Integrand size = 10, antiderivative size = 31 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=-\text {arcsinh}(\tanh (x))+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 399, 221, 385, 212} \[ \int \sqrt {1+\tanh ^2(x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {\tanh ^2(x)+1}}\right )-\text {arcsinh}(\tanh (x)) \]
[In]
[Out]
Rule 212
Rule 221
Rule 385
Rule 399
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{1-x^2} \, dx,x,\tanh (x)\right ) \\ & = 2 \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}{\sqrt {1+x^2}} \, dx,x,\tanh (x)\right ) \\ & = -\text {arcsinh}(\tanh (x))+2 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right ) \\ & = -\text {arcsinh}(\tanh (x))+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=\frac {\left (\sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh (x)\right )-\text {arctanh}\left (\frac {\sinh (x)}{\sqrt {\cosh (2 x)}}\right )\right ) \cosh (x) \sqrt {1+\tanh ^2(x)}}{\sqrt {\cosh (2 x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(25)=50\).
Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.13
method | result | size |
derivativedivides | \(-\frac {\sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}{2}-\operatorname {arcsinh}\left (\tanh \left (x \right )\right )+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}\right )}{2}+\frac {\sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}{2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}\right )}{2}\) | \(97\) |
default | \(-\frac {\sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}{2}-\operatorname {arcsinh}\left (\tanh \left (x \right )\right )+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}\right )}{2}+\frac {\sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}{2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}\right )}{2}\) | \(97\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 679, normalized size of antiderivative = 21.90 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \sqrt {1+\tanh ^2(x)} \, dx=\int \sqrt {\tanh ^{2}{\left (x \right )} + 1}\, dx \]
[In]
[Out]
\[ \int \sqrt {1+\tanh ^2(x)} \, dx=\int { \sqrt {\tanh \left (x\right )^{2} + 1} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.35 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=\frac {\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\left (x\right )+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}-\mathrm {tanh}\left (x\right )+1\right )\right )}{2}-\mathrm {asinh}\left (\mathrm {tanh}\left (x\right )\right )+\frac {\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\left (x\right )+\sqrt {2}\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}+1\right )-\ln \left (\mathrm {tanh}\left (x\right )-1\right )\right )}{2} \]
[In]
[Out]