Optimal. Leaf size=105 \[ \frac{1}{2} a x \left (5 a^2-3 c^2\right )+\frac{1}{6} a \left (15 a^2-4 c^2\right ) \sinh (x)+\frac{1}{6} c \left (15 a^2-4 c^2\right ) \cosh (x)+\frac{5}{6} \left (a^2 \sinh (x)+a c \cosh (x)\right ) (a \cosh (x)+a+c \sinh (x))+\frac{1}{3} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x))^2 \]
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Rubi [A] time = 0.116252, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3120, 3146, 2637, 2638} \[ \frac{1}{2} a x \left (5 a^2-3 c^2\right )+\frac{1}{6} a \left (15 a^2-4 c^2\right ) \sinh (x)+\frac{1}{6} c \left (15 a^2-4 c^2\right ) \cosh (x)+\frac{5}{6} \left (a^2 \sinh (x)+a c \cosh (x)\right ) (a \cosh (x)+a+c \sinh (x))+\frac{1}{3} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x))^2 \]
Antiderivative was successfully verified.
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Rule 3120
Rule 3146
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (a+a \cosh (x)+c \sinh (x))^3 \, dx &=\frac{1}{3} (c \cosh (x)+a \sinh (x)) (a+a \cosh (x)+c \sinh (x))^2+\frac{1}{3} \int (a+a \cosh (x)+c \sinh (x)) \left (5 a^2-2 c^2+5 a^2 \cosh (x)+5 a c \sinh (x)\right ) \, dx\\ &=\frac{5}{6} \left (a c \cosh (x)+a^2 \sinh (x)\right ) (a+a \cosh (x)+c \sinh (x))+\frac{1}{3} (c \cosh (x)+a \sinh (x)) (a+a \cosh (x)+c \sinh (x))^2+\frac{\int \left (3 a^2 \left (5 a^2-3 c^2\right )+a^2 \left (15 a^2-4 c^2\right ) \cosh (x)+a c \left (15 a^2-4 c^2\right ) \sinh (x)\right ) \, dx}{6 a}\\ &=\frac{1}{2} a \left (5 a^2-3 c^2\right ) x+\frac{5}{6} \left (a c \cosh (x)+a^2 \sinh (x)\right ) (a+a \cosh (x)+c \sinh (x))+\frac{1}{3} (c \cosh (x)+a \sinh (x)) (a+a \cosh (x)+c \sinh (x))^2+\frac{1}{6} \left (a \left (15 a^2-4 c^2\right )\right ) \int \cosh (x) \, dx+\frac{1}{6} \left (c \left (15 a^2-4 c^2\right )\right ) \int \sinh (x) \, dx\\ &=\frac{1}{2} a \left (5 a^2-3 c^2\right ) x+\frac{1}{6} c \left (15 a^2-4 c^2\right ) \cosh (x)+\frac{1}{6} a \left (15 a^2-4 c^2\right ) \sinh (x)+\frac{5}{6} \left (a c \cosh (x)+a^2 \sinh (x)\right ) (a+a \cosh (x)+c \sinh (x))+\frac{1}{3} (c \cosh (x)+a \sinh (x)) (a+a \cosh (x)+c \sinh (x))^2\\ \end{align*}
Mathematica [A] time = 0.163128, size = 112, normalized size = 1.07 \[ \frac{1}{12} \left (-9 c \left (c^2-5 a^2\right ) \cosh (x)+18 a^2 c \cosh (2 x)+3 a^2 c \cosh (3 x)+30 a^3 x+45 a^3 \sinh (x)+9 a^3 \sinh (2 x)+a^3 \sinh (3 x)-18 a c^2 x-9 a c^2 \sinh (x)+9 a c^2 \sinh (2 x)+3 a c^2 \sinh (3 x)+c^3 \cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 129, normalized size = 1.2 \begin{align*}{a}^{3}x+3\,{a}^{3}\sinh \left ( x \right ) +3\,{a}^{2}c\cosh \left ( x \right ) +3\,{a}^{3} \left ( 1/2\,\cosh \left ( x \right ) \sinh \left ( x \right ) +x/2 \right ) +3\,{a}^{2}c \left ( \cosh \left ( x \right ) \right ) ^{2}+3\,a{c}^{2} \left ( 1/2\,\cosh \left ( x \right ) \sinh \left ( x \right ) -x/2 \right ) +{a}^{3} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( x \right ) +3\,{a}^{2}c \left ( 1/3\,\cosh \left ( x \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}+1/3\,\cosh \left ( x \right ) \right ) +3\,a{c}^{2} \left ( 1/3\,\sinh \left ( x \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}-1/3\,\sinh \left ( x \right ) \right ) +{c}^{3} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0289, size = 185, normalized size = 1.76 \begin{align*} a^{2} c \cosh \left (x\right )^{3} + a c^{2} \sinh \left (x\right )^{3} + a^{3} x + \frac{1}{24} \, c^{3}{\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + \frac{1}{24} \, a^{3}{\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )} + 3 \,{\left (c \cosh \left (x\right ) + a \sinh \left (x\right )\right )} a^{2} + \frac{3}{8} \,{\left (8 \, a c \cosh \left (x\right )^{2} + a^{2}{\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - c^{2}{\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41206, size = 381, normalized size = 3.63 \begin{align*} \frac{3}{2} \, a^{2} c \cosh \left (x\right )^{2} + \frac{1}{12} \,{\left (3 \, a^{2} c + c^{3}\right )} \cosh \left (x\right )^{3} + \frac{1}{12} \,{\left (a^{3} + 3 \, a c^{2}\right )} \sinh \left (x\right )^{3} + \frac{1}{4} \,{\left (6 \, a^{2} c +{\left (3 \, a^{2} c + c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac{1}{2} \,{\left (5 \, a^{3} - 3 \, a c^{2}\right )} x + \frac{3}{4} \,{\left (5 \, a^{2} c - c^{3}\right )} \cosh \left (x\right ) + \frac{1}{4} \,{\left (15 \, a^{3} - 3 \, a c^{2} +{\left (a^{3} + 3 \, a c^{2}\right )} \cosh \left (x\right )^{2} + 6 \,{\left (a^{3} + a 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 = 1.06227, size = 189, normalized size = 1.8 \begin{align*} - \frac{3 a^{3} x \sinh ^{2}{\left (x \right )}}{2} + \frac{3 a^{3} x \cosh ^{2}{\left (x \right )}}{2} + a^{3} x - \frac{2 a^{3} \sinh ^{3}{\left (x \right )}}{3} + a^{3} \sinh{\left (x \right )} \cosh ^{2}{\left (x \right )} + \frac{3 a^{3} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} + 3 a^{3} \sinh{\left (x \right )} + 3 a^{2} c \sinh ^{2}{\left (x \right )} + a^{2} c \cosh ^{3}{\left (x \right )} + 3 a^{2} c \cosh{\left (x \right )} + \frac{3 a c^{2} x \sinh ^{2}{\left (x \right )}}{2} - \frac{3 a c^{2} x \cosh ^{2}{\left (x \right )}}{2} + a c^{2} \sinh ^{3}{\left (x \right )} + \frac{3 a c^{2} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} + c^{3} \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )} - \frac{2 c^{3} \cosh ^{3}{\left (x \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11794, size = 251, normalized size = 2.39 \begin{align*} \frac{1}{24} \, a^{3} e^{\left (3 \, x\right )} + \frac{1}{8} \, a^{2} c e^{\left (3 \, x\right )} + \frac{1}{8} \, a c^{2} e^{\left (3 \, x\right )} + \frac{1}{24} \, c^{3} e^{\left (3 \, x\right )} + \frac{3}{8} \, a^{3} e^{\left (2 \, x\right )} + \frac{3}{4} \, a^{2} c e^{\left (2 \, x\right )} + \frac{3}{8} \, a c^{2} e^{\left (2 \, x\right )} + \frac{15}{8} \, a^{3} e^{x} + \frac{15}{8} \, a^{2} c e^{x} - \frac{3}{8} \, a c^{2} e^{x} - \frac{3}{8} \, c^{3} e^{x} + \frac{1}{2} \,{\left (5 \, a^{3} - 3 \, a c^{2}\right )} x - \frac{1}{24} \,{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3} + 9 \,{\left (5 \, a^{3} - 5 \, a^{2} c - a c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} + 9 \,{\left (a^{3} - 2 \, a^{2} c + a c^{2}\right )} e^{x}\right )} e^{\left (-3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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