Integrand size = 13, antiderivative size = 57 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b}+\frac {2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b} \]
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Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3979, 4136, 3855, 4004, 3916, 2739, 632, 212} \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b}+\frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b} \]
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Rule 212
Rule 632
Rule 2739
Rule 3855
Rule 3916
Rule 3979
Rule 4004
Rule 4136
Rubi steps \begin{align*} \text {integral}& = -\int \frac {-1-\text {csch}^2(x)}{a+b \text {csch}(x)} \, dx \\ & = \frac {i \int \frac {-i b+i a \text {csch}(x)}{a+b \text {csch}(x)} \, dx}{b}+\frac {\int \text {csch}(x) \, dx}{b} \\ & = \frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b}-\frac {\left (a^2+b^2\right ) \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx}{a b} \\ & = \frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b}-\left (\frac {1}{a}+\frac {a}{b^2}\right ) \int \frac {1}{1+\frac {a \sinh (x)}{b}} \, dx \\ & = \frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b}-\left (2 \left (\frac {1}{a}+\frac {a}{b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right ) \\ & = \frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b}+\left (4 \left (\frac {1}{a}+\frac {a}{b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right ) \\ & = \frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b}+\frac {2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a^2+b^2}}\right )}{a b} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.32 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {b x+2 \sqrt {-a^2-b^2} \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )-a \log \left (\cosh \left (\frac {x}{2}\right )\right )+a \log \left (\sinh \left (\frac {x}{2}\right )\right )}{a b} \]
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Time = 0.46 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.47
method | result | size |
default | \(\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\left (2 a^{2}+2 b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a b \sqrt {a^{2}+b^{2}}}\) | \(84\) |
risch | \(\frac {x}{a}+\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {b +\sqrt {a^{2}+b^{2}}}{a}\right )}{b a}-\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-b +\sqrt {a^{2}+b^{2}}}{a}\right )}{b a}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{b}\) | \(100\) |
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (53) = 106\).
Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.47 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {b x - a \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + a \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right )}{a b} \]
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\[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.58 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{b} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{a b} \]
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Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.56 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a} - \frac {\log \left (e^{x} + 1\right )}{b} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{b} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{a b} \]
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Time = 0.30 (sec) , antiderivative size = 316, normalized size of antiderivative = 5.54 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a}+\frac {\ln \left (32\,a^2\,b+32\,b^3-32\,b^3\,{\mathrm {e}}^x-32\,a^2\,b\,{\mathrm {e}}^x\right )}{b}-\frac {\ln \left (32\,a^2\,b+32\,b^3+32\,b^3\,{\mathrm {e}}^x+32\,a^2\,b\,{\mathrm {e}}^x\right )}{b}+\frac {\ln \left (128\,b^5\,{\mathrm {e}}^x-64\,a^3\,b^2-64\,a\,b^4-128\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^4\,b\,{\mathrm {e}}^x+160\,a^2\,b^3\,{\mathrm {e}}^x+64\,a\,b^3\,\sqrt {a^2+b^2}+32\,a^3\,b\,\sqrt {a^2+b^2}-96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a\,b}-\frac {\ln \left (128\,b^5\,{\mathrm {e}}^x-64\,a^3\,b^2-64\,a\,b^4+128\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^4\,b\,{\mathrm {e}}^x+160\,a^2\,b^3\,{\mathrm {e}}^x-64\,a\,b^3\,\sqrt {a^2+b^2}-32\,a^3\,b\,\sqrt {a^2+b^2}+96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a\,b} \]
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