Integrand size = 13, antiderivative size = 81 \[ \int \frac {\text {csch}^3(x)}{a+b \sinh (x)} \, dx=\frac {\left (a^2-2 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^3}+\frac {2 b^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2}}+\frac {b \coth (x)}{a^2}-\frac {\coth (x) \text {csch}(x)}{2 a} \]
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Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2881, 3134, 3080, 3855, 2739, 632, 212} \[ \int \frac {\text {csch}^3(x)}{a+b \sinh (x)} \, dx=\frac {b \coth (x)}{a^2}+\frac {\left (a^2-2 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^3}+\frac {2 b^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2}}-\frac {\coth (x) \text {csch}(x)}{2 a} \]
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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 {\coth (x) \text {csch}(x)}{2 a}+\frac {i \int \frac {\text {csch}^2(x) \left (2 i b+i a \sinh (x)+i b \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{2 a} \\ & = \frac {b \coth (x)}{a^2}-\frac {\coth (x) \text {csch}(x)}{2 a}-\frac {\int \frac {\text {csch}(x) \left (a^2-2 b^2+a b \sinh (x)\right )}{a+b \sinh (x)} \, dx}{2 a^2} \\ & = \frac {b \coth (x)}{a^2}-\frac {\coth (x) \text {csch}(x)}{2 a}-\frac {b^3 \int \frac {1}{a+b \sinh (x)} \, dx}{a^3}-\frac {\left (a^2-2 b^2\right ) \int \text {csch}(x) \, dx}{2 a^3} \\ & = \frac {\left (a^2-2 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^3}+\frac {b \coth (x)}{a^2}-\frac {\coth (x) \text {csch}(x)}{2 a}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^3} \\ & = \frac {\left (a^2-2 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^3}+\frac {b \coth (x)}{a^2}-\frac {\coth (x) \text {csch}(x)}{2 a}+\frac {\left (4 b^3\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^3} \\ & = \frac {\left (a^2-2 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^3}+\frac {2 b^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2}}+\frac {b \coth (x)}{a^2}-\frac {\coth (x) \text {csch}(x)}{2 a} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.68 \[ \int \frac {\text {csch}^3(x)}{a+b \sinh (x)} \, dx=-\frac {\frac {16 b^3 \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a b \coth \left (\frac {x}{2}\right )+a^2 \text {csch}^2\left (\frac {x}{2}\right )-4 \left (a^2-2 b^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )+4 \left (a^2-2 b^2\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )+a^2 \text {sech}^2\left (\frac {x}{2}\right )-4 a b \tanh \left (\frac {x}{2}\right )}{8 a^3} \]
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Time = 0.82 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {x}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (-2 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {x}{2}\right )}-\frac {2 b^{3} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3} \sqrt {a^{2}+b^{2}}}\) | \(108\) |
risch | \(-\frac {a \,{\mathrm e}^{3 x}-2 b \,{\mathrm e}^{2 x}+{\mathrm e}^{x} a +2 b}{\left ({\mathrm e}^{2 x}-1\right )^{2} a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a^{3}}-\frac {b^{3} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a^{3}}-\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2 a}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{a^{3}}+\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2 a}-\frac {\ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{a^{3}}\) | \(188\) |
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Leaf count of result is larger than twice the leaf count of optimal. 929 vs. \(2 (73) = 146\).
Time = 0.32 (sec) , antiderivative size = 929, normalized size of antiderivative = 11.47 \[ \int \frac {\text {csch}^3(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {csch}^3(x)}{a+b \sinh (x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (73) = 146\).
Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.90 \[ \int \frac {\text {csch}^3(x)}{a+b \sinh (x)} \, dx=-\frac {b^{3} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} + \frac {a e^{\left (-x\right )} + 2 \, b e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )} - 2 \, b}{2 \, a^{2} e^{\left (-2 \, x\right )} - a^{2} e^{\left (-4 \, x\right )} - a^{2}} + \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, a^{3}} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, a^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.69 \[ \int \frac {\text {csch}^3(x)}{a+b \sinh (x)} \, dx=-\frac {b^{3} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} + \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{3}} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{3}} - \frac {a e^{\left (3 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a e^{x} + 2 \, b}{a^{2} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \]
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Time = 1.77 (sec) , antiderivative size = 617, normalized size of antiderivative = 7.62 \[ \int \frac {\text {csch}^3(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^x}{a-a\,{\mathrm {e}}^{2\,x}}-\frac {2\,{\mathrm {e}}^x}{a-2\,a\,{\mathrm {e}}^{2\,x}+a\,{\mathrm {e}}^{4\,x}}-\frac {\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x-24\,b^4\,{\mathrm {e}}^x+20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{2\,a}+\frac {\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^x+24\,b^4\,{\mathrm {e}}^x-20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{2\,a}+\frac {2\,b}{a^2\,{\mathrm {e}}^{2\,x}-a^2}+\frac {b^2\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x-24\,b^4\,{\mathrm {e}}^x+20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^3}-\frac {b^2\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^x+24\,b^4\,{\mathrm {e}}^x-20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^3}-\frac {b^3\,\ln \left (16\,a^5\,b-48\,a\,b^5-24\,b^5\,\sqrt {a^2+b^2}-32\,a^3\,b^3-32\,a^6\,{\mathrm {e}}^x+24\,b^6\,{\mathrm {e}}^x-40\,a^2\,b^3\,\sqrt {a^2+b^2}-32\,a^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+112\,a^2\,b^4\,{\mathrm {e}}^x+56\,a^4\,b^2\,{\mathrm {e}}^x+16\,a^4\,b\,\sqrt {a^2+b^2}+72\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+72\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^5+a^3\,b^2}+\frac {b^3\,\ln \left (24\,b^5\,\sqrt {a^2+b^2}-48\,a\,b^5+16\,a^5\,b-32\,a^3\,b^3-32\,a^6\,{\mathrm {e}}^x+24\,b^6\,{\mathrm {e}}^x+40\,a^2\,b^3\,\sqrt {a^2+b^2}+32\,a^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+112\,a^2\,b^4\,{\mathrm {e}}^x+56\,a^4\,b^2\,{\mathrm {e}}^x-16\,a^4\,b\,\sqrt {a^2+b^2}-72\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-72\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^5+a^3\,b^2} \]
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