Optimal. Leaf size=19 \[ \tanh (x) \sqrt {\coth ^2(x)+1}-\sinh ^{-1}(\coth (x)) \]
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Rubi [A] time = 0.05, antiderivative size = 19, 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 = {3663, 277, 215} \[ \tanh (x) \sqrt {\coth ^2(x)+1}-\sinh ^{-1}(\coth (x)) \]
Antiderivative was successfully verified.
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Rule 215
Rule 277
Rule 3663
Rubi steps
\begin {align*} \int \sqrt {1+\coth ^2(x)} \text {sech}^2(x) \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{x^2} \, dx,x,\coth (x)\right )\\ &=\sqrt {1+\coth ^2(x)} \tanh (x)-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\coth (x)\right )\\ &=-\sinh ^{-1}(\coth (x))+\sqrt {1+\coth ^2(x)} \tanh (x)\\ \end {align*}
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Mathematica [B] time = 0.23, size = 51, normalized size = 2.68 \[ \sinh (x) \sqrt {\coth ^2(x)+1} \text {sech}(2 x) \left (\cosh (x)+\sinh (x) \tanh (x)-\sqrt {-\cosh (2 x)} \tan ^{-1}\left (\frac {\cosh (x)}{\sqrt {-\cosh (2 x)}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 219, normalized size = 11.53 \[ -\frac {{\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) - 4 \, \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{2 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 120, normalized size = 6.32 \[ \frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \, {\left (\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) - \frac {4 \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 1}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\relax (x )^{2} \sqrt {1+\coth ^{2}\relax (x )}\, dx \]
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 \[ \int \sqrt {\coth \relax (x)^{2} + 1} \operatorname {sech}\relax (x)^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\sqrt {{\mathrm {coth}\relax (x)}^2+1}}{{\mathrm {cosh}\relax (x)}^2} \,d x \]
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 \[ \int \sqrt {\coth ^{2}{\relax (x )} + 1} \operatorname {sech}^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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