Optimal. Leaf size=24 \[ x \sqrt{b^2-c^2}+b \sinh (x)+c \cosh (x) \]
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Rubi [A] time = 0.0108072, antiderivative size = 24, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0080919, size = 24, normalized size = 1. \[ x \sqrt{b^2-c^2}+b \sinh (x)+c \cosh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 1. \begin{align*} c\cosh \left ( x \right ) +b\sinh \left ( x \right ) +x\sqrt{{b}^{2}-{c}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00005, size = 30, normalized size = 1.25 \begin{align*} c \cosh \left (x\right ) + b \sinh \left (x\right ) + \sqrt{b^{2} - c^{2}} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.72436, size = 196, normalized size = 8.17 \begin{align*} \frac{{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{b^{2} - c^{2}}{\left (x \cosh \left (x\right ) + x \sinh \left (x\right )\right )} - b + c}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.268791, size = 20, normalized size = 0.83 \begin{align*} b \sinh{\left (x \right )} + c \cosh{\left (x \right )} + x \sqrt{b^{2} - c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14882, size = 49, normalized size = 2.04 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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