Integrand size = 24, antiderivative size = 76 \[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=-\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {16}{27} i \sqrt {\text {csch}(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)} \]
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Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4273, 4274, 3854, 3856, 2720} \[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}-\frac {16}{27} i \sqrt {i \sinh (x)} \sqrt {\text {csch}(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \]
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Rule 2720
Rule 3854
Rule 3856
Rule 4273
Rule 4274
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int x^2 \sqrt {\text {csch}(x)} \, dx+\int \frac {x^2}{\text {csch}^{\frac {3}{2}}(x)} \, dx \\ & = -\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {1}{3} \int x^2 \sqrt {\text {csch}(x)} \, dx+\frac {8}{9} \int \frac {1}{\text {csch}^{\frac {3}{2}}(x)} \, dx+\frac {1}{3} \left (\sqrt {\text {csch}(x)} \sqrt {-\sinh (x)}\right ) \int \frac {x^2}{\sqrt {-\sinh (x)}} \, dx \\ & = -\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {8}{27} \int \sqrt {\text {csch}(x)} \, dx \\ & = -\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {1}{27} \left (8 \sqrt {\text {csch}(x)} \sqrt {i \sinh (x)}\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx \\ & = -\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {16}{27} i \sqrt {\text {csch}(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\frac {4 \left (3+\text {csch}^2(x)\right ) \left (-12 x+\left (8+9 x^2\right ) \coth (x)+\frac {8 \text {csch}(x) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )}{\sqrt {i \sinh (x)}}\right )}{27 (-1+3 \cosh (2 x)) \text {csch}^{\frac {7}{2}}(x)} \]
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\[\int \left (\frac {x^{2}}{\operatorname {csch}\left (x \right )^{\frac {3}{2}}}+\frac {x^{2} \sqrt {\operatorname {csch}\left (x \right )}}{3}\right )d x\]
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Exception generated. \[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\frac {\int \frac {3 x^{2}}{\operatorname {csch}^{\frac {3}{2}}{\left (x \right )}}\, dx + \int x^{2} \sqrt {\operatorname {csch}{\left (x \right )}}\, dx}{3} \]
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\[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\int { \frac {1}{3} \, x^{2} \sqrt {\operatorname {csch}\left (x\right )} + \frac {x^{2}}{\operatorname {csch}\left (x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\int { \frac {1}{3} \, x^{2} \sqrt {\operatorname {csch}\left (x\right )} + \frac {x^{2}}{\operatorname {csch}\left (x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\int \frac {x^2\,\sqrt {\frac {1}{\mathrm {sinh}\left (x\right )}}}{3}+\frac {x^2}{{\left (\frac {1}{\mathrm {sinh}\left (x\right )}\right )}^{3/2}} \,d x \]
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