Optimal. Leaf size=188 \[ \frac{35}{8} x \left (b^2-c^2\right )^2+\frac{35}{8} b \left (b^2-c^2\right )^{3/2} \sinh (x)+\frac{35}{8} c \left (b^2-c^2\right )^{3/2} \cosh (x)+\frac{1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac{7}{12} \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{35}{24} \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]
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Rubi [A] time = 0.147956, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3113, 2637, 2638} \[ \frac{35}{8} x \left (b^2-c^2\right )^2+\frac{35}{8} b \left (b^2-c^2\right )^{3/2} \sinh (x)+\frac{35}{8} c \left (b^2-c^2\right )^{3/2} \cosh (x)+\frac{1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac{7}{12} \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{35}{24} \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]
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 )^4 \, dx &=\frac{1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac{1}{4} \left (7 \sqrt{b^2-c^2}\right ) \int \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3 \, dx\\ &=\frac{7}{12} \sqrt{b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac{1}{12} \left (35 \left (b^2-c^2\right )\right ) \int \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx\\ &=\frac{35}{24} \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac{7}{12} \sqrt{b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac{1}{8} \left (35 \left (b^2-c^2\right )^{3/2}\right ) \int \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \, dx\\ &=\frac{35}{8} \left (b^2-c^2\right )^2 x+\frac{35}{24} \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac{7}{12} \sqrt{b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac{1}{8} \left (35 b \left (b^2-c^2\right )^{3/2}\right ) \int \cosh (x) \, dx+\frac{1}{8} \left (35 c \left (b^2-c^2\right )^{3/2}\right ) \int \sinh (x) \, dx\\ &=\frac{35}{8} \left (b^2-c^2\right )^2 x+\frac{35}{8} c \left (b^2-c^2\right )^{3/2} \cosh (x)+\frac{35}{8} b \left (b^2-c^2\right )^{3/2} \sinh (x)+\frac{35}{24} \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac{7}{12} \sqrt{b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac{1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3\\ \end{align*}
Mathematica [A] time = 0.493392, size = 208, normalized size = 1.11 \[ 7 b (b-c) \sqrt{b^2-c^2} (b+c) \sinh (x)+\frac{7}{4} \left (b^4-c^4\right ) \sinh (2 x)+\frac{1}{3} b \sqrt{b^2-c^2} \left (b^2+3 c^2\right ) \sinh (3 x)+\frac{1}{32} \left (6 b^2 c^2+b^4+c^4\right ) \sinh (4 x)+7 c (b-c) \sqrt{b^2-c^2} (b+c) \cosh (x)+\frac{7}{2} b c \left (b^2-c^2\right ) \cosh (2 x)+\frac{1}{3} c \sqrt{b^2-c^2} \left (3 b^2+c^2\right ) \cosh (3 x)+\frac{1}{8} b c \left (b^2+c^2\right ) \cosh (4 x)+\frac{35}{8} x (b-c)^2 (b+c)^2 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.082, size = 409, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02717, size = 374, normalized size = 1.99 \begin{align*} b^{3} c \cosh \left (x\right )^{4} + b c^{3} \sinh \left (x\right )^{4} + \frac{1}{64} \, b^{4}{\left (24 \, x + e^{\left (4 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} + \frac{1}{64} \, c^{4}{\left (24 \, x + e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} - \frac{3}{32} \, b^{2} c^{2}{\left (8 \, x - e^{\left (4 \, x\right )} + e^{\left (-4 \, x\right )}\right )} +{\left (b^{2} - c^{2}\right )}^{2} x + 4 \,{\left (b^{2} - c^{2}\right )}^{\frac{3}{2}}{\left (c \cosh \left (x\right ) + b \sinh \left (x\right )\right )} + \frac{3}{4} \,{\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 )}{\left (b^{2} - c^{2}\right )} + \frac{1}{6} \,{\left (24 \, b^{2} c \cosh \left (x\right )^{3} + 24 \, b c^{2} \sinh \left (x\right )^{3} + c^{3}{\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + b^{3}{\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\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 = 2.83112, size = 3148, normalized size = 16.74 \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 = 2.63169, size = 626, normalized size = 3.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17548, size = 527, normalized size = 2.8 \begin{align*} \frac{7}{2} \,{\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \sqrt{b^{2} - c^{2}} e^{x} + \frac{35}{8} \,{\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} x + \frac{1}{64} \,{\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} e^{\left (4 \, x\right )} + \frac{1}{6} \,{\left (\sqrt{b^{2} - c^{2}} b^{3} + 3 \, \sqrt{b^{2} - c^{2}} b^{2} c + 3 \, \sqrt{b^{2} - c^{2}} b c^{2} + \sqrt{b^{2} - c^{2}} c^{3}\right )} e^{\left (3 \, x\right )} + \frac{7}{8} \,{\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} e^{\left (2 \, x\right )} - \frac{1}{192} \,{\left (3 \, b^{4} - 12 \, b^{3} c + 18 \, b^{2} c^{2} - 12 \, b c^{3} + 3 \, c^{4} + 672 \,{\left (\sqrt{b^{2} - c^{2}} b^{3} - \sqrt{b^{2} - c^{2}} b^{2} c - \sqrt{b^{2} - c^{2}} b c^{2} + \sqrt{b^{2} - c^{2}} c^{3}\right )} e^{\left (3 \, x\right )} + 168 \,{\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (2 \, x\right )} + 32 \,{\left (\sqrt{b^{2} - c^{2}} b^{3} - 3 \, \sqrt{b^{2} - c^{2}} b^{2} c + 3 \, \sqrt{b^{2} - c^{2}} b c^{2} - \sqrt{b^{2} - c^{2}} c^{3}\right )} e^{x}\right )} e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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