Integrand size = 10, antiderivative size = 61 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx=-\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}-\frac {3 \text {arctanh}(\cosh (x)) \sinh (x)}{8 a^2 \sqrt {a \sinh ^2(x)}} \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3283, 3286, 3855} \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx=-\frac {3 \sinh (x) \text {arctanh}(\cosh (x))}{8 a^2 \sqrt {a \sinh ^2(x)}}+\frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}-\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}} \]
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Rule 3283
Rule 3286
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}-\frac {3 \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx}{4 a} \\ & = -\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}+\frac {3 \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx}{8 a^2} \\ & = -\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}+\frac {(3 \sinh (x)) \int \text {csch}(x) \, dx}{8 a^2 \sqrt {a \sinh ^2(x)}} \\ & = -\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}-\frac {3 \text {arctanh}(\cosh (x)) \sinh (x)}{8 a^2 \sqrt {a \sinh ^2(x)}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx=-\frac {\text {csch}(x) \left (-6 \text {csch}^2\left (\frac {x}{2}\right )+\text {csch}^4\left (\frac {x}{2}\right )+24 \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )-6 \text {sech}^2\left (\frac {x}{2}\right )-\text {sech}^4\left (\frac {x}{2}\right )\right ) \sqrt {a \sinh ^2(x)}}{64 a^3} \]
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Time = 0.92 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.46
method | result | size |
default | \(\frac {\sqrt {a \cosh \left (x \right )^{2}}\, \left (-3 \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \cosh \left (x \right )^{2}}+2 a}{\sinh \left (x \right )}\right ) a \sinh \left (x \right )^{4}+3 \sinh \left (x \right )^{2} \sqrt {a}\, \sqrt {a \cosh \left (x \right )^{2}}-2 \sqrt {a}\, \sqrt {a \cosh \left (x \right )^{2}}\right )}{8 a^{\frac {7}{2}} \sinh \left (x \right )^{3} \cosh \left (x \right ) \sqrt {a \sinh \left (x \right )^{2}}}\) | \(89\) |
risch | \(\frac {3 \,{\mathrm e}^{6 x}-11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}+3}{4 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{3} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}+\frac {3 \left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}-1\right )}{8 a^{2} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}-\frac {3 \left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}+1\right )}{8 a^{2} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}\) | \(123\) |
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Leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (49) = 98\).
Time = 0.31 (sec) , antiderivative size = 875, normalized size of antiderivative = 14.34 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \sinh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx=\frac {3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 6 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 4 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} - a^{\frac {5}{2}} e^{\left (-8 \, x\right )} - a^{\frac {5}{2}}\right )}} + \frac {3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a^{\frac {5}{2}}} - \frac {3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a^{\frac {5}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx=\frac {3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \, {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2} a^{\frac {5}{2}} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \]
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Timed out. \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {sinh}\left (x\right )}^2\right )}^{5/2}} \,d x \]
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