Optimal. Leaf size=17 \[ \frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3} \]
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Rubi [A]
time = 0.08, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3599, 3187,
3186, 2645, 30, 2644} \begin {gather*} \frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rule 2645
Rule 3186
Rule 3187
Rule 3599
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{1+\coth (x)} \, dx &=-\left (i \int \frac {\cosh (x) \sinh (x)}{-i \cosh (x)-i \sinh (x)} \, dx\right )\\ &=-\int \cosh (x) \sinh (x) (-\cosh (x)+\sinh (x)) \, dx\\ &=i \int \left (-i \cosh ^2(x) \sinh (x)+i \cosh (x) \sinh ^2(x)\right ) \, dx\\ &=\int \cosh ^2(x) \sinh (x) \, dx-\int \cosh (x) \sinh ^2(x) \, dx\\ &=-\left (i \text {Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )\right )+\text {Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )\\ &=\frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.12 \begin {gather*} \frac {1}{12} \left (3 \cosh (x)+\cosh (3 x)-4 \sinh ^3(x)\right ) \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. \(41\) vs.
\(2(13)=26\).
time = 0.37, size = 42, normalized size = 2.47
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{-3 x}}{12}\) | \(12\) |
default | \(-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 11, normalized size = 0.65 \begin {gather*} \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 23, normalized size = 1.35 \begin {gather*} \frac {\cosh \left (x\right )^{2} + \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}{3 \, {\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh {\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 11, normalized size = 0.65 \begin {gather*} \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 11, normalized size = 0.65 \begin {gather*} \frac {{\mathrm {e}}^{-3\,x}}{12}+\frac {{\mathrm {e}}^x}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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