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 = 0.048117, antiderivative size = 90, normalized size of antiderivative = 1., 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 3113
Rule 2637
Rule 2638
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*}
Mathematica [A] time = 0.113133, size = 72, normalized size = 0.8 \[ \frac{1}{4} \left (8 b \sqrt{b^2-c^2} \sinh (x)+\left (b^2+c^2\right ) \sinh (2 x)+8 c \sqrt{b^2-c^2} \cosh (x)+6 x (b-c) (b+c)+2 b c \cosh (2 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 80, normalized size = 0.9 \begin{align*}{b}^{2} \left ({\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}}+{\frac{x}{2}} \right ) +b \left ( \cosh \left ( x \right ) \right ) ^{2}c+{c}^{2} \left ({\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}}-{\frac{x}{2}} \right ) +2\,b\sinh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}+2\,c\cosh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}+{b}^{2}x-{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17802, size = 107, normalized size = 1.19 \begin{align*} b c \cosh \left (x\right )^{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 \left (x\right ) + b \sinh \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.59504, size = 656, normalized size = 7.29 \begin{align*} \frac{{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 4 \,{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} +{\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} + 12 \,{\left (b^{2} - c^{2}\right )} x \cosh \left (x\right )^{2} + 6 \,{\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b^{2} - c^{2}\right )} x\right )} \sinh \left (x\right )^{2} - b^{2} + 2 \, b c - c^{2} + 4 \,{\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} + 6 \,{\left (b^{2} - c^{2}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right ) + 8 \,{\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \,{\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} +{\left (b + c\right )} \sinh \left (x\right )^{3} -{\left (b - c\right )} \cosh \left (x\right ) +{\left (3 \,{\left (b + c\right )} \cosh \left (x\right )^{2} - b + c\right )} \sinh \left (x\right )\right )} \sqrt{b^{2} - c^{2}}}{8 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.497405, size = 122, normalized size = 1.36 \begin{align*} - \frac{b^{2} x \sinh ^{2}{\left (x \right )}}{2} + \frac{b^{2} x \cosh ^{2}{\left (x \right )}}{2} + b^{2} x + \frac{b^{2} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} + b c \sinh ^{2}{\left (x \right )} + 2 b \sqrt{b^{2} - c^{2}} \sinh{\left (x \right )} + \frac{c^{2} x \sinh ^{2}{\left (x \right )}}{2} - \frac{c^{2} x \cosh ^{2}{\left (x \right )}}{2} - c^{2} x + \frac{c^{2} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} + 2 c \sqrt{b^{2} - c^{2}} \cosh{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16837, size = 130, normalized size = 1.44 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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