Optimal. Leaf size=27 \[ -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3071} \[ -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin {align*} \int (a \cosh (c+d x)-a \sinh (c+d x))^3 \, dx &=-\frac {(a \cosh (c+d x)-a \sinh (c+d x))^3}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 1.00 \[ -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 62, normalized size = 2.30 \[ -\frac {a^{3}}{3 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 17, normalized size = 0.63 \[ -\frac {a^{3} e^{\left (-3 \, d x - 3 \, c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 26, normalized size = 0.96 \[ -\frac {a^{3} \left (\cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )^{3}}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 147, normalized size = 5.44 \[ -\frac {a^{3} \cosh \left (d x + c\right )^{3}}{d} + \frac {a^{3} \sinh \left (d x + c\right )^{3}}{d} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} - \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.85, size = 17, normalized size = 0.63 \[ -\frac {a^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 83, normalized size = 3.07 \[ \begin {cases} \frac {a^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} + \frac {a^{3} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {a^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (- a \sinh {\relax (c )} + a \cosh {\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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