Optimal. Leaf size=136 \[ \frac{5}{2} x \left (b^2-c^2\right )^{3/2}+\frac{5}{2} b \left (b^2-c^2\right ) \sinh (x)+\frac{5}{2} c \left (b^2-c^2\right ) \cosh (x)+\frac{1}{3} (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{5}{6} \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]
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Rubi [A] time = 0.0896551, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3113, 2637, 2638} \[ \frac{5}{2} x \left (b^2-c^2\right )^{3/2}+\frac{5}{2} b \left (b^2-c^2\right ) \sinh (x)+\frac{5}{2} c \left (b^2-c^2\right ) \cosh (x)+\frac{1}{3} (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{5}{6} \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]
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 )^3 \, dx &=\frac{1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{1}{3} \left (5 \sqrt{b^2-c^2}\right ) \int \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx\\ &=\frac{5}{6} \sqrt{b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac{1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{1}{2} \left (5 \left (b^2-c^2\right )\right ) \int \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \, dx\\ &=\frac{5}{2} \left (b^2-c^2\right )^{3/2} x+\frac{5}{6} \sqrt{b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac{1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{1}{2} \left (5 b \left (b^2-c^2\right )\right ) \int \cosh (x) \, dx+\frac{1}{2} \left (5 c \left (b^2-c^2\right )\right ) \int \sinh (x) \, dx\\ &=\frac{5}{2} \left (b^2-c^2\right )^{3/2} x+\frac{5}{2} c \left (b^2-c^2\right ) \cosh (x)+\frac{5}{2} b \left (b^2-c^2\right ) \sinh (x)+\frac{5}{6} \sqrt{b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac{1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.257189, size = 134, normalized size = 0.99 \[ \frac{1}{12} \left (30 x (b-c) (b+c) \sqrt{b^2-c^2}+45 b \left (b^2-c^2\right ) \sinh (x)+9 \sqrt{b^2-c^2} \left (b^2+c^2\right ) \sinh (2 x)+b \left (b^2+3 c^2\right ) \sinh (3 x)+45 c \left (b^2-c^2\right ) \cosh (x)+18 b c \sqrt{b^2-c^2} \cosh (2 x)+c \left (3 b^2+c^2\right ) \cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 202, normalized size = 1.5 \begin{align*}{b}^{3} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( x \right ) +3\,c{b}^{2} \left ( 1/3\,\cosh \left ( x \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}+1/3\,\cosh \left ( x \right ) \right ) +3\,\sqrt{{b}^{2}-{c}^{2}}{b}^{2} \left ( 1/2\,\cosh \left ( x \right ) \sinh \left ( x \right ) +x/2 \right ) +3\,b{c}^{2} \left ( 1/3\,\sinh \left ( x \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}-1/3\,\sinh \left ( x \right ) \right ) +3\,\sqrt{{b}^{2}-{c}^{2}}bc \left ( \cosh \left ( x \right ) \right ) ^{2}+3\,{b}^{3}\sinh \left ( x \right ) -3\,b{c}^{2}\sinh \left ( x \right ) +{c}^{3} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( x \right ) +3\,\sqrt{{b}^{2}-{c}^{2}}{c}^{2} \left ( 1/2\,\cosh \left ( x \right ) \sinh \left ( x \right ) -x/2 \right ) +3\,{b}^{2}\cosh \left ( x \right ) c-3\,{c}^{3}\cosh \left ( x \right ) +\sqrt{{b}^{2}-{c}^{2}}{b}^{2}x-\sqrt{{b}^{2}-{c}^{2}}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03399, size = 217, normalized size = 1.6 \begin{align*} b^{2} c \cosh \left (x\right )^{3} + b c^{2} \sinh \left (x\right )^{3} + \frac{1}{24} \, c^{3}{\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + \frac{1}{24} \, b^{3}{\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )} +{\left (b^{2} - c^{2}\right )}^{\frac{3}{2}} x + 3 \,{\left (b^{2} - c^{2}\right )}{\left (c \cosh \left (x\right ) + b \sinh \left (x\right )\right )} + \frac{3}{8} \,{\left (8 \, b c \cosh \left (x\right )^{2} + b^{2}{\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - c^{2}{\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}\right )} \sqrt{b^{2} - c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.64613, size = 1665, normalized size = 12.24 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.25663, size = 298, normalized size = 2.19 \begin{align*} - \frac{2 b^{3} \sinh ^{3}{\left (x \right )}}{3} + b^{3} \sinh{\left (x \right )} \cosh ^{2}{\left (x \right )} + 3 b^{3} \sinh{\left (x \right )} + b^{2} c \cosh ^{3}{\left (x \right )} + 3 b^{2} c \cosh{\left (x \right )} - \frac{3 b^{2} x \sqrt{b^{2} - c^{2}} \sinh ^{2}{\left (x \right )}}{2} + \frac{3 b^{2} x \sqrt{b^{2} - c^{2}} \cosh ^{2}{\left (x \right )}}{2} + b^{2} x \sqrt{b^{2} - c^{2}} + \frac{3 b^{2} \sqrt{b^{2} - c^{2}} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} + b c^{2} \sinh ^{3}{\left (x \right )} - 3 b c^{2} \sinh{\left (x \right )} + 3 b c \sqrt{b^{2} - c^{2}} \sinh ^{2}{\left (x \right )} + c^{3} \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )} - \frac{2 c^{3} \cosh ^{3}{\left (x \right )}}{3} - 3 c^{3} \cosh{\left (x \right )} + \frac{3 c^{2} x \sqrt{b^{2} - c^{2}} \sinh ^{2}{\left (x \right )}}{2} - \frac{3 c^{2} x \sqrt{b^{2} - c^{2}} \cosh ^{2}{\left (x \right )}}{2} - c^{2} x \sqrt{b^{2} - c^{2}} + \frac{3 c^{2} \sqrt{b^{2} - c^{2}} \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15932, size = 262, normalized size = 1.93 \begin{align*} \frac{5}{2} \,{\left (b^{2} - c^{2}\right )}^{\frac{3}{2}} x + \frac{3}{8} \,{\left (b^{2} + 2 \, b c + c^{2}\right )} \sqrt{b^{2} - c^{2}} e^{\left (2 \, x\right )} + \frac{1}{24} \,{\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} e^{\left (3 \, x\right )} - \frac{1}{24} \,{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3} + 45 \,{\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} + 9 \,{\left (\sqrt{b^{2} - c^{2}} b^{2} - 2 \, \sqrt{b^{2} - c^{2}} b c + \sqrt{b^{2} - c^{2}} c^{2}\right )} e^{x}\right )} e^{\left (-3 \, x\right )} + \frac{15}{8} \,{\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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