Optimal. Leaf size=119 \[ \frac{1}{2} a x \left (2 a^2+3 b^2-3 c^2\right )+\frac{1}{6} b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+\frac{1}{6} c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )+\frac{1}{3} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^2+\frac{5}{6} (a b \sinh (x)+a c \cosh (x)) (a+b \cosh (x)+c \sinh (x)) \]
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Rubi [A] time = 0.131999, antiderivative size = 119, 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} \[ \frac{1}{2} a x \left (2 a^2+3 b^2-3 c^2\right )+\frac{1}{6} b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+\frac{1}{6} c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )+\frac{1}{3} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^2+\frac{5}{6} (a b \sinh (x)+a c \cosh (x)) (a+b \cosh (x)+c \sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3120
Rule 3146
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (a+b \cosh (x)+c \sinh (x))^3 \, dx &=\frac{1}{3} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^2+\frac{1}{3} \int (a+b \cosh (x)+c \sinh (x)) \left (3 a^2+2 b^2-2 c^2+5 a b \cosh (x)+5 a c \sinh (x)\right ) \, dx\\ &=\frac{5}{6} (a c \cosh (x)+a b \sinh (x)) (a+b \cosh (x)+c \sinh (x))+\frac{1}{3} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^2+\frac{\int \left (3 a^2 \left (2 a^2+3 b^2-3 c^2\right )+a b \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+a c \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)\right ) \, dx}{6 a}\\ &=\frac{1}{2} a \left (2 a^2+3 b^2-3 c^2\right ) x+\frac{5}{6} (a c \cosh (x)+a b \sinh (x)) (a+b \cosh (x)+c \sinh (x))+\frac{1}{3} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^2+\frac{1}{6} \left (b \left (11 a^2+4 b^2-4 c^2\right )\right ) \int \cosh (x) \, dx+\frac{1}{6} \left (c \left (11 a^2+4 b^2-4 c^2\right )\right ) \int \sinh (x) \, dx\\ &=\frac{1}{2} a \left (2 a^2+3 b^2-3 c^2\right ) x+\frac{1}{6} c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+\frac{1}{6} b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)+\frac{5}{6} (a c \cosh (x)+a b \sinh (x)) (a+b \cosh (x)+c \sinh (x))+\frac{1}{3} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^2\\ \end{align*}
Mathematica [A] time = 0.186675, size = 116, normalized size = 0.97 \[ \frac{1}{12} \left (6 a x \left (2 a^2+3 b^2-3 c^2\right )+9 b \sinh (x) \left (4 a^2+b^2-c^2\right )+9 c \cosh (x) \left (4 a^2+b^2-c^2\right )+9 a \left (b^2+c^2\right ) \sinh (2 x)+18 a b c \cosh (2 x)+b \left (b^2+3 c^2\right ) \sinh (3 x)+c \left (3 b^2+c^2\right ) \cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 130, normalized size = 1.1 \begin{align*}{a}^{3}x+3\,{a}^{2}b\sinh \left ( x \right ) +3\,{a}^{2}c\cosh \left ( x \right ) +3\,a{b}^{2} \left ( 1/2\,\cosh \left ( x \right ) \sinh \left ( x \right ) +x/2 \right ) +3\,abc \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 ) +{b}^{3} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( x \right ) +3\,c{b}^{2} \left ( 1/3\,\cosh \left ( x \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}+1/3\,\cosh \left ( x \right ) \right ) +3\,b{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 = 1.02551, size = 185, normalized size = 1.55 \begin{align*} b^{2} c \cosh \left (x\right )^{3} + b 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} \, b^{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 ) + b \sinh \left (x\right )\right )} a^{2} + \frac{3}{8} \,{\left (8 \, b c \cosh \left (x\right )^{2} + b^{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 = 1.98616, size = 421, normalized size = 3.54 \begin{align*} \frac{3}{2} \, a b c \cosh \left (x\right )^{2} + \frac{1}{12} \,{\left (3 \, b^{2} c + c^{3}\right )} \cosh \left (x\right )^{3} + \frac{1}{12} \,{\left (b^{3} + 3 \, b c^{2}\right )} \sinh \left (x\right )^{3} + \frac{1}{4} \,{\left (6 \, a b c +{\left (3 \, b^{2} c + c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac{1}{2} \,{\left (2 \, a^{3} + 3 \, a b^{2} - 3 \, a c^{2}\right )} x - \frac{3}{4} \,{\left (c^{3} -{\left (4 \, a^{2} + b^{2}\right )} c\right )} \cosh \left (x\right ) + \frac{1}{4} \,{\left (12 \, a^{2} b + 3 \, b^{3} - 3 \, b c^{2} +{\left (b^{3} + 3 \, b c^{2}\right )} \cosh \left (x\right )^{2} + 6 \,{\left (a b^{2} + 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 = 0.722612, size = 196, normalized size = 1.65 \begin{align*} a^{3} x + 3 a^{2} b \sinh{\left (x \right )} + 3 a^{2} c \cosh{\left (x \right )} - \frac{3 a b^{2} x \sinh ^{2}{\left (x \right )}}{2} + \frac{3 a b^{2} x \cosh ^{2}{\left (x \right )}}{2} + \frac{3 a b^{2} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} + 3 a b c \sinh ^{2}{\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} + \frac{3 a c^{2} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} - \frac{2 b^{3} \sinh ^{3}{\left (x \right )}}{3} + b^{3} \sinh{\left (x \right )} \cosh ^{2}{\left (x \right )} + b^{2} c \cosh ^{3}{\left (x \right )} + b c^{2} \sinh ^{3}{\left (x \right )} + 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 [B] time = 1.12245, size = 296, normalized size = 2.49 \begin{align*} \frac{1}{24} \, b^{3} e^{\left (3 \, x\right )} + \frac{1}{8} \, b^{2} c e^{\left (3 \, x\right )} + \frac{1}{8} \, b c^{2} e^{\left (3 \, x\right )} + \frac{1}{24} \, c^{3} e^{\left (3 \, x\right )} + \frac{3}{8} \, a b^{2} e^{\left (2 \, x\right )} + \frac{3}{4} \, a b c e^{\left (2 \, x\right )} + \frac{3}{8} \, a c^{2} e^{\left (2 \, x\right )} + \frac{3}{2} \, a^{2} b e^{x} + \frac{3}{8} \, b^{3} e^{x} + \frac{3}{2} \, a^{2} c e^{x} + \frac{3}{8} \, b^{2} c e^{x} - \frac{3}{8} \, b c^{2} e^{x} - \frac{3}{8} \, c^{3} e^{x} + \frac{1}{2} \,{\left (2 \, a^{3} + 3 \, a b^{2} - 3 \, a c^{2}\right )} x - \frac{1}{24} \,{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3} + 9 \,{\left (4 \, a^{2} b + b^{3} - 4 \, a^{2} c - b^{2} c - b c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} + 9 \,{\left (a b^{2} - 2 \, a b 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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