Optimal. Leaf size=59 \[ \frac {8 a^2 \sinh (c+d x)}{3 d \sqrt {a \cosh (c+d x)+a}}+\frac {2 a \sinh (c+d x) \sqrt {a \cosh (c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac {8 a^2 \sinh (c+d x)}{3 d \sqrt {a \cosh (c+d x)+a}}+\frac {2 a \sinh (c+d x) \sqrt {a \cosh (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rubi steps
\begin {align*} \int (a+a \cosh (c+d x))^{3/2} \, dx &=\frac {2 a \sqrt {a+a \cosh (c+d x)} \sinh (c+d x)}{3 d}+\frac {1}{3} (4 a) \int \sqrt {a+a \cosh (c+d x)} \, dx\\ &=\frac {8 a^2 \sinh (c+d x)}{3 d \sqrt {a+a \cosh (c+d x)}}+\frac {2 a \sqrt {a+a \cosh (c+d x)} \sinh (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 55, normalized size = 0.93 \[ \frac {a \left (9 \sinh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {3}{2} (c+d x)\right )\right ) \text {sech}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cosh (c+d x)+1)}}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 140, normalized size = 2.37 \[ \frac {\sqrt {\frac {1}{2}} {\left (a \cosh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{3} + 9 \, a \cosh \left (d x + c\right )^{2} + 3 \, {\left (a \cosh \left (d x + c\right ) + 3 \, a\right )} \sinh \left (d x + c\right )^{2} - 9 \, a \cosh \left (d x + c\right ) + 3 \, {\left (a \cosh \left (d x + c\right )^{2} + 6 \, a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right ) - a\right )} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{3 \, {\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 75, normalized size = 1.27 \[ -\frac {\sqrt {2} {\left ({\left (9 \, a^{\frac {3}{2}} e^{\left (d x + \frac {3}{2} \, c\right )} + a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, c\right )}\right )} e^{\left (-\frac {3}{2} \, d x - 2 \, c\right )} - {\left (a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, d x + \frac {15}{2} \, c\right )} + 9 \, a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, d x + \frac {13}{2} \, c\right )}\right )} e^{\left (-6 \, c\right )}\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 58, normalized size = 0.98 \[ \frac {4 a^{2} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+2\right ) \sqrt {2}}{3 \sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 81, normalized size = 1.37 \[ \frac {\sqrt {2} a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}}{6 \, d} + \frac {3 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} - \frac {3 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{2 \, d} - \frac {\sqrt {2} a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, d x - \frac {3}{2} \, c\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int \left (a \cosh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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