Integrand size = 13, antiderivative size = 158 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\left (a^2-6 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^4}+\frac {2 b^3 \left (4 a^2+3 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \left (a^2+b^2\right )^{3/2}}+\frac {b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]
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Time = 0.46 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2881, 3134, 3080, 3855, 2739, 632, 212} \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\left (a^2-6 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^4}+\frac {2 b^3 \left (4 a^2+3 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \left (a^2+b^2\right )^{3/2}}+\frac {b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )} \]
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Rule 212
Rule 632
Rule 2739
Rule 2881
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\int \frac {\text {csch}^3(x) \left (a^2+3 b^2-a b \sinh (x)+2 b^2 \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{a \left (a^2+b^2\right )} \\ & = -\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {i \int \frac {\text {csch}^2(x) \left (2 i b \left (2 a^2+3 b^2\right )+i a \left (a^2-b^2\right ) \sinh (x)+i b \left (a^2+3 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{2 a^2 \left (a^2+b^2\right )} \\ & = \frac {b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\int \frac {\text {csch}(x) \left (a^4-5 a^2 b^2-6 b^4+a b \left (a^2+3 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{2 a^3 \left (a^2+b^2\right )} \\ & = \frac {b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (a^2-6 b^2\right ) \int \text {csch}(x) \, dx}{2 a^4}-\frac {\left (b^3 \left (4 a^2+3 b^2\right )\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{a^4 \left (a^2+b^2\right )} \\ & = \frac {\left (a^2-6 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^4}+\frac {b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (2 b^3 \left (4 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^4 \left (a^2+b^2\right )} \\ & = \frac {\left (a^2-6 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^4}+\frac {b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\left (4 b^3 \left (4 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^4 \left (a^2+b^2\right )} \\ & = \frac {\left (a^2-6 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^4}+\frac {2 b^3 \left (4 a^2+3 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \left (a^2+b^2\right )^{3/2}}+\frac {b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \\ \end{align*}
Time = 0.96 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {16 b^3 \left (4 a^2+3 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+8 a b \coth \left (\frac {x}{2}\right )-a^2 \text {csch}^2\left (\frac {x}{2}\right )+4 \left (a^2-6 b^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )-4 \left (a^2-6 b^2\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )-a^2 \text {sech}^2\left (\frac {x}{2}\right )+\frac {8 a b^4 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+8 a b \tanh \left (\frac {x}{2}\right )}{8 a^4} \]
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Time = 0.97 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2} a}{2}+4 b \tanh \left (\frac {x}{2}\right )}{4 a^{3}}+\frac {4 b^{3} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{2 \left (a^{2}+b^{2}\right )}-\frac {a b}{2 \left (a^{2}+b^{2}\right )}}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (4 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{4}}-\frac {1}{8 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (-2 a^{2}+12 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tanh \left (\frac {x}{2}\right )}\) | \(175\) |
risch | \(-\frac {a^{3} b \,{\mathrm e}^{5 x}+3 a \,b^{3} {\mathrm e}^{5 x}+2 a^{4} {\mathrm e}^{4 x}-2 a^{2} b^{2} {\mathrm e}^{4 x}-6 b^{4} {\mathrm e}^{4 x}-8 a^{3} b \,{\mathrm e}^{3 x}-12 a \,b^{3} {\mathrm e}^{3 x}+2 a^{4} {\mathrm e}^{2 x}+10 a^{2} b^{2} {\mathrm e}^{2 x}+12 b^{4} {\mathrm e}^{2 x}+7 a^{3} b \,{\mathrm e}^{x}+9 b^{3} {\mathrm e}^{x} a -4 a^{2} b^{2}-6 b^{4}}{a^{3} \left ({\mathrm e}^{2 x}-1\right )^{2} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2 a^{2}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{a^{4}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{2}}+\frac {3 b^{5} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{4}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {3 b^{5} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{4}}-\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2 a^{2}}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{a^{4}}\) | \(464\) |
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Leaf count of result is larger than twice the leaf count of optimal. 3754 vs. \(2 (150) = 300\).
Time = 0.51 (sec) , antiderivative size = 3754, normalized size of antiderivative = 23.76 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (150) = 300\).
Time = 0.29 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.30 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + a^{4} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {4 \, a^{2} b^{2} + 6 \, b^{4} + {\left (7 \, a^{3} b + 9 \, a b^{3}\right )} e^{\left (-x\right )} - 2 \, {\left (a^{4} + 5 \, a^{2} b^{2} + 6 \, b^{4}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} e^{\left (-3 \, x\right )} - 2 \, {\left (a^{4} - a^{2} b^{2} - 3 \, b^{4}\right )} e^{\left (-4 \, x\right )} + {\left (a^{3} b + 3 \, a b^{3}\right )} e^{\left (-5 \, x\right )}}{a^{5} b + a^{3} b^{3} + 2 \, {\left (a^{6} + a^{4} b^{2}\right )} e^{\left (-x\right )} - 3 \, {\left (a^{5} b + a^{3} b^{3}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (a^{6} + a^{4} b^{2}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (a^{5} b + a^{3} b^{3}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (a^{6} + a^{4} b^{2}\right )} e^{\left (-5 \, x\right )} - {\left (a^{5} b + a^{3} b^{3}\right )} e^{\left (-6 \, x\right )}} + \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, a^{4}} - \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, a^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.28 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + a^{4} b^{2}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (a b^{3} e^{x} - b^{4}\right )}}{{\left (a^{5} + a^{3} b^{2}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} + \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{4}} - \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{4}} - \frac {a e^{\left (3 \, x\right )} - 4 \, b e^{\left (2 \, x\right )} + a e^{x} + 4 \, b}{a^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \]
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Time = 4.44 (sec) , antiderivative size = 977, normalized size of antiderivative = 6.18 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {4\,b}{a^3}-\frac {{\mathrm {e}}^x}{a^2}}{{\mathrm {e}}^{2\,x}-1}+\frac {\frac {2\,b^7}{a^3\,\left (a^2\,b^3+b^5\right )}-\frac {2\,b^6\,{\mathrm {e}}^x}{a^2\,\left (a^2\,b^3+b^5\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (a^2-6\,b^2\right )}{2\,a^4}+\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (a^2-6\,b^2\right )}{2\,a^4}-\frac {2\,{\mathrm {e}}^x}{a^2\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}+\frac {b^3\,\ln \left (\frac {8\,\left (4\,a^2+3\,b^2\right )\,\left (20\,a^9\,b^5-72\,b^{11}\,\sqrt {{\left (a^2+b^2\right )}^3}-9\,a^3\,b^{11}-30\,a^5\,b^9-18\,a^7\,b^7-2\,a^{13}\,b+15\,a^{11}\,b^3+4\,a^{14}\,{\mathrm {e}}^x-192\,a^2\,b^9\,\sqrt {{\left (a^2+b^2\right )}^3}-128\,a^4\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}+27\,a^4\,b^{10}\,{\mathrm {e}}^x+72\,a^6\,b^8\,{\mathrm {e}}^x+30\,a^8\,b^6\,{\mathrm {e}}^x-48\,a^{10}\,b^4\,{\mathrm {e}}^x-29\,a^{12}\,b^2\,{\mathrm {e}}^x+312\,a^3\,b^8\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+206\,a^5\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+8\,a\,b^4\,{\mathrm {e}}^x\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}+118\,a\,b^{10}\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^9\,b^2\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}-\frac {8\,\left (-4\,a^4+21\,a^2\,b^2+18\,b^4\right )\,\left (-4\,{\mathrm {e}}^x\,a^5+2\,a^4\,b+19\,{\mathrm {e}}^x\,a^3\,b^2-10\,a^2\,b^3+21\,{\mathrm {e}}^x\,a\,b^4-12\,b^5\right )}{a^9\,b^2\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (4\,a^2+3\,b^2\right )}{a^{10}+3\,a^8\,b^2+3\,a^6\,b^4+a^4\,b^6}-\frac {b^3\,\ln \left (\frac {8\,\left (4\,a^2+3\,b^2\right )\,\left (2\,a^{13}\,b-72\,b^{11}\,\sqrt {{\left (a^2+b^2\right )}^3}+9\,a^3\,b^{11}+30\,a^5\,b^9+18\,a^7\,b^7-20\,a^9\,b^5-15\,a^{11}\,b^3-4\,a^{14}\,{\mathrm {e}}^x-192\,a^2\,b^9\,\sqrt {{\left (a^2+b^2\right )}^3}-128\,a^4\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}-27\,a^4\,b^{10}\,{\mathrm {e}}^x-72\,a^6\,b^8\,{\mathrm {e}}^x-30\,a^8\,b^6\,{\mathrm {e}}^x+48\,a^{10}\,b^4\,{\mathrm {e}}^x+29\,a^{12}\,b^2\,{\mathrm {e}}^x+312\,a^3\,b^8\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+206\,a^5\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+8\,a\,b^4\,{\mathrm {e}}^x\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}+118\,a\,b^{10}\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^9\,b^2\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}-\frac {8\,\left (-4\,a^4+21\,a^2\,b^2+18\,b^4\right )\,\left (-4\,{\mathrm {e}}^x\,a^5+2\,a^4\,b+19\,{\mathrm {e}}^x\,a^3\,b^2-10\,a^2\,b^3+21\,{\mathrm {e}}^x\,a\,b^4-12\,b^5\right )}{a^9\,b^2\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (4\,a^2+3\,b^2\right )}{a^{10}+3\,a^8\,b^2+3\,a^6\,b^4+a^4\,b^6} \]
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