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 = 0.12, antiderivative size = 105, 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} \[ \frac {1}{2} a x \left (5 a^2-3 c^2\right )+\frac {1}{6} a \left (15 a^2-4 c^2\right ) \sinh (x)+\frac {1}{6} c \left (15 a^2-4 c^2\right ) \cosh (x)+\frac {5}{6} \left (a^2 \sinh (x)+a c \cosh (x)\right ) (a \cosh (x)+a+c \sinh (x))+\frac {1}{3} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x))^2 \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3120
Rule 3146
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*}
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Mathematica [A] time = 0.17, size = 112, normalized size = 1.07 \[ \frac {1}{12} \left (30 a^3 x+45 a^3 \sinh (x)+9 a^3 \sinh (2 x)+a^3 \sinh (3 x)-9 c \left (c^2-5 a^2\right ) \cosh (x)+18 a^2 c \cosh (2 x)+3 a^2 c \cosh (3 x)-18 a c^2 x-9 a c^2 \sinh (x)+9 a c^2 \sinh (2 x)+3 a c^2 \sinh (3 x)+c^3 \cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 144, normalized size = 1.37 \[ \frac {3}{2} \, a^{2} c \cosh \relax (x)^{2} + \frac {1}{12} \, {\left (3 \, a^{2} c + c^{3}\right )} \cosh \relax (x)^{3} + \frac {1}{12} \, {\left (a^{3} + 3 \, a c^{2}\right )} \sinh \relax (x)^{3} + \frac {1}{4} \, {\left (6 \, a^{2} c + {\left (3 \, a^{2} c + c^{3}\right )} \cosh \relax (x)\right )} \sinh \relax (x)^{2} + \frac {1}{2} \, {\left (5 \, a^{3} - 3 \, a c^{2}\right )} x + \frac {3}{4} \, {\left (5 \, a^{2} c - c^{3}\right )} \cosh \relax (x) + \frac {1}{4} \, {\left (15 \, a^{3} - 3 \, a c^{2} + {\left (a^{3} + 3 \, a c^{2}\right )} \cosh \relax (x)^{2} + 6 \, {\left (a^{3} + a c^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 186, normalized size = 1.77 \[ \frac {1}{24} \, a^{3} e^{\left (3 \, x\right )} + \frac {1}{8} \, a^{2} c e^{\left (3 \, x\right )} + \frac {1}{8} \, a c^{2} e^{\left (3 \, x\right )} + \frac {1}{24} \, c^{3} e^{\left (3 \, x\right )} + \frac {3}{8} \, a^{3} e^{\left (2 \, x\right )} + \frac {3}{4} \, a^{2} c e^{\left (2 \, x\right )} + \frac {3}{8} \, a c^{2} e^{\left (2 \, x\right )} + \frac {15}{8} \, a^{3} e^{x} + \frac {15}{8} \, a^{2} c e^{x} - \frac {3}{8} \, a c^{2} e^{x} - \frac {3}{8} \, c^{3} e^{x} + \frac {1}{2} \, {\left (5 \, a^{3} - 3 \, a c^{2}\right )} x - \frac {1}{24} \, {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3} + 9 \, {\left (5 \, a^{3} - 5 \, a^{2} c - a c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} + 9 \, {\left (a^{3} - 2 \, a^{2} c + a c^{2}\right )} e^{x}\right )} e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 109, normalized size = 1.04 \[ a^{3} x +3 a^{3} \sinh \relax (x )+3 a^{2} c \cosh \relax (x )+3 a^{3} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}+\frac {x}{2}\right )+3 a^{2} c \left (\cosh ^{2}\relax (x )\right )+3 a \,c^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}-\frac {x}{2}\right )+a^{3} \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\relax (x )\right )}{3}\right ) \sinh \relax (x )+a^{2} c \left (\cosh ^{3}\relax (x )\right )+a \,c^{2} \left (\sinh ^{3}\relax (x )\right )+c^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\relax (x )\right )}{3}\right ) \cosh \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 137, normalized size = 1.30 \[ a^{2} c \cosh \relax (x)^{3} + a c^{2} \sinh \relax (x)^{3} + a^{3} x + \frac {1}{24} \, c^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + \frac {1}{24} \, a^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )} + 3 \, {\left (c \cosh \relax (x) + a \sinh \relax (x)\right )} a^{2} + \frac {3}{8} \, {\left (8 \, a c \cosh \relax (x)^{2} + a^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 131, normalized size = 1.25 \[ 3\,a^3\,\mathrm {sinh}\relax (x)+a^3\,x+{\mathrm {cosh}\relax (x)}^3\,\left (a^2\,c-\frac {2\,c^3}{3}\right )+{\mathrm {sinh}\relax (x)}^3\,\left (a\,c^2-\frac {2\,a^3}{3}\right )+a^3\,{\mathrm {cosh}\relax (x)}^2\,\mathrm {sinh}\relax (x)+c^3\,\mathrm {cosh}\relax (x)\,{\mathrm {sinh}\relax (x)}^2+3\,a^2\,c\,\mathrm {cosh}\relax (x)+3\,a^2\,c\,{\mathrm {cosh}\relax (x)}^2+\frac {3\,a\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)\,\left (a^2+c^2\right )}{2}+\frac {3\,a\,x\,{\mathrm {cosh}\relax (x)}^2\,\left (a^2-c^2\right )}{2}-\frac {3\,a\,x\,{\mathrm {sinh}\relax (x)}^2\,\left (a^2-c^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 189, normalized size = 1.80 \[ - \frac {3 a^{3} x \sinh ^{2}{\relax (x )}}{2} + \frac {3 a^{3} x \cosh ^{2}{\relax (x )}}{2} + a^{3} x - \frac {2 a^{3} \sinh ^{3}{\relax (x )}}{3} + a^{3} \sinh {\relax (x )} \cosh ^{2}{\relax (x )} + \frac {3 a^{3} \sinh {\relax (x )} \cosh {\relax (x )}}{2} + 3 a^{3} \sinh {\relax (x )} + a^{2} c \cosh ^{3}{\relax (x )} + 3 a^{2} c \cosh ^{2}{\relax (x )} + 3 a^{2} c \cosh {\relax (x )} + \frac {3 a c^{2} x \sinh ^{2}{\relax (x )}}{2} - \frac {3 a c^{2} x \cosh ^{2}{\relax (x )}}{2} + a c^{2} \sinh ^{3}{\relax (x )} + \frac {3 a c^{2} \sinh {\relax (x )} \cosh {\relax (x )}}{2} + c^{3} \sinh ^{2}{\relax (x )} \cosh {\relax (x )} - \frac {2 c^{3} \cosh ^{3}{\relax (x )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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