Optimal. Leaf size=9 \[ \frac {1}{2} \text {ArcSin}(2 \tanh (x)) \]
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Rubi [A]
time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3756, 222}
\begin {gather*} \frac {1}{2} \text {ArcSin}(2 \tanh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 3756
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x)}{\sqrt {1-4 \tanh ^2(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt {1-4 x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \sin ^{-1}(2 \tanh (x))\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(9)=18\).
time = 0.04, size = 47, normalized size = 5.22 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {-1+3 \sinh ^2(x)}}\right ) \sqrt {-5+3 \cosh (2 x)} \text {sech}(x)}{2 \sqrt {2-8 \tanh ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.22, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {sech}\left (x \right )^{2}}{\sqrt {1-4 \left (\tanh ^{2}\left (x \right )\right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs.
\(2 (7) = 14\).
time = 0.35, size = 118, normalized size = 13.11 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {-\frac {3 \, \cosh \left (x\right )^{2} + 3 \, \sinh \left (x\right )^{2} - 5}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{3 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} + 2 \, {\left (9 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 10 \, \cosh \left (x\right )^{2} + 4 \, {\left (3 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{2}{\left (x \right )}}{\sqrt {- \left (2 \tanh {\left (x \right )} - 1\right ) \left (2 \tanh {\left (x \right )} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (7) = 14\).
time = 0.40, size = 44, normalized size = 4.89 \begin {gather*} -\arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (\sqrt {3} \sqrt {-3 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 3} - 4\right )}}{3 \, e^{\left (2 \, x\right )} - 5} - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.11 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (x\right )}^2\,\sqrt {1-4\,{\mathrm {tanh}\left (x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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