Optimal. Leaf size=92 \[ \frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))+\frac {8 \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3113, 3112} \[ \frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))+\frac {8 \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 3112
Rule 3113
Rubi steps
\begin {align*} \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx &=\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {1}{3} \left (4 \sqrt {b^2-c^2}\right ) \int \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx\\ &=\frac {8 \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))}{3 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}\\ \end {align*}
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Mathematica [C] time = 73.41, size = 10141, normalized size = 110.23 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.44, size = 329, normalized size = 3.58 \[ \frac {\sqrt {\frac {1}{2}} {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \relax (x)^{4} - 18 \, {\left (b^{2} - c^{2}\right )} \cosh \relax (x)^{2} + 6 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{2} - 3 \, b^{2} + 3 \, c^{2}\right )} \sinh \relax (x)^{2} + b^{2} - 2 \, b c + c^{2} + 4 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{3} - 9 \, {\left (b^{2} - c^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 8 \, {\left ({\left (b + c\right )} \cosh \relax (x)^{3} + 3 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (b + c\right )} \sinh \relax (x)^{3} + {\left (b - c\right )} \cosh \relax (x) + {\left (3 \, {\left (b + c\right )} \cosh \relax (x)^{2} + b - c\right )} \sinh \relax (x)\right )} \sqrt {b^{2} - c^{2}}\right )} \sqrt {\frac {{\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + b - c}{\cosh \relax (x) + \sinh \relax (x)}}}{3 \, {\left ({\left (b + c\right )} \cosh \relax (x)^{3} + 3 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (b + c\right )} \sinh \relax (x)^{3} - {\left (b - c\right )} \cosh \relax (x) + {\left (3 \, {\left (b + c\right )} \cosh \relax (x)^{2} - b + c\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 303, normalized size = 3.29 \[ -\frac {\sqrt {2} {\left ({\left (\sqrt {b^{2} - c^{2}} b \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + \sqrt {b^{2} - c^{2}} c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (\frac {3}{2} \, x\right )} + 9 \, {\left (b^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (\frac {1}{2} \, x\right )} - 9 \, {\left (\sqrt {b^{2} - c^{2}} b \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - \sqrt {b^{2} - c^{2}} c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} - {\left (b^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - 2 \, b c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {3}{2} \, x\right )}\right )}}{6 \, \sqrt {b - c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.81, size = 190, normalized size = 2.07 \[ \frac {\left (-2 b^{2}+2 c^{2}\right ) \cosh \relax (x )}{\sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right ) \left (\sinh ^{2}\relax (x )\right )}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right )}\, \cosh \relax (x )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right ) \left (\sinh ^{2}\relax (x )\right )}}\right ) \left (b^{2}-c^{2}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right )}\, \sinh \relax (x ) \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.94, size = 640, normalized size = 6.96 \[ \frac {\sqrt {2} {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c\right )} {\left (2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c\right )}^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c + 3 \, {\left (b^{2} - c^{2}\right )} e^{\left (-x\right )} + 3 \, {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} e^{\left (-2 \, x\right )} + {\left (b^{2} - 2 \, b c + c^{2}\right )} e^{\left (-3 \, x\right )}\right )}} + \frac {3 \, \sqrt {2} {\left (b^{2} - c^{2}\right )} {\left (2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c\right )}^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )}}{2 \, {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c + 3 \, {\left (b^{2} - c^{2}\right )} e^{\left (-x\right )} + 3 \, {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} e^{\left (-2 \, x\right )} + {\left (b^{2} - 2 \, b c + c^{2}\right )} e^{\left (-3 \, x\right )}\right )}} - \frac {3 \, \sqrt {2} {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} {\left (2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c\right )}^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c + 3 \, {\left (b^{2} - c^{2}\right )} e^{\left (-x\right )} + 3 \, {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} e^{\left (-2 \, x\right )} + {\left (b^{2} - 2 \, b c + c^{2}\right )} e^{\left (-3 \, x\right )}\right )}} - \frac {\sqrt {2} {\left (b^{2} - 2 \, b c + c^{2}\right )} {\left (2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c\right )}^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )}}{6 \, {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c + 3 \, {\left (b^{2} - c^{2}\right )} e^{\left (-x\right )} + 3 \, {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} e^{\left (-2 \, x\right )} + {\left (b^{2} - 2 \, b c + c^{2}\right )} e^{\left (-3 \, x\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,\mathrm {cosh}\relax (x)+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \cosh {\relax (x )} + c \sinh {\relax (x )} + \sqrt {b^{2} - c^{2}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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