Optimal. Leaf size=25 \[ -\frac {1}{3} \cosh ^3(x)+\frac {\cosh ^5(x)}{5}-\frac {\sinh ^5(x)}{5} \]
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Rubi [A]
time = 0.12, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3599, 3187,
3186, 2645, 14, 2644, 30} \begin {gather*} -\frac {\sinh ^5(x)}{5}+\frac {\cosh ^5(x)}{5}-\frac {\cosh ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 3186
Rule 3187
Rule 3599
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx &=\int \frac {\cosh (x) \sinh ^3(x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \cosh (x) (-i \cosh (x)+i \sinh (x)) \sinh ^3(x) \, dx\\ &=-\int \left (-\cosh ^2(x) \sinh ^3(x)+\cosh (x) \sinh ^4(x)\right ) \, dx\\ &=\int \cosh ^2(x) \sinh ^3(x) \, dx-\int \cosh (x) \sinh ^4(x) \, dx\\ &=i \text {Subst}\left (\int x^4 \, dx,x,i \sinh (x)\right )-\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{5} \sinh ^5(x)-\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{3} \cosh ^3(x)+\frac {\cosh ^5(x)}{5}-\frac {\sinh ^5(x)}{5}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 34, normalized size = 1.36 \begin {gather*} \frac {1}{120} (\cosh (x)-\sinh (x)) (-20 \cosh (2 x)+4 \cosh (4 x)-10 \sinh (2 x)+\sinh (4 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs.
\(2(19)=38\).
time = 0.49, size = 72, normalized size = 2.88
method | result | size |
risch | \(\frac {{\mathrm e}^{3 x}}{48}-\frac {{\mathrm e}^{x}}{8}-\frac {{\mathrm e}^{-3 x}}{24}+\frac {{\mathrm e}^{-5 x}}{80}\) | \(24\) |
default | \(-\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )-8}+\frac {2}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 27, normalized size = 1.08 \begin {gather*} -\frac {1}{48} \, {\left (6 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{80} \, e^{\left (-5 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (19) = 38\).
time = 0.44, size = 56, normalized size = 2.24 \begin {gather*} \frac {\cosh \left (x\right )^{4} + \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 5 \, \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )}{30 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs.
\(2 (19) = 38\).
time = 0.37, size = 134, normalized size = 5.36 \begin {gather*} \frac {3 \sinh ^{3}{\left (x \right )} \tanh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} - \frac {3 \sinh ^{3}{\left (x \right )}}{15 \tanh {\left (x \right )} + 15} + \frac {6 \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )} \tanh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} + \frac {9 \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} - \frac {6 \sinh {\left (x \right )} \cosh ^{2}{\left (x \right )} \tanh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} + \frac {6 \sinh {\left (x \right )} \cosh ^{2}{\left (x \right )}}{15 \tanh {\left (x \right )} + 15} - \frac {8 \cosh ^{3}{\left (x \right )} \tanh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} - \frac {2 \cosh ^{3}{\left (x \right )}}{15 \tanh {\left (x \right )} + 15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{240} \, {\left (10 \, e^{\left (2 \, x\right )} - 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{48} \, e^{\left (3 \, x\right )} - \frac {1}{8} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 23, normalized size = 0.92 \begin {gather*} \frac {{\mathrm {e}}^{3\,x}}{48}-\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{-5\,x}}{80}-\frac {{\mathrm {e}}^x}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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