Optimal. Leaf size=76 \[ \frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}-\frac {16}{27} i \sqrt {i \sinh (x)} \sqrt {\text {csch}(x)} F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \]
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Rubi [A] time = 0.21, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4188, 4189, 3769, 3771, 2641} \[ \frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}-\frac {16}{27} i \sqrt {i \sinh (x)} \sqrt {\text {csch}(x)} F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4188
Rule 4189
Rubi steps
\begin {align*} \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx &=\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)} F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {i \sinh (x)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 63, normalized size = 0.83 \[ \frac {1}{27} \sqrt {\text {csch}(x)} \left (9 x^2 \sinh (2 x)+12 x+8 \sinh (2 x)-12 x \cosh (2 x)-16 i \sqrt {i \sinh (x)} F\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{3} \, x^{2} \sqrt {\operatorname {csch}\relax (x)} + \frac {x^{2}}{\operatorname {csch}\relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\mathrm {csch}\relax (x )^{\frac {3}{2}}}+\frac {x^{2} \sqrt {\mathrm {csch}\relax (x )}}{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{3} \, x^{2} \sqrt {\operatorname {csch}\relax (x)} + \frac {x^{2}}{\operatorname {csch}\relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\sqrt {\frac {1}{\mathrm {sinh}\relax (x)}}}{3}+\frac {x^2}{{\left (\frac {1}{\mathrm {sinh}\relax (x)}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {3 x^{2}}{\operatorname {csch}^{\frac {3}{2}}{\relax (x )}}\, dx + \int x^{2} \sqrt {\operatorname {csch}{\relax (x )}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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