Optimal. Leaf size=90 \[ \frac {3}{2} x \left (b^2-c^2\right )+\frac {3}{2} b \sqrt {b^2-c^2} \sinh (x)+\frac {3}{2} c \sqrt {b^2-c^2} \cosh (x)+\frac {1}{2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3113, 2637, 2638} \[ \frac {3}{2} x \left (b^2-c^2\right )+\frac {3}{2} b \sqrt {b^2-c^2} \sinh (x)+\frac {3}{2} c \sqrt {b^2-c^2} \cosh (x)+\frac {1}{2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3113
Rubi steps
\begin {align*} \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx &=\frac {1}{2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {1}{2} \left (3 \sqrt {b^2-c^2}\right ) \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \, dx\\ &=\frac {3}{2} \left (b^2-c^2\right ) x+\frac {1}{2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {1}{2} \left (3 b \sqrt {b^2-c^2}\right ) \int \cosh (x) \, dx+\frac {1}{2} \left (3 c \sqrt {b^2-c^2}\right ) \int \sinh (x) \, dx\\ &=\frac {3}{2} \left (b^2-c^2\right ) x+\frac {3}{2} c \sqrt {b^2-c^2} \cosh (x)+\frac {3}{2} b \sqrt {b^2-c^2} \sinh (x)+\frac {1}{2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 72, normalized size = 0.80 \[ \frac {1}{4} \left (8 b \sqrt {b^2-c^2} \sinh (x)+\left (b^2+c^2\right ) \sinh (2 x)+8 c \sqrt {b^2-c^2} \cosh (x)+6 x (b-c) (b+c)+2 b c \cosh (2 x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 238, normalized size = 2.64 \[ \frac {{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \relax (x)^{4} + 12 \, {\left (b^{2} - c^{2}\right )} x \cosh \relax (x)^{2} + 6 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (b^{2} - c^{2}\right )} x\right )} \sinh \relax (x)^{2} - b^{2} + 2 \, b c - c^{2} + 4 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{3} + 6 \, {\left (b^{2} - c^{2}\right )} x \cosh \relax (x)\right )} \sinh \relax (x) + 8 \, {\left ({\left (b + c\right )} \cosh \relax (x)^{3} + 3 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (b + c\right )} \sinh \relax (x)^{3} - {\left (b - c\right )} \cosh \relax (x) + {\left (3 \, {\left (b + c\right )} \cosh \relax (x)^{2} - b + c\right )} \sinh \relax (x)\right )} \sqrt {b^{2} - c^{2}}}{8 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 96, normalized size = 1.07 \[ \sqrt {b^{2} - c^{2}} {\left (b + c\right )} e^{x} + \frac {3}{2} \, {\left (b^{2} - c^{2}\right )} x + \frac {1}{8} \, {\left (b^{2} + 2 \, b c + c^{2}\right )} e^{\left (2 \, x\right )} - \frac {1}{8} \, {\left (b^{2} - 2 \, b c + c^{2} + 8 \, {\left (\sqrt {b^{2} - c^{2}} b - \sqrt {b^{2} - c^{2}} c\right )} e^{x}\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 80, normalized size = 0.89 \[ b^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}+\frac {x}{2}\right )+c b \left (\cosh ^{2}\relax (x )\right )+c^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}-\frac {x}{2}\right )+2 b \sinh \relax (x ) \sqrt {b^{2}-c^{2}}+2 c \cosh \relax (x ) \sqrt {b^{2}-c^{2}}+b^{2} x -c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 79, normalized size = 0.88 \[ b c \cosh \relax (x)^{2} + \frac {1}{8} \, b^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - \frac {1}{8} \, c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + b^{2} x - c^{2} x + 2 \, \sqrt {b^{2} - c^{2}} {\left (c \cosh \relax (x) + b \sinh \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 70, normalized size = 0.78 \[ \frac {3\,b^2\,x}{2}-\frac {3\,c^2\,x}{2}+2\,c\,\mathrm {cosh}\relax (x)\,\sqrt {b^2-c^2}+2\,b\,\mathrm {sinh}\relax (x)\,\sqrt {b^2-c^2}+b\,c\,{\mathrm {cosh}\relax (x)}^2+\frac {b^2\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)}{2}+\frac {c^2\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 122, normalized size = 1.36 \[ - \frac {b^{2} x \sinh ^{2}{\relax (x )}}{2} + \frac {b^{2} x \cosh ^{2}{\relax (x )}}{2} + b^{2} x + \frac {b^{2} \sinh {\relax (x )} \cosh {\relax (x )}}{2} + b c \cosh ^{2}{\relax (x )} + 2 b \sqrt {b^{2} - c^{2}} \sinh {\relax (x )} + \frac {c^{2} x \sinh ^{2}{\relax (x )}}{2} - \frac {c^{2} x \cosh ^{2}{\relax (x )}}{2} - c^{2} x + \frac {c^{2} \sinh {\relax (x )} \cosh {\relax (x )}}{2} + 2 c \sqrt {b^{2} - c^{2}} \cosh {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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