Optimal. Leaf size=11 \[ \frac{1}{3} \tan ^{-1}\left (\frac{\tanh (x)}{3}\right ) \]
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Rubi [A] time = 0.0340017, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3675, 203} \[ \frac{1}{3} \tan ^{-1}\left (\frac{\tanh (x)}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 203
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{9+\tanh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{9+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{3} \tan ^{-1}\left (\frac{\tanh (x)}{3}\right )\\ \end{align*}
Mathematica [F] time = 0.0196953, size = 0, normalized size = 0. \[ \int \frac{\text{sech}^2(x)}{9+\tanh ^2(x)} \, dx \]
Verification is Not applicable to the result.
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Maple [B] time = 0.071, size = 116, normalized size = 10.6 \begin{align*} -2\,{\frac{\sqrt{10}}{6\,\sqrt{10}+6}\arctan \left ( 18\,{\frac{\tanh \left ( x/2 \right ) }{6\,\sqrt{10}+6}} \right ) }-2\,{\frac{1}{6\,\sqrt{10}+6}\arctan \left ( 18\,{\frac{\tanh \left ( x/2 \right ) }{6\,\sqrt{10}+6}} \right ) }+2\,{\frac{\sqrt{10}}{6\,\sqrt{10}-6}\arctan \left ( 18\,{\frac{\tanh \left ( x/2 \right ) }{6\,\sqrt{10}-6}} \right ) }-2\,{\frac{1}{6\,\sqrt{10}-6}\arctan \left ( 18\,{\frac{\tanh \left ( x/2 \right ) }{6\,\sqrt{10}-6}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57431, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{3} \, \arctan \left (\frac{5}{3} \, e^{\left (-2 \, x\right )} + \frac{4}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.33005, size = 82, normalized size = 7.45 \begin{align*} -\frac{1}{3} \, \arctan \left (-\frac{9 \, \cosh \left (x\right ) + \sinh \left (x\right )}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) \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 \frac{\operatorname{sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} + 9}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1588, size = 15, normalized size = 1.36 \begin{align*} \frac{1}{3} \, \arctan \left (\frac{5}{3} \, e^{\left (2 \, x\right )} + \frac{4}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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