Optimal. Leaf size=20 \[ \frac{2 \tanh ^{-1}\left (\frac{2 \tanh (x)+3}{\sqrt{17}}\right )}{\sqrt{17}} \]
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Rubi [A] time = 0.124094, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {618, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{2 \tanh (x)+3}{\sqrt{17}}\right )}{\sqrt{17}} \]
Antiderivative was successfully verified.
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Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{1+\text{sech}^2(x)-3 \tanh (x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{2-3 x-x^2} \, dx,x,\tanh (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{17-x^2} \, dx,x,-3-2 \tanh (x)\right )\right )\\ &=\frac{2 \tanh ^{-1}\left (\frac{3+2 \tanh (x)}{\sqrt{17}}\right )}{\sqrt{17}}\\ \end{align*}
Mathematica [F] time = 0.0291272, size = 0, normalized size = 0. \[ \int \frac{\text{sech}^2(x)}{1+\text{sech}^2(x)-3 \tanh (x)} \, dx \]
Verification is Not applicable to the result.
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Maple [B] time = 0.069, size = 63, normalized size = 3.2 \begin{align*}{\frac{\sqrt{17}}{17}\ln \left ( \sqrt{17}\tanh \left ({\frac{x}{2}} \right ) +2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-3\,\tanh \left ( x/2 \right ) +2 \right ) }-{\frac{\sqrt{17}}{17}\ln \left ( -\sqrt{17}\tanh \left ({\frac{x}{2}} \right ) +2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-3\,\tanh \left ( x/2 \right ) +2 \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}}{\operatorname{sech}\left (x\right )^{2} - 3 \, \tanh \left (x\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35321, size = 244, normalized size = 12.2 \begin{align*} \frac{1}{17} \, \sqrt{17} \log \left (\frac{3 \,{\left (\sqrt{17} - 5\right )} \cosh \left (x\right )^{2} - 2 \,{\left (3 \, \sqrt{17} - 11\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \,{\left (\sqrt{17} - 5\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{17} + 6}{\cosh \left (x\right )^{2} - 6 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 3}\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 )}}{- 3 \tanh{\left (x \right )} + \operatorname{sech}^{2}{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15557, size = 47, normalized size = 2.35 \begin{align*} -\frac{1}{17} \, \sqrt{17} \log \left (\frac{{\left | -\sqrt{17} + 2 \, e^{\left (2 \, x\right )} - 3 \right |}}{{\left | \sqrt{17} + 2 \, e^{\left (2 \, x\right )} - 3 \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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