Optimal. Leaf size=47 \[ -\frac {4}{49 \text {csch}^{\frac {7}{2}}(x)}+\frac {2 x \cosh (x)}{7 \text {csch}^{\frac {5}{2}}(x)}+\frac {20}{63 \text {csch}^{\frac {3}{2}}(x)}-\frac {10 x \cosh (x)}{21 \sqrt {\text {csch}(x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274}
\begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4272
Rule 4274
Rubi steps
\begin {align*} \int \left (\frac {x}{\text {csch}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {csch}(x)}\right ) \, dx &=-\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*}
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Mathematica [A]
time = 0.09, size = 45, normalized size = 0.96 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.57, size = 0, normalized size = 0.00 \[\int \frac {x}{\mathrm {csch}\left (x \right )^{\frac {7}{2}}}-\frac {5 x \sqrt {\mathrm {csch}\left (x \right )}}{21}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} - \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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