Optimal. Leaf size=24 \[ \frac {1}{2} \tanh (x) \sqrt {\tanh ^2(x)+1}+\frac {1}{2} \sinh ^{-1}(\tanh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3675, 195, 215} \[ \frac {1}{2} \tanh (x) \sqrt {\tanh ^2(x)+1}+\frac {1}{2} \sinh ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 3675
Rubi steps
\begin {align*} \int \text {sech}^2(x) \sqrt {1+\tanh ^2(x)} \, dx &=\operatorname {Subst}\left (\int \sqrt {1+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \sinh ^{-1}(\tanh (x))+\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}\\ \end {align*}
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Mathematica [B] time = 0.10, size = 55, normalized size = 2.29 \[ \frac {1}{4} \sqrt {\tanh ^2(x)+1} \text {sech}(x) \text {sech}(2 x) \left (-\sinh (x)+\sinh (3 x)+2 \sqrt {\cosh (2 x)} \cosh ^2(x) \tanh ^{-1}\left (\frac {\sinh (x)}{\sqrt {\cosh (2 x)}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 334, normalized size = 13.92 \[ \frac {{\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} - 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) - {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} - 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) + 4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{4 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 145, normalized size = 6.04 \[ \frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) - \frac {4 \, {\left (3 \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} - {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} - 1\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\relax (x )^{2} \sqrt {1+\tanh ^{2}\relax (x )}\, dx \]
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 \[ \int \sqrt {\tanh \relax (x)^{2} + 1} \operatorname {sech}\relax (x)^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {{\mathrm {tanh}\relax (x)}^2+1}}{{\mathrm {cosh}\relax (x)}^2} \,d x \]
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 \[ \int \sqrt {\tanh ^{2}{\relax (x )} + 1} \operatorname {sech}^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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