Integrand size = 10, antiderivative size = 25 \[ \int \frac {1}{\sqrt {1+\tanh ^2(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right )}{\sqrt {2}} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3742, 385, 212} \[ \int \frac {1}{\sqrt {1+\tanh ^2(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {\tanh ^2(x)+1}}\right )}{\sqrt {2}} \]
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Rule 212
Rule 385
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\tanh (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {1+\tanh ^2(x)}} \, dx=\frac {\text {arcsinh}\left (\sqrt {2} \sinh (x)\right ) \sqrt {\cosh (2 x)} \text {sech}(x)}{\sqrt {2} \sqrt {1+\tanh ^2(x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(20)=40\).
Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}\right )}{4}\) | \(62\) |
default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}\right )}{4}\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 543, normalized size of antiderivative = 21.72 \[ \int \frac {1}{\sqrt {1+\tanh ^2(x)}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\sqrt {1+\tanh ^2(x)}} \, dx=\int \frac {1}{\sqrt {\tanh ^{2}{\left (x \right )} + 1}}\, dx \]
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\[ \int \frac {1}{\sqrt {1+\tanh ^2(x)}} \, dx=\int { \frac {1}{\sqrt {\tanh \left (x\right )^{2} + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {1}{\sqrt {1+\tanh ^2(x)}} \, dx=-\frac {1}{4} \, \sqrt {2} {\left (\log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \]
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Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {1}{\sqrt {1+\tanh ^2(x)}} \, dx=\frac {\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\left (x\right )+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}-\mathrm {tanh}\left (x\right )+1\right )\right )}{4}+\frac {\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\left (x\right )+\sqrt {2}\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}+1\right )-\ln \left (\mathrm {tanh}\left (x\right )-1\right )\right )}{4} \]
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