Optimal. Leaf size=31 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {\tanh ^2(x)+1}}\right )-\sinh ^{-1}(\tanh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3661, 402, 215, 377, 206} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {\tanh ^2(x)+1}}\right )-\sinh ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 377
Rule 402
Rule 3661
Rubi steps
\begin {align*} \int \sqrt {1+\tanh ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\tanh (x)\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tanh (x)\right )\\ &=-\sinh ^{-1}(\tanh (x))+2 \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right )\\ &=-\sinh ^{-1}(\tanh (x))+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 51, normalized size = 1.65 \[ \frac {\cosh (x) \sqrt {\tanh ^2(x)+1} \left (\sqrt {2} \sinh ^{-1}\left (\sqrt {2} \sinh (x)\right )-\tanh ^{-1}\left (\frac {\sinh (x)}{\sqrt {\cosh (2 x)}}\right )\right )}{\sqrt {\cosh (2 x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 679, normalized size = 21.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 104, normalized size = 3.35 \[ -\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 97, normalized size = 3.13 \[ -\frac {\sqrt {\left (\tanh \relax (x )-1\right )^{2}+2 \tanh \relax (x )}}{2}-\arcsinh \left (\tanh \relax (x )\right )+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \relax (x )+2\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \relax (x )-1\right )^{2}+2 \tanh \relax (x )}}\right )}{2}+\frac {\sqrt {\left (1+\tanh \relax (x )\right )^{2}-2 \tanh \relax (x )}}{2}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2-2 \tanh \relax (x )\right ) \sqrt {2}}{4 \sqrt {\left (1+\tanh \relax (x )\right )^{2}-2 \tanh \relax (x )}}\right )}{2} \]
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 \[ \int \sqrt {\tanh \relax (x)^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 68, normalized size = 2.19 \[ \frac {\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\relax (x)+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {tanh}\relax (x)}^2+1}-\mathrm {tanh}\relax (x)+1\right )\right )}{2}-\mathrm {asinh}\left (\mathrm {tanh}\relax (x)\right )+\frac {\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\relax (x)+\sqrt {2}\,\sqrt {{\mathrm {tanh}\relax (x)}^2+1}+1\right )-\ln \left (\mathrm {tanh}\relax (x)-1\right )\right )}{2} \]
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 \[ \int \sqrt {\tanh ^{2}{\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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