Integrand size = 13, antiderivative size = 94 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=-\frac {(a-b \cosh (x)) \text {csch}^2(x)}{2 \left (a^2-b^2\right )}+\frac {(2 a+b) \log (1-\cosh (x))}{4 (a+b)^2}+\frac {(2 a-b) \log (1+\cosh (x))}{4 (a-b)^2}-\frac {a^3 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^2} \]
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Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2800, 1661, 815} \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=-\frac {\text {csch}^2(x) (a-b \cosh (x))}{2 \left (a^2-b^2\right )}-\frac {a^3 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^2}+\frac {(2 a+b) \log (1-\cosh (x))}{4 (a+b)^2}+\frac {(2 a-b) \log (\cosh (x)+1)}{4 (a-b)^2} \]
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Rule 815
Rule 1661
Rule 2800
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^3}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \cosh (x)\right ) \\ & = -\frac {(a-b \cosh (x)) \text {csch}^2(x)}{2 \left (a^2-b^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {a b^4}{a^2-b^2}-\frac {b^2 \left (2 a^2-b^2\right ) x}{a^2-b^2}}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \cosh (x)\right )}{2 b^2} \\ & = -\frac {(a-b \cosh (x)) \text {csch}^2(x)}{2 \left (a^2-b^2\right )}+\frac {\text {Subst}\left (\int \left (-\frac {b^2 (2 a+b)}{2 (a+b)^2 (b-x)}-\frac {2 a^3 b^2}{(a-b)^2 (a+b)^2 (a+x)}+\frac {(2 a-b) b^2}{2 (a-b)^2 (b+x)}\right ) \, dx,x,b \cosh (x)\right )}{2 b^2} \\ & = -\frac {(a-b \cosh (x)) \text {csch}^2(x)}{2 \left (a^2-b^2\right )}+\frac {(2 a+b) \log (1-\cosh (x))}{4 (a+b)^2}+\frac {(2 a-b) \log (1+\cosh (x))}{4 (a-b)^2}-\frac {a^3 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^2} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\frac {1}{8} \left (-\frac {\text {csch}^2\left (\frac {x}{2}\right )}{a+b}+\frac {4 (2 a-b) \log \left (\cosh \left (\frac {x}{2}\right )\right )}{(a-b)^2}-\frac {8 a^3 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^2}+\frac {4 (2 a+b) \log \left (\sinh \left (\frac {x}{2}\right )\right )}{(a+b)^2}+\frac {\text {sech}^2\left (\frac {x}{2}\right )}{a-b}\right ) \]
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Time = 0.64 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8 \left (a -b \right )}-\frac {1}{8 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a +2 b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 \left (a +b \right )^{2}}-\frac {a^{3} \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b -a -b \right )}{\left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(91\) |
risch | \(-\frac {x a}{a^{2}-2 a b +b^{2}}+\frac {b x}{2 a^{2}-4 a b +2 b^{2}}-\frac {x a}{a^{2}+2 a b +b^{2}}-\frac {x b}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 x \,a^{3}}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {{\mathrm e}^{x} \left (-b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} \left (a^{2}-b^{2}\right )}+\frac {\ln \left ({\mathrm e}^{x}+1\right ) a}{a^{2}-2 a b +b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+1\right ) b}{2 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) a}{a^{2}+2 a b +b^{2}}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) b}{2 a^{2}+4 a b +2 b^{2}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}\) | \(251\) |
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Leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (89) = 178\).
Time = 0.28 (sec) , antiderivative size = 839, normalized size of antiderivative = 8.93 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.66 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=-\frac {a^{3} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (2 \, a - b\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {{\left (2 \, a + b\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {b e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} + b e^{\left (-3 \, x\right )}}{a^{2} - b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} - b^{2}\right )} e^{\left (-4 \, x\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.89 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=-\frac {a^{3} b \log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} + \frac {{\left (2 \, a - b\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {{\left (2 \, a + b\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} - 4 \, a b^{2}}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}} \]
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Time = 2.29 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.10 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\frac {\frac {2\,\left (a\,b^2-a^3\right )}{{\left (a^2-b^2\right )}^2}+\frac {{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {2\,a}{a^2-b^2}-\frac {2\,b\,{\mathrm {e}}^x}{a^2-b^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (2\,a-b\right )}{2\,a^2-4\,a\,b+2\,b^2}-\frac {a^3\,\ln \left (b^7\,{\mathrm {e}}^{2\,x}-16\,a^6\,b+b^7-6\,a^2\,b^5+9\,a^4\,b^3-32\,a^7\,{\mathrm {e}}^x-6\,a^2\,b^5\,{\mathrm {e}}^{2\,x}+9\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+2\,a\,b^6\,{\mathrm {e}}^x-16\,a^6\,b\,{\mathrm {e}}^{2\,x}-12\,a^3\,b^4\,{\mathrm {e}}^x+18\,a^5\,b^2\,{\mathrm {e}}^x\right )}{a^4-2\,a^2\,b^2+b^4}+\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (2\,a+b\right )}{2\,a^2+4\,a\,b+2\,b^2} \]
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