Integrand size = 10, antiderivative size = 50 \[ \int \left (1+\coth ^2(x)\right )^{3/2} \, dx=-\frac {5}{2} \text {arcsinh}(\coth (x))+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {1+\coth ^2(x)}}\right )-\frac {1}{2} \coth (x) \sqrt {1+\coth ^2(x)} \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3742, 427, 537, 221, 385, 212} \[ \int \left (1+\coth ^2(x)\right )^{3/2} \, dx=-\frac {5}{2} \text {arcsinh}(\coth (x))+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {\coth ^2(x)+1}}\right )-\frac {1}{2} \coth (x) \sqrt {\coth ^2(x)+1} \]
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Rule 212
Rule 221
Rule 385
Rule 427
Rule 537
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right ) \\ & = -\frac {1}{2} \coth (x) \sqrt {1+\coth ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {-3-5 x^2}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\coth (x)\right ) \\ & = -\frac {1}{2} \coth (x) \sqrt {1+\coth ^2(x)}-\frac {5}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\coth (x)\right )+4 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\coth (x)\right ) \\ & = -\frac {5}{2} \text {arcsinh}(\coth (x))-\frac {1}{2} \coth (x) \sqrt {1+\coth ^2(x)}+4 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\coth (x)}{\sqrt {1+\coth ^2(x)}}\right ) \\ & = -\frac {5}{2} \text {arcsinh}(\coth (x))+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {1+\coth ^2(x)}}\right )-\frac {1}{2} \coth (x) \sqrt {1+\coth ^2(x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(116\) vs. \(2(50)=100\).
Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.32 \[ \int \left (1+\coth ^2(x)\right )^{3/2} \, dx=-\frac {1}{8} \left (1+\coth ^2(x)\right )^{3/2} \text {sech}^2(2 x) \left (16 \text {arctanh}\left (\frac {\cosh (x)}{\sqrt {\cosh (2 x)}}\right ) \sqrt {\cosh (2 x)} \sinh ^3(x)+4 \left (\arctan \left (\frac {\cosh (x)}{\sqrt {-\cosh (2 x)}}\right ) \sqrt {-\cosh (2 x)}-4 \sqrt {2} \sqrt {\cosh (2 x)} \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )\right ) \sinh ^3(x)+\sinh (4 x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(38)=76\).
Time = 0.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 3.16
method | result | size |
derivativedivides | \(\frac {\left (\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}-\frac {\coth \left (x \right ) \sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}}{4}-\frac {5 \,\operatorname {arcsinh}\left (\coth \left (x \right )\right )}{2}+\sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}}\right )-\frac {\left (\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}-\frac {\coth \left (x \right ) \sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}}{4}-\sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}}\right )\) | \(158\) |
default | \(\frac {\left (\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}-\frac {\coth \left (x \right ) \sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}}{4}-\frac {5 \,\operatorname {arcsinh}\left (\coth \left (x \right )\right )}{2}+\sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}}\right )-\frac {\left (\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}-\frac {\coth \left (x \right ) \sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}}{4}-\sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}}\right )\) | \(158\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1043 vs. \(2 (38) = 76\).
Time = 0.28 (sec) , antiderivative size = 1043, normalized size of antiderivative = 20.86 \[ \int \left (1+\coth ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]
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\[ \int \left (1+\coth ^2(x)\right )^{3/2} \, dx=\int \left (\coth ^{2}{\left (x \right )} + 1\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (1+\coth ^2(x)\right )^{3/2} \, dx=\int { {\left (\coth \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (38) = 76\).
Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 5.30 \[ \int \left (1+\coth ^2(x)\right )^{3/2} \, dx=\frac {1}{4} \, \sqrt {2} {\left (5 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \, {\left (\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 4 \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - 4 \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - 4 \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - \frac {4 \, {\left (3 \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}}\right )} \]
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Time = 2.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.56 \[ \int \left (1+\coth ^2(x)\right )^{3/2} \, dx=\sqrt {2}\,\left (\ln \left (\mathrm {coth}\left (x\right )+\sqrt {2}\,\sqrt {{\mathrm {coth}\left (x\right )}^2+1}+1\right )-\ln \left (\mathrm {coth}\left (x\right )-1\right )\right )-\frac {\mathrm {coth}\left (x\right )\,\sqrt {{\mathrm {coth}\left (x\right )}^2+1}}{2}-\frac {5\,\mathrm {asinh}\left (\mathrm {coth}\left (x\right )\right )}{2}+\sqrt {2}\,\left (\ln \left (\mathrm {coth}\left (x\right )+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {coth}\left (x\right )}^2+1}-\mathrm {coth}\left (x\right )+1\right )\right ) \]
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