Integrand size = 13, antiderivative size = 52 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {b \text {csch}(x)}{a^2}-\frac {\text {csch}^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3} \]
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Time = 0.06 (sec) , antiderivative size = 52, 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 = {2800, 908} \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {b \text {csch}(x)}{a^2}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3}-\frac {\text {csch}^2(x)}{2 a} \]
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Rule 908
Rule 2800
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {-b^2-x^2}{x^3 (a+x)} \, dx,x,b \sinh (x)\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {b^2}{a x^3}+\frac {b^2}{a^2 x^2}+\frac {-a^2-b^2}{a^3 x}+\frac {a^2+b^2}{a^3 (a+x)}\right ) \, dx,x,b \sinh (x)\right ) \\ & = \frac {b \text {csch}(x)}{a^2}-\frac {\text {csch}^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {2 a b \text {csch}(x)-a^2 \text {csch}^2(x)+2 \left (a^2+b^2\right ) (\log (\sinh (x))-\log (a+b \sinh (x)))}{2 a^3} \]
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Time = 0.88 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.88
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{x} \left (-b \,{\mathrm e}^{2 x}+{\mathrm e}^{x} a +b \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} a^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) b^{2}}{a^{3}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) b^{2}}{a^{3}}\) | \(98\) |
default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {x}{2}\right )}{4 a^{2}}+\frac {\left (-4 a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{4 a^{3}}-\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {x}{2}\right )}\) | \(104\) |
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (50) = 100\).
Time = 0.29 (sec) , antiderivative size = 427, normalized size of antiderivative = 8.21 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {2 \, a b \cosh \left (x\right )^{3} + 2 \, a b \sinh \left (x\right )^{3} - 2 \, a^{2} \cosh \left (x\right )^{2} - 2 \, a b \cosh \left (x\right ) + 2 \, {\left (3 \, a b \cosh \left (x\right ) - a^{2}\right )} \sinh \left (x\right )^{2} - {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{3} - {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{3} - {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (x\right )^{2} - 2 \, a^{2} \cosh \left (x\right ) - a b\right )} \sinh \left (x\right )}{a^{3} \cosh \left (x\right )^{4} + 4 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{3} \sinh \left (x\right )^{4} - 2 \, a^{3} \cosh \left (x\right )^{2} + a^{3} + 2 \, {\left (3 \, a^{3} \cosh \left (x\right )^{2} - a^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{3} \cosh \left (x\right )^{3} - a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
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\[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (50) = 100\).
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.23 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=-\frac {2 \, {\left (b e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - b e^{\left (-3 \, x\right )}\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} - a^{2} e^{\left (-4 \, x\right )} - a^{2}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (50) = 100\).
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.40 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, a b {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a^{2}}{2 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \]
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Time = 1.97 (sec) , antiderivative size = 1163, normalized size of antiderivative = 22.37 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {\left (2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{2\,a^3\,{\left (a^2+b^2\right )}^2}+\frac {\left (a^7+a^5\,b^2\right )\,\sqrt {-a^6}}{2\,a^6\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {a^6\,b^2\,{\mathrm {e}}^x\,\sqrt {-a^6}\,\left (\frac {8\,\left (a^4+2\,a^2\,b^2+b^4\right )}{a^8\,b\,{\left (a^2+b^2\right )}^2}-\frac {4\,\left (2\,a^6\,b+2\,a^4\,b^3\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{12}\,b^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}+\frac {2\,\left (a^7+a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {2\,\left (a^2+2\,b^2\right )\,\left (a^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{10}\,b^3\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2}\right )}{8\,\sqrt {a^4+2\,a^2\,b^2+b^4}}-\frac {a^6\,b^2\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^6}\,\left (\frac {4\,\left (a^2+2\,b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{a^9\,b^2\,{\left (a^2+b^2\right )}^2}+\frac {4\,\left (a^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^9\,b^2\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2}+\frac {2\,\left (2\,a^6\,b+2\,a^4\,b^3\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}+\frac {4\,\left (a^7+a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{12}\,b^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}\right )}{8\,\sqrt {a^4+2\,a^2\,b^2+b^4}}+\frac {a^6\,b^2\,{\mathrm {e}}^{3\,x}\,\left (\frac {2\,\left (a^7+a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {2\,\left (a^2+2\,b^2\right )\,\left (a^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{10}\,b^3\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {-a^6}}{8\,\sqrt {a^4+2\,a^2\,b^2+b^4}}\right )-2\,\mathrm {atan}\left (\left (4\,a^6\,b\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2+4\,a^4\,b^3\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2\right )\,\left (\frac {1}{8\,a^5\,b\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}-{\mathrm {e}}^x\,\left (\frac {1}{16\,a^4\,b^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^8\,b^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}\right )+\frac {a^2+2\,b^2}{8\,a^7\,b\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}\right )\right )\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{\sqrt {-a^6}}-\frac {2}{a\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\frac {2}{a}-\frac {2\,b\,{\mathrm {e}}^x}{a^2}}{{\mathrm {e}}^{2\,x}-1} \]
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