Optimal. Leaf size=38 \[ \frac{1}{16} \tan ^{-1}(\sinh (x))-\frac{1}{6} \tanh ^3(x) \text{sech}^3(x)-\frac{1}{8} \tanh (x) \text{sech}^3(x)+\frac{1}{16} \tanh (x) \text{sech}(x) \]
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Rubi [A] time = 0.053362, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2611, 3768, 3770} \[ \frac{1}{16} \tan ^{-1}(\sinh (x))-\frac{1}{6} \tanh ^3(x) \text{sech}^3(x)-\frac{1}{8} \tanh (x) \text{sech}^3(x)+\frac{1}{16} \tanh (x) \text{sech}(x) \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \text{sech}^3(x) \tanh ^4(x) \, dx &=-\frac{1}{6} \text{sech}^3(x) \tanh ^3(x)+\frac{1}{2} \int \text{sech}^3(x) \tanh ^2(x) \, dx\\ &=-\frac{1}{8} \text{sech}^3(x) \tanh (x)-\frac{1}{6} \text{sech}^3(x) \tanh ^3(x)+\frac{1}{8} \int \text{sech}^3(x) \, dx\\ &=\frac{1}{16} \text{sech}(x) \tanh (x)-\frac{1}{8} \text{sech}^3(x) \tanh (x)-\frac{1}{6} \text{sech}^3(x) \tanh ^3(x)+\frac{1}{16} \int \text{sech}(x) \, dx\\ &=\frac{1}{16} \tan ^{-1}(\sinh (x))+\frac{1}{16} \text{sech}(x) \tanh (x)-\frac{1}{8} \text{sech}^3(x) \tanh (x)-\frac{1}{6} \text{sech}^3(x) \tanh ^3(x)\\ \end{align*}
Mathematica [A] time = 0.009206, size = 48, normalized size = 1.26 \[ \frac{1}{16} \tan ^{-1}(\sinh (x))-\frac{1}{3} \tanh ^3(x) \text{sech}^3(x)-\frac{1}{6} \tanh (x) \text{sech}^5(x)+\frac{1}{24} \tanh (x) \text{sech}^3(x)+\frac{1}{16} \tanh (x) \text{sech}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 46, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{3}}{3\, \left ( \cosh \left ( x \right ) \right ) ^{6}}}-{\frac{\sinh \left ( x \right ) }{5\, \left ( \cosh \left ( x \right ) \right ) ^{6}}}+{\frac{\tanh \left ( x \right ) }{5} \left ({\frac{ \left ({\rm sech} \left (x\right ) \right ) ^{5}}{6}}+{\frac{5\, \left ({\rm sech} \left (x\right ) \right ) ^{3}}{24}}+{\frac{5\,{\rm sech} \left (x\right )}{16}} \right ) }+{\frac{\arctan \left ({{\rm e}^{x}} \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.72808, size = 115, normalized size = 3.03 \begin{align*} \frac{3 \, e^{\left (-x\right )} - 47 \, e^{\left (-3 \, x\right )} + 78 \, e^{\left (-5 \, x\right )} - 78 \, e^{\left (-7 \, x\right )} + 47 \, e^{\left (-9 \, x\right )} - 3 \, e^{\left (-11 \, x\right )}}{24 \,{\left (6 \, e^{\left (-2 \, x\right )} + 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} + e^{\left (-12 \, x\right )} + 1\right )}} - \frac{1}{8} \, \arctan \left (e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.80916, size = 3186, normalized size = 83.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh ^{4}{\left (x \right )} \operatorname{sech}^{3}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1813, size = 99, normalized size = 2.61 \begin{align*} \frac{1}{32} \, \pi - \frac{3 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{5} - 32 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 48 \, e^{\left (-x\right )} + 48 \, e^{x}}{24 \,{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{3}} + \frac{1}{16} \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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