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 = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2637
Rule 2638
Rule 3120
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.42, size = 57, normalized size = 1.00 \[ \frac {1}{2} \, a c \cosh \relax (x)^{2} + \frac {1}{2} \, a c \sinh \relax (x)^{2} + 2 \, a c \cosh \relax (x) + \frac {1}{2} \, {\left (3 \, a^{2} - c^{2}\right )} x + \frac {1}{2} \, {\left (4 \, a^{2} + {\left (a^{2} + c^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 81, normalized size = 1.42 \[ \frac {1}{8} \, a^{2} e^{\left (2 \, x\right )} + \frac {1}{4} \, a c e^{\left (2 \, x\right )} + \frac {1}{8} \, c^{2} e^{\left (2 \, x\right )} + a^{2} e^{x} + a c e^{x} + \frac {1}{2} \, {\left (3 \, a^{2} - c^{2}\right )} x - \frac {1}{8} \, {\left (a^{2} - 2 \, a c + c^{2} + 8 \, {\left (a^{2} - a c\right )} e^{x}\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 55, normalized size = 0.96 \[ a^{2} x +2 a^{2} \sinh \relax (x )+2 a c \cosh \relax (x )+a^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}+\frac {x}{2}\right )+a c \left (\cosh ^{2}\relax (x )\right )+c^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}-\frac {x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 63, normalized size = 1.11 \[ a c \cosh \relax (x)^{2} + \frac {1}{8} \, a^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - \frac {1}{8} \, c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + a^{2} x + 2 \, {\left (c \cosh \relax (x) + a \sinh \relax (x)\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 51, normalized size = 0.89 \[ 2\,a^2\,\mathrm {sinh}\relax (x)+\frac {3\,a^2\,x}{2}-\frac {c^2\,x}{2}+a\,c\,{\mathrm {cosh}\relax (x)}^2+\frac {a^2\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)}{2}+\frac {c^2\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)}{2}+2\,a\,c\,\mathrm {cosh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 100, normalized size = 1.75 \[ - \frac {a^{2} x \sinh ^{2}{\relax (x )}}{2} + \frac {a^{2} x \cosh ^{2}{\relax (x )}}{2} + a^{2} x + \frac {a^{2} \sinh {\relax (x )} \cosh {\relax (x )}}{2} + 2 a^{2} \sinh {\relax (x )} + a c \cosh ^{2}{\relax (x )} + 2 a c \cosh {\relax (x )} + \frac {c^{2} x \sinh ^{2}{\relax (x )}}{2} - \frac {c^{2} x \cosh ^{2}{\relax (x )}}{2} + \frac {c^{2} \sinh {\relax (x )} \cosh {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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