Integrand size = 10, antiderivative size = 55 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{5/2}} \, dx=\frac {\coth (x)}{5 \left (a \text {csch}^2(x)\right )^{5/2}}-\frac {4 \coth (x)}{15 a \left (a \text {csch}^2(x)\right )^{3/2}}+\frac {8 \coth (x)}{15 a^2 \sqrt {a \text {csch}^2(x)}} \]
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Time = 0.02 (sec) , antiderivative size = 55, 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 = {4207, 198, 197} \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{5/2}} \, dx=\frac {8 \coth (x)}{15 a^2 \sqrt {a \text {csch}^2(x)}}-\frac {4 \coth (x)}{15 a \left (a \text {csch}^2(x)\right )^{3/2}}+\frac {\coth (x)}{5 \left (a \text {csch}^2(x)\right )^{5/2}} \]
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Rule 197
Rule 198
Rule 4207
Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \frac {1}{\left (-a+a x^2\right )^{7/2}} \, dx,x,\coth (x)\right )\right ) \\ & = \frac {\coth (x)}{5 \left (a \text {csch}^2(x)\right )^{5/2}}+\frac {4}{5} \text {Subst}\left (\int \frac {1}{\left (-a+a x^2\right )^{5/2}} \, dx,x,\coth (x)\right ) \\ & = \frac {\coth (x)}{5 \left (a \text {csch}^2(x)\right )^{5/2}}-\frac {4 \coth (x)}{15 a \left (a \text {csch}^2(x)\right )^{3/2}}-\frac {8 \text {Subst}\left (\int \frac {1}{\left (-a+a x^2\right )^{3/2}} \, dx,x,\coth (x)\right )}{15 a} \\ & = \frac {\coth (x)}{5 \left (a \text {csch}^2(x)\right )^{5/2}}-\frac {4 \coth (x)}{15 a \left (a \text {csch}^2(x)\right )^{3/2}}+\frac {8 \coth (x)}{15 a^2 \sqrt {a \text {csch}^2(x)}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{5/2}} \, dx=\frac {(150 \cosh (x)-25 \cosh (3 x)+3 \cosh (5 x)) \sqrt {a \text {csch}^2(x)} \sinh (x)}{240 a^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(195\) vs. \(2(43)=86\).
Time = 0.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 3.56
method | result | size |
risch | \(\frac {{\mathrm e}^{6 x}}{160 a^{2} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {5 \,{\mathrm e}^{4 x}}{96 a^{2} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {5 \,{\mathrm e}^{2 x}}{16 a^{2} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {5}{16 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right ) a^{2}}-\frac {5 \,{\mathrm e}^{-2 x}}{96 a^{2} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {{\mathrm e}^{-4 x}}{160 a^{2} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}\) | \(196\) |
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Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (43) = 86\).
Time = 0.27 (sec) , antiderivative size = 590, normalized size of antiderivative = 10.73 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
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none
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{5/2}} \, dx=-\frac {e^{\left (5 \, x\right )}}{160 \, a^{\frac {5}{2}}} + \frac {5 \, e^{\left (3 \, x\right )}}{96 \, a^{\frac {5}{2}}} - \frac {5 \, e^{\left (-x\right )}}{16 \, a^{\frac {5}{2}}} + \frac {5 \, e^{\left (-3 \, x\right )}}{96 \, a^{\frac {5}{2}}} - \frac {e^{\left (-5 \, x\right )}}{160 \, a^{\frac {5}{2}}} - \frac {5 \, e^{x}}{16 \, a^{\frac {5}{2}}} \]
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none
Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{5/2}} \, dx=\frac {{\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + 3 \, e^{\left (5 \, x\right )} - 25 \, e^{\left (3 \, x\right )} + 150 \, e^{x}}{480 \, a^{\frac {5}{2}} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \]
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Timed out. \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^2}\right )}^{5/2}} \,d x \]
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