Optimal. Leaf size=22 \[ -\frac{2 \tan ^{-1}\left (\sqrt{\frac{5}{39}} (1-2 \tanh (x))\right )}{\sqrt{195}} \]
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Rubi [A] time = 0.0680059, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4342, 618, 204} \[ -\frac{2 \tan ^{-1}\left (\sqrt{\frac{5}{39}} (1-2 \tanh (x))\right )}{\sqrt{195}} \]
Antiderivative was successfully verified.
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Rule 4342
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{11-5 \tanh (x)+5 \tanh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{11-5 x+5 x^2} \, dx,x,\tanh (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-195-x^2} \, dx,x,-5+10 \tanh (x)\right )\right )\\ &=-\frac{2 \tan ^{-1}\left (\sqrt{\frac{5}{39}} (1-2 \tanh (x))\right )}{\sqrt{195}}\\ \end{align*}
Mathematica [F] time = 0.0307312, size = 0, normalized size = 0. \[ \int \frac{\text{sech}^2(x)}{11-5 \tanh (x)+5 \tanh ^2(x)} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 0.069, size = 62, normalized size = 2.8 \begin{align*}{\frac{i}{195}}\sqrt{195}\ln \left ( 11\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+ \left ( -i\sqrt{195}-5 \right ) \tanh \left ({\frac{x}{2}} \right ) +11 \right ) -{\frac{i}{195}}\sqrt{195}\ln \left ( 11\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+ \left ( i\sqrt{195}-5 \right ) \tanh \left ({\frac{x}{2}} \right ) +11 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (x\right )^{2}}{5 \, \tanh \left (x\right )^{2} - 5 \, \tanh \left (x\right ) + 11}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09234, size = 132, normalized size = 6. \begin{align*} -\frac{2}{195} \, \sqrt{195} \arctan \left (-\frac{17 \, \sqrt{195} \cosh \left (x\right ) + 5 \, \sqrt{195} \sinh \left (x\right )}{195 \,{\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 )}}{5 \tanh ^{2}{\left (x \right )} - 5 \tanh{\left (x \right )} + 11}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15518, size = 26, normalized size = 1.18 \begin{align*} \frac{2}{195} \, \sqrt{195} \arctan \left (\frac{1}{195} \, \sqrt{195}{\left (11 \, e^{\left (2 \, x\right )} + 6\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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