Optimal. Leaf size=14 \[ x-\frac{1}{3} \tanh ^3(x)-\tanh (x) \]
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Rubi [A] time = 0.0111177, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3473, 8} \[ x-\frac{1}{3} \tanh ^3(x)-\tanh (x) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \tanh ^4(x) \, dx &=-\frac{1}{3} \tanh ^3(x)+\int \tanh ^2(x) \, dx\\ &=-\tanh (x)-\frac{\tanh ^3(x)}{3}+\int 1 \, dx\\ &=x-\tanh (x)-\frac{\tanh ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0030748, size = 18, normalized size = 1.29 \[ x-\frac{4 \tanh (x)}{3}+\frac{1}{3} \tanh (x) \text{sech}^2(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 26, normalized size = 1.9 \begin{align*} -{\frac{ \left ( \tanh \left ( x \right ) \right ) ^{3}}{3}}-\tanh \left ( x \right ) -{\frac{\ln \left ( -1+\tanh \left ( x \right ) \right ) }{2}}+{\frac{\ln \left ( 1+\tanh \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.948404, size = 51, normalized size = 3.64 \begin{align*} x - \frac{4 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 2\right )}}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10164, size = 221, normalized size = 15.79 \begin{align*} \frac{{\left (3 \, x + 4\right )} \cosh \left (x\right )^{3} + 3 \,{\left (3 \, x + 4\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} - 12 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) - 4 \, \sinh \left (x\right )^{3} + 3 \,{\left (3 \, x + 4\right )} \cosh \left (x\right )}{3 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.216398, size = 10, normalized size = 0.71 \begin{align*} x - \frac{\tanh ^{3}{\left (x \right )}}{3} - \tanh{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13156, size = 35, normalized size = 2.5 \begin{align*} x + \frac{4 \,{\left (3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 2\right )}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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