Optimal. Leaf size=24 \[ x \sqrt {b^2-c^2}+b \sinh (x)+c \cosh (x) \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2637, 2638} \[ x \sqrt {b^2-c^2}+b \sinh (x)+c \cosh (x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin {align*} \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \, dx &=\sqrt {b^2-c^2} x+b \int \cosh (x) \, dx+c \int \sinh (x) \, dx\\ &=\sqrt {b^2-c^2} x+c \cosh (x)+b \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ x \sqrt {b^2-c^2}+b \sinh (x)+c \cosh (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 61, normalized size = 2.54 \[ \frac {{\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {b^{2} - c^{2}} {\left (x \cosh \relax (x) + x \sinh \relax (x)\right )} - b + c}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 36, normalized size = 1.50 \[ \frac {1}{2} \, c {\left (e^{\left (-x\right )} + e^{x}\right )} - \frac {1}{2} \, b {\left (e^{\left (-x\right )} - e^{x}\right )} + \sqrt {b^{2} - c^{2}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 23, normalized size = 0.96 \[ c \cosh \relax (x )+b \sinh \relax (x )+x \sqrt {b^{2}-c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 22, normalized size = 0.92 \[ c \cosh \relax (x) + b \sinh \relax (x) + \sqrt {b^{2} - c^{2}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 22, normalized size = 0.92 \[ x\,\sqrt {b^2-c^2}+c\,\mathrm {cosh}\relax (x)+b\,\mathrm {sinh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.83 \[ b \sinh {\relax (x )} + c \cosh {\relax (x )} + x \sqrt {b^{2} - c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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