Integrand size = 13, antiderivative size = 32 \[ \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {\text {csch}(x)}{b}+\left (\frac {1}{a}+\frac {a}{b^2}\right ) \log (a+b \text {csch}(x))+\frac {\log (\sinh (x))}{a} \]
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Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3970, 908} \[ \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx=\left (\frac {a}{b^2}+\frac {1}{a}\right ) \log (a+b \text {csch}(x))+\frac {\log (\sinh (x))}{a}-\frac {\text {csch}(x)}{b} \]
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Rule 908
Rule 3970
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-b^2-x^2}{x (a+x)} \, dx,x,b \text {csch}(x)\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int \left (-1-\frac {b^2}{a x}+\frac {a^2+b^2}{a (a+x)}\right ) \, dx,x,b \text {csch}(x)\right )}{b^2} \\ & = -\frac {\text {csch}(x)}{b}+\left (\frac {1}{a}+\frac {a}{b^2}\right ) \log (a+b \text {csch}(x))+\frac {\log (\sinh (x))}{a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {-a b \text {csch}(x)-a^2 \log (\sinh (x))+\left (a^2+b^2\right ) \log (b+a \sinh (x))}{a b^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(32)=64\).
Time = 0.67 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34
method | result | size |
risch | \(-\frac {x}{a}-\frac {2 \,{\mathrm e}^{x}}{b \left ({\mathrm e}^{2 x}-1\right )}+\frac {a \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a}-\frac {a \ln \left ({\mathrm e}^{2 x}-1\right )}{b^{2}}\) | \(75\) |
default | \(\frac {\tanh \left (\frac {x}{2}\right )}{2 b}+\frac {\left (2 a^{2}+2 b^{2}\right ) \ln \left (-\tanh \left (\frac {x}{2}\right )^{2} b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{2 a \,b^{2}}-\frac {1}{2 b \tanh \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}\) | \(95\) |
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 6.22 \[ \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^{2} x \cosh \left (x\right )^{2} + b^{2} x \sinh \left (x\right )^{2} - b^{2} x + 2 \, a b \cosh \left (x\right ) - {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} - a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (b^{2} x \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )}{a b^{2} \cosh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a b^{2} \sinh \left (x\right )^{2} - a b^{2}} \]
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\[ \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (32) = 64\).
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a} + \frac {2 \, e^{\left (-x\right )}}{b e^{\left (-2 \, x\right )} - b} - \frac {a \log \left (e^{\left (-x\right )} + 1\right )}{b^{2}} - \frac {a \log \left (e^{\left (-x\right )} - 1\right )}{b^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {a \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{b^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a b^{2}} + \frac {a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b}{b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}} \]
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Time = 2.44 (sec) , antiderivative size = 261, normalized size of antiderivative = 8.16 \[ \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {2\,{\mathrm {e}}^x}{b-b\,{\mathrm {e}}^{2\,x}}-\frac {x}{a}+\frac {\ln \left (16\,a^5\,{\mathrm {e}}^{2\,x}-4\,a\,b^4-16\,a^5-16\,a^3\,b^2+8\,b^5\,{\mathrm {e}}^x+16\,a^3\,b^2\,{\mathrm {e}}^{2\,x}+32\,a^4\,b\,{\mathrm {e}}^x+4\,a\,b^4\,{\mathrm {e}}^{2\,x}+32\,a^2\,b^3\,{\mathrm {e}}^x\right )}{a}+\frac {a\,\ln \left (16\,a^5\,{\mathrm {e}}^{2\,x}-4\,a\,b^4-16\,a^5-16\,a^3\,b^2+8\,b^5\,{\mathrm {e}}^x+16\,a^3\,b^2\,{\mathrm {e}}^{2\,x}+32\,a^4\,b\,{\mathrm {e}}^x+4\,a\,b^4\,{\mathrm {e}}^{2\,x}+32\,a^2\,b^3\,{\mathrm {e}}^x\right )}{b^2}-\frac {a\,\ln \left (16\,a^6\,{\mathrm {e}}^{2\,x}+4\,b^6\,{\mathrm {e}}^{2\,x}-16\,a^6-4\,b^6-20\,a^2\,b^4-32\,a^4\,b^2+20\,a^2\,b^4\,{\mathrm {e}}^{2\,x}+32\,a^4\,b^2\,{\mathrm {e}}^{2\,x}\right )}{b^2} \]
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