Integrand size = 11, antiderivative size = 54 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=x-\frac {7 \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )} \]
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Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {481, 592, 536, 212} \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=-\frac {7 \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}+x-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2} \]
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Rule 212
Rule 481
Rule 536
Rule 592
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right ) \left (2-x^2\right )^3} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}+\frac {1}{4} \text {Subst}\left (\int \frac {x^2 \left (6-2 x^2\right )}{\left (1-x^2\right ) \left (2-x^2\right )^2} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {1}{8} \text {Subst}\left (\int \frac {-2-6 x^2}{\left (1-x^2\right ) \left (2-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {7}{4} \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-\frac {7 \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\frac {76 x+48 x \cosh (2 x)-7 \sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) (3+\cosh (2 x))^2+4 x \cosh (4 x)-2 \sinh (2 x)-3 \sinh (4 x)}{8 (3+\cosh (2 x))^2} \]
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Time = 37.53 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.04
method | result | size |
parallelrisch | \(0\) | \(2\) |
risch | \(x +\frac {17 \,{\mathrm e}^{6 x}+57 \,{\mathrm e}^{4 x}+19 \,{\mathrm e}^{2 x}+3}{2 \left ({\mathrm e}^{4 x}+6 \,{\mathrm e}^{2 x}+1\right )^{2}}+\frac {7 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3+2 \sqrt {2}\right )}{16}-\frac {7 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-2 \sqrt {2}+3\right )}{16}\) | \(73\) |
default | \(\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )+\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{7}}{4}-\frac {5 \tanh \left (\frac {x}{2}\right )^{5}}{4}-\frac {5 \tanh \left (\frac {x}{2}\right )^{3}}{4}-\frac {\tanh \left (\frac {x}{2}\right )}{4}}{\left (\tanh \left (\frac {x}{2}\right )^{4}+1\right )^{2}}-\frac {7 \sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{32}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{32}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(199\) |
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Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (41) = 82\).
Time = 0.28 (sec) , antiderivative size = 715, normalized size of antiderivative = 13.24 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\int \frac {1}{\left (\coth ^{2}{\left (x \right )} + \operatorname {csch}^{2}{\left (x \right )}\right )^{3}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (41) = 82\).
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\frac {7}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) + x - \frac {19 \, e^{\left (-2 \, x\right )} + 57 \, e^{\left (-4 \, x\right )} + 17 \, e^{\left (-6 \, x\right )} + 3}{2 \, {\left (12 \, e^{\left (-2 \, x\right )} + 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=-\frac {7}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + x + \frac {17 \, e^{\left (6 \, x\right )} + 57 \, e^{\left (4 \, x\right )} + 19 \, e^{\left (2 \, x\right )} + 3}{2 \, {\left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
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Time = 2.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.07 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=x+\frac {136\,{\mathrm {e}}^{2\,x}+24}{12\,{\mathrm {e}}^{2\,x}+38\,{\mathrm {e}}^{4\,x}+12\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}-\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{16}\right )}{16}-\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}+\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{16}\right )}{16}+\frac {\frac {17\,{\mathrm {e}}^{2\,x}}{2}-\frac {45}{2}}{6\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]
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