∫ (−√ ____ b2 − c2 + b cosh(x) + c sinh(x))5∕2 dx [774]

Optimal. Leaf size=146 \[ \frac {64 \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x))}{15 \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}-\frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \]

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Rubi [A]
time = 0.10, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3192, 3191} \begin {gather*} \frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}-\frac {16}{15} \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {64 \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x))}{15 \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \end {gather*}

\begin {align*} \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx &=\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}-\frac {1}{5} \left (8 \sqrt {b^2-c^2}\right ) \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx\\ &=-\frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {1}{15} \left (32 \left (b^2-c^2\right )\right ) \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx\\ &=\frac {64 \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x))}{15 \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}-\frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 96.87, size = 4368, normalized size = 29.92 \begin {gather*} \text {Result too large to show} \end {gather*}

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(274\) vs. \(2(126)=252\).
time = 3.20, size = 275, normalized size = 1.88


method	result	size

default	\(-\frac {\sqrt {\left (b -c \right ) \left (b +c \right )}\, \left (b -c \right ) \left (b +c \right ) \left (\frac {\left (\cosh ^{3}\left (x \right )\right )}{3}+2 \cosh \left (x \right )\right )}{\sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}\, \left (\frac {\left (b^{2}-c^{2}\right )^{2} \cosh \left (x \right ) \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}}{2 \sinh \left (x \right ) b^{2}-2 \sinh \left (x \right ) c^{2}+2 b^{2}-2 c^{2}}-\frac {\left (b^{2}-c^{2}\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}}\right )}{2 \sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}}\right )}{\sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}\)	\(275\)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1789 vs. \(2 (126) = 252\).
time = 2.47, size = 1789, normalized size = 12.25 \begin {gather*} \text {Too large to display} \end {gather*}

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (126) = 252\).
time = 0.37, size = 784, normalized size = 5.37 \begin {gather*} \frac {\sqrt {\frac {1}{2}} {\left (3 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{6} + 18 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + 3 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \sinh \left (x\right )^{6} + 125 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{4} + 5 \, {\left (25 \, b^{3} + 25 \, b^{2} c - 25 \, b c^{2} - 25 \, c^{3} + 9 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} + 20 \, {\left (3 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{3} + 25 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, b^{3} - 9 \, b^{2} c + 9 \, b c^{2} - 3 \, c^{3} + 125 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \cosh \left (x\right )^{2} + 5 \, {\left (9 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{4} + 25 \, b^{3} - 25 \, b^{2} c - 25 \, b c^{2} + 25 \, c^{3} + 150 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (9 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{5} + 250 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{3} + 125 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 2 \, {\left (11 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{5} + 55 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} + 11 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{5} - 150 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{3} + 10 \, {\left (11 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} - 15 \, b^{2} + 15 \, c^{2}\right )} \sinh \left (x\right )^{3} + 10 \, {\left (11 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} - 45 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 11 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} \cosh \left (x\right ) + {\left (55 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} - 450 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{2} + 11 \, b^{2} - 22 \, b c + 11 \, c^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}\right )} \sqrt {\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 (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + b - c}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{30 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b + c\right )} \sinh \left (x\right )^{4} - {\left (b - c\right )} \cosh \left (x\right )^{2} + {\left (6 \, {\left (b + c\right )} \cosh \left (x\right )^{2} - b + c\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, {\left (b + c\right )} \cosh \left (x\right )^{3} - {\left (b - c\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end {gather*}

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (126) = 252\).
time = 0.45, size = 315, normalized size = 2.16 \begin {gather*} -\frac {\sqrt {2} {\left (150 \, {\left (b^{2} - c^{2}\right )}^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) + 3 \, {\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^{\left (\frac {5}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - 25 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} e^{\left (\frac {3}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - {\left (25 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} \sqrt {b^{2} - c^{2}} e^{x} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - 150 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - 3 \, {\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )\right )} e^{\left (-\frac {5}{2} \, x\right )}\right )}}{60 \, \sqrt {b - c}} \end {gather*}

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,\mathrm {cosh}\left (x\right )-\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )\right )}^{5/2} \,d x \end {gather*}

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3.8.74 \(\int (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{5/2} \, dx\) [774]