Optimal. Leaf size=34 \[ -\frac{5 \sinh ^3(x)}{6}+\frac{5 \sinh (x)}{2}+\frac{1}{2} \sinh ^3(x) \tanh ^2(x)-\frac{5}{2} \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.0475908, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4397, 2592, 288, 302, 203} \[ -\frac{5 \sinh ^3(x)}{6}+\frac{5 \sinh (x)}{2}+\frac{1}{2} \sinh ^3(x) \tanh ^2(x)-\frac{5}{2} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2592
Rule 288
Rule 302
Rule 203
Rubi steps
\begin{align*} \int (-\cosh (x)+\text{sech}(x))^3 \, dx &=-\int \sinh ^3(x) \tanh ^3(x) \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^2} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{2} \sinh ^3(x) \tanh ^2(x)-\frac{5}{2} \operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{2} \sinh ^3(x) \tanh ^2(x)-\frac{5}{2} \operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,\sinh (x)\right )\\ &=\frac{5 \sinh (x)}{2}-\frac{5 \sinh ^3(x)}{6}+\frac{1}{2} \sinh ^3(x) \tanh ^2(x)-\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{5}{2} \tan ^{-1}(\sinh (x))+\frac{5 \sinh (x)}{2}-\frac{5 \sinh ^3(x)}{6}+\frac{1}{2} \sinh ^3(x) \tanh ^2(x)\\ \end{align*}
Mathematica [A] time = 0.0281505, size = 37, normalized size = 1.09 \[ -\frac{1}{48} \text{sech}^2(x) \left (-50 \sinh (x)-25 \sinh (3 x)+\sinh (5 x)+60 \tan ^{-1}(\sinh (x))+60 \cosh (2 x) \tan ^{-1}(\sinh (x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 29, normalized size = 0.9 \begin{align*} - \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( x \right ) +3\,\sinh \left ( x \right ) -5\,\arctan \left ({{\rm e}^{x}} \right ) +{\frac{{\rm sech} \left (x\right )\tanh \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.75132, size = 76, normalized size = 2.24 \begin{align*} \frac{e^{\left (-x\right )} - e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 5 \, \arctan \left (e^{\left (-x\right )}\right ) - \frac{1}{24} \, e^{\left (3 \, x\right )} - \frac{9}{8} \, e^{\left (-x\right )} + \frac{1}{24} \, e^{\left (-3 \, x\right )} + \frac{9}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07347, size = 1648, normalized size = 48.47 \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 3 \cosh{\left (x \right )} \operatorname{sech}^{2}{\left (x \right )}\, dx - \int - 3 \cosh ^{2}{\left (x \right )} \operatorname{sech}{\left (x \right )}\, dx - \int \cosh ^{3}{\left (x \right )}\, dx - \int - \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 = 1.15187, size = 89, normalized size = 2.62 \begin{align*} -\frac{5}{4} \, \pi + \frac{1}{24} \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - \frac{e^{\left (-x\right )} - e^{x}}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4} - \frac{5}{2} \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - e^{\left (-x\right )} + e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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