Integrand size = 11, antiderivative size = 18 \[ \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx=x-\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1144, 213} \[ \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx=x-\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]
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Rule 213
Rule 1144
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^2}{2-3 x^2+x^4} \, dx,x,\tanh (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\tanh (x)\right )-\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (x)\right ) \\ & = x-\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx=x-\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(36\) vs. \(2(15)=30\).
Time = 0.83 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06
method | result | size |
risch | \(x +\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3+2 \sqrt {2}\right )}{2}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-2 \sqrt {2}+3\right )}{2}\) | \(37\) |
default | \(-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{4}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{4}\) | \(156\) |
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (15) = 30\).
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.72 \[ \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) + x \]
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\[ \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx=\int \frac {1}{\coth ^{2}{\left (x \right )} + \operatorname {csch}^{2}{\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).
Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + x \]
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Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.00 \[ \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx=x+\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{2}\right )}{2}-\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{2}\right )}{2} \]
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