Optimal. Leaf size=188 \[ \frac {35}{8} \left (b^2-c^2\right )^2 x+\frac {35}{8} c \left (b^2-c^2\right )^{3/2} \cosh (x)+\frac {35}{8} b \left (b^2-c^2\right )^{3/2} \sinh (x)+\frac {35}{24} \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {7}{12} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3 \]
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Rubi [A]
time = 0.11, antiderivative size = 188, normalized size of antiderivative = 1.00, 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 = {3192, 2717,
2718} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 2718
Rule 3192
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*}
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Mathematica [A]
time = 0.36, size = 208, normalized size = 1.11 \begin {gather*} \frac {35}{8} (b-c)^2 (b+c)^2 x+7 (b-c) c (b+c) \sqrt {b^2-c^2} \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)+7 b (b-c) (b+c) \sqrt {b^2-c^2} \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 (b^4+6 b^2 c^2+c^4\right ) \sinh (4 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(518\) vs.
\(2(164)=328\).
time = 1.31, size = 519, normalized size = 2.76
method | result | size |
risch | \(\frac {35 c^{4} x}{8}-\frac {\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-3 x} b^{3}}{6}+\frac {\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-3 x} c^{3}}{6}+\frac {{\mathrm e}^{-4 x} b^{3} c}{16}-\frac {3 \,{\mathrm e}^{-4 x} b^{2} c^{2}}{32}+\frac {{\mathrm e}^{-4 x} b \,c^{3}}{16}+\frac {\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-3 x} b^{2} c}{2}-\frac {\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-3 x} b \,c^{2}}{2}+\frac {\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{3 x} b^{2} c}{2}+\frac {\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{3 x} b \,c^{2}}{2}+\frac {\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{3 x} b^{3}}{6}+\frac {\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{3 x} c^{3}}{6}+\frac {7 \,{\mathrm e}^{2 x} b^{3} c}{4}-\frac {7 \,{\mathrm e}^{2 x} b \,c^{3}}{4}+\frac {7 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x} b^{3}}{2}-\frac {7 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x} c^{3}}{2}-\frac {7 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-x} b^{3}}{2}-\frac {7 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-x} c^{3}}{2}+\frac {7 \,{\mathrm e}^{-2 x} b^{3} c}{4}-\frac {7 \,{\mathrm e}^{-2 x} b \,c^{3}}{4}-\frac {35 x \,b^{2} c^{2}}{4}+\frac {{\mathrm e}^{4 x} b^{3} c}{16}+\frac {3 \,{\mathrm e}^{4 x} b^{2} c^{2}}{32}+\frac {{\mathrm e}^{4 x} b \,c^{3}}{16}+\frac {7 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x} b^{2} c}{2}-\frac {7 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x} b \,c^{2}}{2}+\frac {7 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-x} b^{2} c}{2}+\frac {7 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-x} b \,c^{2}}{2}+\frac {35 b^{4} x}{8}+\frac {{\mathrm e}^{4 x} b^{4}}{64}+\frac {{\mathrm e}^{4 x} c^{4}}{64}+\frac {7 \,{\mathrm e}^{2 x} b^{4}}{8}-\frac {7 \,{\mathrm e}^{2 x} c^{4}}{8}-\frac {7 \,{\mathrm e}^{-2 x} b^{4}}{8}+\frac {7 \,{\mathrm e}^{-2 x} c^{4}}{8}-\frac {{\mathrm e}^{-4 x} b^{4}}{64}-\frac {{\mathrm e}^{-4 x} c^{4}}{64}\) | \(519\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 277, normalized size = 1.47 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1293 vs.
\(2 (164) = 328\).
time = 0.52, size = 1293, normalized size = 6.88 \begin {gather*} \frac {3 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{8} + 24 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{7} + 3 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \sinh \left (x\right )^{8} + 168 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{6} + 84 \, {\left (2 \, b^{4} + 4 \, b^{3} c - 4 \, b c^{3} - 2 \, c^{4} + {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{6} + 840 \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} x \cosh \left (x\right )^{4} + 168 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{3} + 6 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 210 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{4} + 12 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} x\right )} \sinh \left (x\right )^{4} - 3 \, b^{4} + 12 \, b^{3} c - 18 \, b^{2} c^{2} + 12 \, b c^{3} - 3 \, c^{4} + 168 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{5} + 20 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{3} + 20 \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 168 \, {\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{2} + 84 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{6} + 30 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{4} - 2 \, b^{4} + 4 \, b^{3} c - 4 \, b c^{3} + 2 \, c^{4} + 60 \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} x \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 24 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{7} + 42 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{5} + 140 \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} x \cosh \left (x\right )^{3} - 14 \, {\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 32 \, {\left ({\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{7} + 7 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{6} + {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \sinh \left (x\right )^{7} + 21 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{5} + 21 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3} + {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{5} + 35 \, {\left ({\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} - 21 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \cosh \left (x\right )^{3} + 7 \, {\left (5 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{4} - 3 \, b^{3} + 3 \, b^{2} c + 3 \, b c^{2} - 3 \, c^{3} + 30 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{3} + 21 \, {\left ({\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{5} + 10 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - {\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} \cosh \left (x\right ) + {\left (7 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{6} + 105 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{4} - b^{3} + 3 \, b^{2} c - 3 \, b c^{2} + c^{3} - 63 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}}{192 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 626 vs.
\(2 (178) = 356\).
time = 0.37, size = 626, normalized size = 3.33 \begin {gather*} \frac {3 b^{4} x \sinh ^{4}{\left (x \right )}}{8} - \frac {3 b^{4} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{4} - 3 b^{4} x \sinh ^{2}{\left (x \right )} + \frac {3 b^{4} x \cosh ^{4}{\left (x \right )}}{8} + 3 b^{4} x \cosh ^{2}{\left (x \right )} + b^{4} x - \frac {3 b^{4} \sinh ^{3}{\left (x \right )} \cosh {\left (x \right )}}{8} + \frac {5 b^{4} \sinh {\left (x \right )} \cosh ^{3}{\left (x \right )}}{8} + 3 b^{4} \sinh {\left (x \right )} \cosh {\left (x \right )} + b^{3} c \cosh ^{4}{\left (x \right )} + 6 b^{3} c \cosh ^{2}{\left (x \right )} - \frac {8 b^{3} \sqrt {b^{2} - c^{2}} \sinh ^{3}{\left (x \right )}}{3} + 4 b^{3} \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} \cosh ^{2}{\left (x \right )} + 4 b^{3} \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} - \frac {3 b^{2} c^{2} x \sinh ^{4}{\left (x \right )}}{4} + \frac {3 b^{2} c^{2} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{2} + 6 b^{2} c^{2} x \sinh ^{2}{\left (x \right )} - \frac {3 b^{2} c^{2} x \cosh ^{4}{\left (x \right )}}{4} - 6 b^{2} c^{2} x \cosh ^{2}{\left (x \right )} - 2 b^{2} c^{2} x + \frac {3 b^{2} c^{2} \sinh ^{3}{\left (x \right )} \cosh {\left (x \right )}}{4} + \frac {3 b^{2} c^{2} \sinh {\left (x \right )} \cosh ^{3}{\left (x \right )}}{4} + 4 b^{2} c \sqrt {b^{2} - c^{2}} \cosh ^{3}{\left (x \right )} + 4 b^{2} c \sqrt {b^{2} - c^{2}} \cosh {\left (x \right )} + b c^{3} \sinh ^{4}{\left (x \right )} - 6 b c^{3} \cosh ^{2}{\left (x \right )} + 4 b c^{2} \sqrt {b^{2} - c^{2}} \sinh ^{3}{\left (x \right )} - 4 b c^{2} \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} + \frac {3 c^{4} x \sinh ^{4}{\left (x \right )}}{8} - \frac {3 c^{4} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{4} - 3 c^{4} x \sinh ^{2}{\left (x \right )} + \frac {3 c^{4} x \cosh ^{4}{\left (x \right )}}{8} + 3 c^{4} x \cosh ^{2}{\left (x \right )} + c^{4} x + \frac {5 c^{4} \sinh ^{3}{\left (x \right )} \cosh {\left (x \right )}}{8} - \frac {3 c^{4} \sinh {\left (x \right )} \cosh ^{3}{\left (x \right )}}{8} - 3 c^{4} \sinh {\left (x \right )} \cosh {\left (x \right )} + 4 c^{3} \sqrt {b^{2} - c^{2}} \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )} - \frac {8 c^{3} \sqrt {b^{2} - c^{2}} \cosh ^{3}{\left (x \right )}}{3} - 4 c^{3} \sqrt {b^{2} - c^{2}} \cosh {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 390 vs.
\(2 (164) = 328\).
time = 0.43, size = 390, normalized size = 2.07 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.45, size = 361, normalized size = 1.92 \begin {gather*} x\,{\left (b^2-c^2\right )}^2-{\mathrm {cosh}\left (x\right )}^2\,\left (6\,b\,c^3-6\,b^3\,c\right )-{\mathrm {cosh}\left (x\right )}^4\,\left (b\,c^3-b^3\,c\right )+\mathrm {cosh}\left (x\right )\,{\mathrm {sinh}\left (x\right )}^3\,\left (-\frac {3\,b^4}{8}+\frac {3\,b^2\,c^2}{4}+\frac {5\,c^4}{8}\right )+{\mathrm {cosh}\left (x\right )}^3\,\mathrm {sinh}\left (x\right )\,\left (\frac {5\,b^4}{8}+\frac {3\,b^2\,c^2}{4}-\frac {3\,c^4}{8}\right )+4\,c\,\mathrm {cosh}\left (x\right )\,{\left (b^2-c^2\right )}^{3/2}+4\,b\,\mathrm {sinh}\left (x\right )\,{\left (b^2-c^2\right )}^{3/2}+3\,x\,{\mathrm {cosh}\left (x\right )}^2\,{\left (b^2-c^2\right )}^2+\frac {3\,x\,{\mathrm {cosh}\left (x\right )}^4\,{\left (b^2-c^2\right )}^2}{8}-3\,x\,{\mathrm {sinh}\left (x\right )}^2\,{\left (b^2-c^2\right )}^2+\frac {3\,x\,{\mathrm {sinh}\left (x\right )}^4\,{\left (b^2-c^2\right )}^2}{8}+\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\left (3\,b^4-3\,c^4\right )+2\,b\,c^3\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {sinh}\left (x\right )}^2+\frac {4\,c\,{\mathrm {cosh}\left (x\right )}^3\,\sqrt {b^2-c^2}\,\left (3\,b^2-2\,c^2\right )}{3}-\frac {4\,b\,{\mathrm {sinh}\left (x\right )}^3\,\sqrt {b^2-c^2}\,\left (2\,b^2-3\,c^2\right )}{3}+4\,b^3\,{\mathrm {cosh}\left (x\right )}^2\,\mathrm {sinh}\left (x\right )\,\sqrt {b^2-c^2}+4\,c^3\,\mathrm {cosh}\left (x\right )\,{\mathrm {sinh}\left (x\right )}^2\,\sqrt {b^2-c^2}-\frac {3\,x\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {sinh}\left (x\right )}^2\,{\left (b^2-c^2\right )}^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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