Integrand size = 13, antiderivative size = 77 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}-\frac {a \coth (x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2} \]
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Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2806, 3852, 8, 2686, 2738, 214} \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}-\frac {a \coth (x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2} \]
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Rule 8
Rule 214
Rule 2686
Rule 2738
Rule 2806
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {a \int \text {csch}^2(x) \, dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a+b \cosh (x)} \, dx}{a^2-b^2}-\frac {b \int \coth (x) \text {csch}(x) \, dx}{a^2-b^2} \\ & = -\frac {(i a) \text {Subst}(\int 1 \, dx,x,-i \coth (x))}{a^2-b^2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2}+\frac {(i b) \text {Subst}(\int 1 \, dx,x,-i \text {csch}(x))}{a^2-b^2} \\ & = \frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}-\frac {a \coth (x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\frac {2 a^2 \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}-\frac {\coth \left (\frac {x}{2}\right )}{2 (a+b)}-\frac {\tanh \left (\frac {x}{2}\right )}{2 (a-b)} \]
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Time = 0.38 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {\tanh \left (\frac {x}{2}\right )}{2 \left (a -b \right )}-\frac {1}{2 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )}+\frac {2 a^{2} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(78\) |
risch | \(-\frac {2 \left (-{\mathrm e}^{x} b +a \right )}{\left ({\mathrm e}^{2 x}-1\right ) \left (a^{2}-b^{2}\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right )}-\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right )}\) | \(167\) |
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (67) = 134\).
Time = 0.28 (sec) , antiderivative size = 470, normalized size of antiderivative = 6.10 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\left [\frac {2 \, a^{3} - 2 \, a b^{2} + {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) - 2 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}}, \frac {2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) - {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) - {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )\right )}}{a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}}\right ] \]
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\[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\frac {2 \, a^{2} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (b e^{x} - a\right )}}{{\left (a^{2} - b^{2}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
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Time = 2.11 (sec) , antiderivative size = 337, normalized size of antiderivative = 4.38 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=-\frac {\frac {2\,a}{a^2-b^2}-\frac {2\,b\,{\mathrm {e}}^x}{a^2-b^2}}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,a^2}{b^2\,{\left (a^2-b^2\right )}^2\,\sqrt {a^4}}+\frac {2\,\left (a^3\,\sqrt {a^4}-a\,b^2\,\sqrt {a^4}\right )}{a\,b^2\,\left (a^2-b^2\right )\,\sqrt {-{\left (a^2-b^2\right )}^3}\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}\right )-\frac {2\,\left (b^3\,\sqrt {a^4}-a^2\,b\,\sqrt {a^4}\right )}{a\,b^2\,\left (a^2-b^2\right )\,\sqrt {-{\left (a^2-b^2\right )}^3}\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}\right )\,\left (\frac {b^3\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{2}-\frac {a^2\,b\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{2}\right )\right )\,\sqrt {a^4}}{\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}} \]
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