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 = 0.13, antiderivative size = 119, normalized size of antiderivative = 1.00, 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 2637
Rule 2638
Rule 3120
Rule 3146
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*}
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Mathematica [A] time = 0.20, 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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fricas [A] time = 0.50, size = 160, normalized size = 1.34 \[ \frac {3}{2} \, a b c \cosh \relax (x)^{2} + \frac {1}{12} \, {\left (3 \, b^{2} c + c^{3}\right )} \cosh \relax (x)^{3} + \frac {1}{12} \, {\left (b^{3} + 3 \, b c^{2}\right )} \sinh \relax (x)^{3} + \frac {1}{4} \, {\left (6 \, a b c + {\left (3 \, b^{2} c + c^{3}\right )} \cosh \relax (x)\right )} \sinh \relax (x)^{2} + \frac {1}{2} \, {\left (2 \, a^{3} + 3 \, a b^{2} - 3 \, a c^{2}\right )} x - \frac {3}{4} \, {\left (c^{3} - {\left (4 \, a^{2} + b^{2}\right )} c\right )} \cosh \relax (x) + \frac {1}{4} \, {\left (12 \, a^{2} b + 3 \, b^{3} - 3 \, b c^{2} + {\left (b^{3} + 3 \, b c^{2}\right )} \cosh \relax (x)^{2} + 6 \, {\left (a b^{2} + a c^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 219, normalized size = 1.84 \[ \frac {1}{24} \, b^{3} e^{\left (3 \, x\right )} + \frac {1}{8} \, b^{2} c e^{\left (3 \, x\right )} + \frac {1}{8} \, b c^{2} e^{\left (3 \, x\right )} + \frac {1}{24} \, c^{3} e^{\left (3 \, x\right )} + \frac {3}{8} \, a b^{2} e^{\left (2 \, x\right )} + \frac {3}{4} \, a b c e^{\left (2 \, x\right )} + \frac {3}{8} \, a c^{2} e^{\left (2 \, x\right )} + \frac {3}{2} \, a^{2} b e^{x} + \frac {3}{8} \, b^{3} e^{x} + \frac {3}{2} \, a^{2} c e^{x} + \frac {3}{8} \, b^{2} c e^{x} - \frac {3}{8} \, b c^{2} e^{x} - \frac {3}{8} \, c^{3} e^{x} + \frac {1}{2} \, {\left (2 \, a^{3} + 3 \, a b^{2} - 3 \, a c^{2}\right )} x - \frac {1}{24} \, {\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3} + 9 \, {\left (4 \, a^{2} b + b^{3} - 4 \, a^{2} c - b^{2} c - b c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} + 9 \, {\left (a b^{2} - 2 \, a b c + a c^{2}\right )} e^{x}\right )} e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 110, normalized size = 0.92 \[ a^{3} x +3 a^{2} b \sinh \relax (x )+3 a^{2} c \cosh \relax (x )+3 a \,b^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}+\frac {x}{2}\right )+3 a b c \left (\cosh ^{2}\relax (x )\right )+3 a \,c^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}-\frac {x}{2}\right )+b^{3} \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\relax (x )\right )}{3}\right ) \sinh \relax (x )+\left (\cosh ^{3}\relax (x )\right ) b^{2} c +b \,c^{2} \left (\sinh ^{3}\relax (x )\right )+c^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\relax (x )\right )}{3}\right ) \cosh \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 137, normalized size = 1.15 \[ b^{2} c \cosh \relax (x)^{3} + b c^{2} \sinh \relax (x)^{3} + a^{3} x + \frac {1}{24} \, c^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + \frac {1}{24} \, b^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )} + 3 \, {\left (c \cosh \relax (x) + b \sinh \relax (x)\right )} a^{2} + \frac {3}{8} \, {\left (8 \, b c \cosh \relax (x)^{2} + b^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 131, normalized size = 1.10 \[ a^3\,x+{\mathrm {cosh}\relax (x)}^3\,\left (b^2\,c-\frac {2\,c^3}{3}\right )+{\mathrm {sinh}\relax (x)}^3\,\left (b\,c^2-\frac {2\,b^3}{3}\right )+b^3\,{\mathrm {cosh}\relax (x)}^2\,\mathrm {sinh}\relax (x)+c^3\,\mathrm {cosh}\relax (x)\,{\mathrm {sinh}\relax (x)}^2+3\,a^2\,c\,\mathrm {cosh}\relax (x)+3\,a^2\,b\,\mathrm {sinh}\relax (x)+\frac {3\,a\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)\,\left (b^2+c^2\right )}{2}+\frac {3\,a\,x\,{\mathrm {cosh}\relax (x)}^2\,\left (b^2-c^2\right )}{2}+3\,a\,b\,c\,{\mathrm {cosh}\relax (x)}^2-\frac {3\,a\,x\,{\mathrm {sinh}\relax (x)}^2\,\left (b^2-c^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 196, normalized size = 1.65 \[ a^{3} x + 3 a^{2} b \sinh {\relax (x )} + 3 a^{2} c \cosh {\relax (x )} - \frac {3 a b^{2} x \sinh ^{2}{\relax (x )}}{2} + \frac {3 a b^{2} x \cosh ^{2}{\relax (x )}}{2} + \frac {3 a b^{2} \sinh {\relax (x )} \cosh {\relax (x )}}{2} + 3 a b c \cosh ^{2}{\relax (x )} + \frac {3 a c^{2} x \sinh ^{2}{\relax (x )}}{2} - \frac {3 a c^{2} x \cosh ^{2}{\relax (x )}}{2} + \frac {3 a c^{2} \sinh {\relax (x )} \cosh {\relax (x )}}{2} - \frac {2 b^{3} \sinh ^{3}{\relax (x )}}{3} + b^{3} \sinh {\relax (x )} \cosh ^{2}{\relax (x )} + b^{2} c \cosh ^{3}{\relax (x )} + b c^{2} \sinh ^{3}{\relax (x )} + c^{3} \sinh ^{2}{\relax (x )} \cosh {\relax (x )} - \frac {2 c^{3} \cosh ^{3}{\relax (x )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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