Optimal. Leaf size=23 \[ \frac {\cosh ^3(x)}{3 a}-\frac {\sinh ^2(x)}{2 a} \]
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Rubi [A] time = 0.12, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3872, 2835, 2564, 30, 2565} \[ \frac {\cosh ^3(x)}{3 a}-\frac {\sinh ^2(x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2565
Rule 2835
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\cosh (x) \sinh ^3(x)}{-a-a \cosh (x)} \, dx\\ &=-\frac {\int \cosh (x) \sinh (x) \, dx}{a}+\frac {\int \cosh ^2(x) \sinh (x) \, dx}{a}\\ &=\frac {\operatorname {Subst}(\int x \, dx,x,i \sinh (x))}{a}+\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )}{a}\\ &=\frac {\cosh ^3(x)}{3 a}-\frac {\sinh ^2(x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 23, normalized size = 1.00 \[ \frac {3 \cosh (x)-3 \cosh (2 x)+\cosh (3 x)-7}{12 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 30, normalized size = 1.30 \[ \frac {\cosh \relax (x)^{3} + 3 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x)^{2} - 3 \, \cosh \relax (x)^{2} + 3 \, \cosh \relax (x)}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 37, normalized size = 1.61 \[ \frac {{\left (3 \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 1\right )} e^{\left (-3 \, x\right )} + e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} + 3 \, e^{x}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 67, normalized size = 2.91 \[ \frac {-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {8}{8 \tanh \left (\frac {x}{2}\right )+8}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 46, normalized size = 2.00 \[ -\frac {{\left (3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )}}{24 \, a} + \frac {3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 53, normalized size = 2.30 \[ \frac {{\mathrm {e}}^{-x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {{\mathrm {e}}^{2\,x}}{8\,a}+\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}+\frac {{\mathrm {e}}^x}{8\,a} \]
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 \[ \frac {\int \frac {\sinh ^{3}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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