Integrand size = 10, antiderivative size = 86 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\frac {5 \coth (x)}{16 a \sqrt {a \text {csch}^4(x)}}-\frac {5 x \text {csch}^2(x)}{16 a \sqrt {a \text {csch}^4(x)}}-\frac {5 \cosh (x) \sinh (x)}{24 a \sqrt {a \text {csch}^4(x)}}+\frac {\cosh (x) \sinh ^3(x)}{6 a \sqrt {a \text {csch}^4(x)}} \]
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Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 2715, 8} \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=-\frac {5 x \text {csch}^2(x)}{16 a \sqrt {a \text {csch}^4(x)}}+\frac {5 \coth (x)}{16 a \sqrt {a \text {csch}^4(x)}}+\frac {\sinh ^3(x) \cosh (x)}{6 a \sqrt {a \text {csch}^4(x)}}-\frac {5 \sinh (x) \cosh (x)}{24 a \sqrt {a \text {csch}^4(x)}} \]
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Rule 8
Rule 2715
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {\text {csch}^2(x) \int \sinh ^6(x) \, dx}{a \sqrt {a \text {csch}^4(x)}} \\ & = \frac {\cosh (x) \sinh ^3(x)}{6 a \sqrt {a \text {csch}^4(x)}}-\frac {\left (5 \text {csch}^2(x)\right ) \int \sinh ^4(x) \, dx}{6 a \sqrt {a \text {csch}^4(x)}} \\ & = -\frac {5 \cosh (x) \sinh (x)}{24 a \sqrt {a \text {csch}^4(x)}}+\frac {\cosh (x) \sinh ^3(x)}{6 a \sqrt {a \text {csch}^4(x)}}+\frac {\left (5 \text {csch}^2(x)\right ) \int \sinh ^2(x) \, dx}{8 a \sqrt {a \text {csch}^4(x)}} \\ & = \frac {5 \coth (x)}{16 a \sqrt {a \text {csch}^4(x)}}-\frac {5 \cosh (x) \sinh (x)}{24 a \sqrt {a \text {csch}^4(x)}}+\frac {\cosh (x) \sinh ^3(x)}{6 a \sqrt {a \text {csch}^4(x)}}-\frac {\left (5 \text {csch}^2(x)\right ) \int 1 \, dx}{16 a \sqrt {a \text {csch}^4(x)}} \\ & = \frac {5 \coth (x)}{16 a \sqrt {a \text {csch}^4(x)}}-\frac {5 x \text {csch}^2(x)}{16 a \sqrt {a \text {csch}^4(x)}}-\frac {5 \cosh (x) \sinh (x)}{24 a \sqrt {a \text {csch}^4(x)}}+\frac {\cosh (x) \sinh ^3(x)}{6 a \sqrt {a \text {csch}^4(x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\frac {\text {csch}^6(x) (-60 x+45 \sinh (2 x)-9 \sinh (4 x)+\sinh (6 x))}{192 \left (a \text {csch}^4(x)\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(70)=140\).
Time = 0.16 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.67
method | result | size |
risch | \(-\frac {5 \,{\mathrm e}^{2 x} x}{16 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}+\frac {{\mathrm e}^{8 x}}{384 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {3 \,{\mathrm e}^{6 x}}{128 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}+\frac {15 \,{\mathrm e}^{4 x}}{128 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {15}{128 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 x}-1\right )^{2} a}+\frac {3 \,{\mathrm e}^{-2 x}}{128 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {{\mathrm e}^{-4 x}}{384 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}\) | \(230\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1141 vs. \(2 (70) = 140\).
Time = 0.27 (sec) , antiderivative size = 1141, normalized size of antiderivative = 13.27 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=-\frac {{\left (9 \, e^{\left (-2 \, x\right )} - 45 \, e^{\left (-4 \, x\right )} + 45 \, e^{\left (-8 \, x\right )} - 9 \, e^{\left (-10 \, x\right )} + e^{\left (-12 \, x\right )} - 1\right )} e^{\left (6 \, x\right )}}{384 \, a^{\frac {3}{2}}} - \frac {5 \, x}{16 \, a^{\frac {3}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\frac {{\left (110 \, e^{\left (6 \, x\right )} - 45 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-6 \, x\right )} - 120 \, x + e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} + 45 \, e^{\left (2 \, x\right )}}{384 \, a^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^4}\right )}^{3/2}} \,d x \]
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