85, normalized size = 1.49 \begin{align*} a c \cosh \left (x\right )^{2} + \frac{1}{8} \, a^{2}{\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - \frac{1}{8} \, c^{2}{\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + a^{2} x + 2 \,{\left (c \cosh \left (x\right ) + a \sinh \left (x\right )\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32193, size = 163, normalized size = 2.86 \begin{align*} \frac{1}{2} \, a c \cosh \left (x\right )^{2} + \frac{1}{2} \, a c \sinh \left (x\right )^{2} + 2 \, a c \cosh \left (x\right ) + \frac{1}{2} \,{\left (3 \, a^{2} - c^{2}\right )} x + \frac{1}{2} \,{\left (4 \, a^{2} +{\left (a^{2} + c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.433168, size = 100, normalized size = 1.75 \begin{align*} - \frac{a^{2} x \sinh ^{2}{\left (x \right )}}{2} + \frac{a^{2} x \cosh ^{2}{\left (x \right )}}{2} + a^{2} x + \frac{a^{2} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} + 2 a^{2} \sinh{\left (x \right )} + a c \sinh ^{2}{\left (x \right )} + 2 a c \cosh{\left (x \right )} + \frac{c^{2} x \sinh ^{2}{\left (x \right )}}{2} - \frac{c^{2} x \cosh ^{2}{\left (x \right )}}{2} + \frac{c^{2} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14582, size = 109, normalized size = 1.91 \begin{align*} \frac{1}{8} \, a^{2} e^{\left (2 \, x\right )} + \frac{1}{4} \, a c e^{\left (2 \, x\right )} + \frac{1}{8} \, c^{2} e^{\left (2 \, x\right )} + a^{2} e^{x} + a c e^{x} + \frac{1}{2} \,{\left (3 \, a^{2} - c^{2}\right )} x - \frac{1}{8} \,{\left (a^{2} - 2 \, a c + c^{2} + 8 \,{\left (a^{2} - a c\right )} e^{x}\right )} e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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