Integrand size = 10, antiderivative size = 49 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx=\frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {4 \coth (x)}{15 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {8 \coth (x)}{15 \sqrt {-\text {csch}^2(x)}} \]
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Time = 0.01 (sec) , antiderivative size = 49, 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} \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx=\frac {8 \coth (x)}{15 \sqrt {-\text {csch}^2(x)}}+\frac {4 \coth (x)}{15 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}} \]
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Rule 197
Rule 198
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{7/2}} \, dx,x,\coth (x)\right ) \\ & = \frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {4}{5} \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{5/2}} \, dx,x,\coth (x)\right ) \\ & = \frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {4 \coth (x)}{15 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {8}{15} \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right ) \\ & = \frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {4 \coth (x)}{15 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {8 \coth (x)}{15 \sqrt {-\text {csch}^2(x)}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx=\frac {(150 \cosh (x)-25 \cosh (3 x)+3 \cosh (5 x)) \text {csch}(x)}{240 \sqrt {-\text {csch}^2(x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(177\) vs. \(2(37)=74\).
Time = 0.15 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.63
method | result | size |
risch | \(\frac {{\mathrm e}^{6 x}}{160 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {5 \,{\mathrm e}^{4 x}}{96 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {5 \,{\mathrm e}^{2 x}}{16 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {5}{16 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {5 \,{\mathrm e}^{-2 x}}{96 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {{\mathrm e}^{-4 x}}{160 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}\) | \(178\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx=\frac {1}{480} \, {\left (-3 i \, e^{\left (10 \, x\right )} + 25 i \, e^{\left (8 \, x\right )} - 150 i \, e^{\left (6 \, x\right )} - 150 i \, e^{\left (4 \, x\right )} + 25 i \, e^{\left (2 \, x\right )} - 3 i\right )} e^{\left (-5 \, x\right )} \]
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\[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (- \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx=\frac {1}{160} i \, e^{\left (5 \, x\right )} - \frac {5}{96} i \, e^{\left (3 \, x\right )} + \frac {5}{16} i \, e^{\left (-x\right )} - \frac {5}{96} i \, e^{\left (-3 \, x\right )} + \frac {1}{160} i \, e^{\left (-5 \, x\right )} + \frac {5}{16} i \, e^{x} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx=-\frac {{\left (-150 i \, e^{\left (4 \, x\right )} + 25 i \, e^{\left (2 \, x\right )} - 3 i\right )} e^{\left (-5 \, x\right )} - 3 i \, e^{\left (5 \, x\right )} + 25 i \, e^{\left (3 \, x\right )} - 150 i \, e^{x}}{480 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]
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Timed out. \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (-\frac {1}{{\mathrm {sinh}\left (x\right )}^2}\right )}^{5/2}} \,d x \]
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