Optimal. Leaf size=38 \[ \frac{x}{2}+\frac{1}{6 (\tanh (x)+1)}+\frac{2 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.153718, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2074, 207, 618, 204} \[ \frac{x}{2}+\frac{1}{6 (\tanh (x)+1)}+\frac{2 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2074
Rule 207
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3}{-1+x^2-x^3+x^5} \, dx,x,\tanh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{6 (1+x)^2}+\frac{1}{2 \left (-1+x^2\right )}+\frac{1}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (x)\right )\\ &=\frac{1}{6 (1+\tanh (x))}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\tanh (x)\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{x}{2}+\frac{1}{6 (1+\tanh (x))}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \tanh (x)\right )\\ &=\frac{x}{2}+\frac{2 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{1}{6 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.131348, size = 40, normalized size = 1.05 \[ \frac{1}{36} \left (18 x-3 \sinh (2 x)+3 \cosh (2 x)-8 \sqrt{3} \tan ^{-1}\left (\frac{2 \tanh (x)-1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.071, size = 96, normalized size = 2.5 \begin{align*}{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{i}{9}}\sqrt{3}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( i\sqrt{3}-1 \right ) \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{i}{9}}\sqrt{3}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( -i\sqrt{3}-1 \right ) \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63195, size = 99, normalized size = 2.61 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, \sqrt{3} e^{\left (-x\right )} + 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, \sqrt{3} e^{\left (-x\right )} - 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{2} \, x + \frac{1}{12} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7515, size = 340, normalized size = 8.95 \begin{align*} \frac{18 \, x \cosh \left (x\right )^{2} + 36 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 18 \, x \sinh \left (x\right )^{2} + 8 \,{\left (\sqrt{3} \cosh \left (x\right )^{2} + 2 \, \sqrt{3} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{3} \sinh \left (x\right )^{2}\right )} \arctan \left (-\frac{\sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) + 3}{36 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.42531, size = 136, normalized size = 3.58 \begin{align*} \frac{9 x \sinh{\left (x \right )}}{18 \sinh{\left (x \right )} + 18 \cosh{\left (x \right )}} + \frac{9 x \cosh{\left (x \right )}}{18 \sinh{\left (x \right )} + 18 \cosh{\left (x \right )}} - \frac{4 \sqrt{3} \sinh{\left (x \right )} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sinh{\left (x \right )}}{3 \cosh{\left (x \right )}} - \frac{\sqrt{3}}{3} \right )}}{18 \sinh{\left (x \right )} + 18 \cosh{\left (x \right )}} - \frac{4 \sqrt{3} \cosh{\left (x \right )} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sinh{\left (x \right )}}{3 \cosh{\left (x \right )}} - \frac{\sqrt{3}}{3} \right )}}{18 \sinh{\left (x \right )} + 18 \cosh{\left (x \right )}} + \frac{3 \cosh{\left (x \right )}}{18 \sinh{\left (x \right )} + 18 \cosh{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16556, size = 45, normalized size = 1.18 \begin{align*} -\frac{1}{12} \,{\left (3 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} e^{\left (2 \, x\right )}\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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