Integrand size = 20, antiderivative size = 47 \[ \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx=-\frac {4}{49 \text {csch}^{\frac {7}{2}}(x)}+\frac {2 x \cosh (x)}{7 \text {csch}^{\frac {5}{2}}(x)}+\frac {20}{63 \text {csch}^{\frac {3}{2}}(x)}-\frac {10 x \cosh (x)}{21 \sqrt {\text {csch}(x)}} \]
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Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274} \[ \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx=\frac {20}{63 \text {csch}^{\frac {3}{2}}(x)}-\frac {4}{49 \text {csch}^{\frac {7}{2}}(x)}+\frac {2 x \cosh (x)}{7 \text {csch}^{\frac {5}{2}}(x)}-\frac {10 x \cosh (x)}{21 \sqrt {\text {csch}(x)}} \]
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Rule 4272
Rule 4274
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {5}{21} \int x \sqrt {\text {csch}(x)} \, dx\right )+\int \frac {x}{\text {csch}^{\frac {7}{2}}(x)} \, dx \\ & = -\frac {4}{49 \text {csch}^{\frac {7}{2}}(x)}+\frac {2 x \cosh (x)}{7 \text {csch}^{\frac {5}{2}}(x)}-\frac {5}{7} \int \frac {x}{\text {csch}^{\frac {3}{2}}(x)} \, dx-\frac {1}{21} \left (5 \sqrt {\text {csch}(x)} \sqrt {-\sinh (x)}\right ) \int \frac {x}{\sqrt {-\sinh (x)}} \, dx \\ & = -\frac {4}{49 \text {csch}^{\frac {7}{2}}(x)}+\frac {2 x \cosh (x)}{7 \text {csch}^{\frac {5}{2}}(x)}+\frac {20}{63 \text {csch}^{\frac {3}{2}}(x)}-\frac {10 x \cosh (x)}{21 \sqrt {\text {csch}(x)}}+\frac {5}{21} \int x \sqrt {\text {csch}(x)} \, dx-\frac {1}{21} \left (5 \sqrt {\text {csch}(x)} \sqrt {-\sinh (x)}\right ) \int \frac {x}{\sqrt {-\sinh (x)}} \, dx \\ & = -\frac {4}{49 \text {csch}^{\frac {7}{2}}(x)}+\frac {2 x \cosh (x)}{7 \text {csch}^{\frac {5}{2}}(x)}+\frac {20}{63 \text {csch}^{\frac {3}{2}}(x)}-\frac {10 x \cosh (x)}{21 \sqrt {\text {csch}(x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx=\sqrt {\text {csch}(x)} \left (-\frac {167}{882}+\frac {88}{441} \cosh (2 x)-\frac {1}{98} \cosh (4 x)-\frac {13}{42} x \sinh (2 x)+\frac {1}{28} x \sinh (4 x)\right ) \]
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\[\int \left (\frac {x}{\operatorname {csch}\left (x \right )^{\frac {7}{2}}}-\frac {5 x \sqrt {\operatorname {csch}\left (x \right )}}{21}\right )d x\]
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Exception generated. \[ \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx=- \frac {\int \left (- \frac {21 x}{\operatorname {csch}^{\frac {7}{2}}{\left (x \right )}}\right )\, dx + \int 5 x \sqrt {\operatorname {csch}{\left (x \right )}}\, dx}{21} \]
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\[ \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx=\int { -\frac {5}{21} \, x \sqrt {\operatorname {csch}\left (x\right )} + \frac {x}{\operatorname {csch}\left (x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx=\int { -\frac {5}{21} \, x \sqrt {\operatorname {csch}\left (x\right )} + \frac {x}{\operatorname {csch}\left (x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx=-\int \frac {5\,x\,\sqrt {\frac {1}{\mathrm {sinh}\left (x\right )}}}{21}-\frac {x}{{\left (\frac {1}{\mathrm {sinh}\left (x\right )}\right )}^{7/2}} \,d x \]
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