Optimal. Leaf size=33 \[ -\frac{1}{3 (\tanh (x)+1)}-\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.136227, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2074, 618, 204} \[ -\frac{1}{3 (\tanh (x)+1)}-\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2074
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)-\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x+x^2}{1+x+x^3+x^4} \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{3 (1+x)^2}+\frac{2}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{3 (1+\tanh (x))}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{3 (1+\tanh (x))}-\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \tanh (x)\right )\\ &=-\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{3 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.128968, size = 37, normalized size = 1.12 \[ \frac{1}{18} \left (3 \sinh (2 x)-3 \cosh (2 x)+8 \sqrt{3} \tan ^{-1}\left (\frac{2 \tanh (x)-1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.084, size = 78, normalized size = 2.4 \begin{align*} -{\frac{2}{3} \left ( 1+\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{2}{3} \left ( 1+\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{2\,i}{9}}\sqrt{3}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( -1-i\sqrt{3} \right ) \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{2\,i}{9}}\sqrt{3}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( -1+i\sqrt{3} \right ) \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 = 1.46238, size = 95, normalized size = 2.88 \begin{align*} \frac{4}{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{4}{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}{6} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.22686, size = 265, normalized size = 8.03 \begin{align*} -\frac{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}{18 \,{\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 = 1.99149, size = 102, normalized size = 3.09 \begin{align*} \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 )}}{9 \sinh{\left (x \right )} + 9 \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 )}}{9 \sinh{\left (x \right )} + 9 \cosh{\left (x \right )}} - \frac{3 \cosh{\left (x \right )}}{9 \sinh{\left (x \right )} + 9 \cosh{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08848, size = 30, normalized size = 0.91 \begin{align*} \frac{4}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} e^{\left (2 \, x\right )}\right ) - \frac{1}{6} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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