Optimal. Leaf size=26 \[ \frac{1}{2} \tan ^{-1}(\tanh (x))+\frac{\tanh (x) \text{sech}^2(x)}{2 \left (\tanh ^2(x)+1\right )^2} \]
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Rubi [A] time = 0.0287418, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {413, 21, 203} \[ \frac{1}{2} \tan ^{-1}(\tanh (x))+\frac{\tanh (x) \text{sech}^2(x)}{2 \left (\tanh ^2(x)+1\right )^2} \]
Antiderivative was successfully verified.
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Rule 413
Rule 21
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^3} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{\left (1+x^2\right )^3} \, dx,x,\tanh (x)\right )\\ &=\frac{\text{sech}^2(x) \tanh (x)}{2 \left (1+\tanh ^2(x)\right )^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{2+2 x^2}{\left (1+x^2\right )^2} \, dx,x,\tanh (x)\right )\\ &=\frac{\text{sech}^2(x) \tanh (x)}{2 \left (1+\tanh ^2(x)\right )^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} \tan ^{-1}(\tanh (x))+\frac{\text{sech}^2(x) \tanh (x)}{2 \left (1+\tanh ^2(x)\right )^2}\\ \end{align*}
Mathematica [A] time = 0.006351, size = 22, normalized size = 0.85 \[ \frac{1}{4} \tan ^{-1}(\sinh (2 x))+\frac{1}{4} \tanh (2 x) \text{sech}(2 x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 166, normalized size = 6.4 \begin{align*} -2\,{\frac{-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{7}+1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{5}+1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-1/2\,\tanh \left ( x/2 \right ) }{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+6\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{\sqrt{2}}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }-{\frac{1}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }+{\frac{\sqrt{2}}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) }-{\frac{1}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.7561, size = 86, normalized size = 3.31 \begin{align*} \frac{e^{\left (-2 \, x\right )} - e^{\left (-6 \, x\right )}}{2 \,{\left (2 \, e^{\left (-4 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac{1}{2} \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \frac{1}{2} \, \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-x\right )}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24376, size = 1022, normalized size = 39.31 \begin{align*} \frac{\cosh \left (x\right )^{6} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} +{\left (15 \, \cosh \left (x\right )^{4} - 1\right )} \sinh \left (x\right )^{2} -{\left (\cosh \left (x\right )^{8} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \,{\left (35 \, \cosh \left (x\right )^{4} + 1\right )} \sinh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{4} + 8 \,{\left (7 \, \cosh \left (x\right )^{5} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \,{\left (7 \, \cosh \left (x\right )^{6} + 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 8 \,{\left (\cosh \left (x\right )^{7} + \cosh \left (x\right )^{3}\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (-\frac{\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \cosh \left (x\right )^{2} + 2 \,{\left (3 \, \cosh \left (x\right )^{5} - \cosh \left (x\right )\right )} \sinh \left (x\right )}{2 \,{\left (\cosh \left (x\right )^{8} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \,{\left (35 \, \cosh \left (x\right )^{4} + 1\right )} \sinh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{4} + 8 \,{\left (7 \, \cosh \left (x\right )^{5} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \,{\left (7 \, \cosh \left (x\right )^{6} + 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 8 \,{\left (\cosh \left (x\right )^{7} + \cosh \left (x\right )^{3}\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16879, size = 62, normalized size = 2.38 \begin{align*} \frac{e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}}{2 \,{\left ({\left (e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )}^{2} + 4\right )}} + \frac{1}{4} \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (4 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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