Optimal. Leaf size=47 \[ -\frac{20}{63 \text{sech}^{\frac{3}{2}}(x)}-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{10 x \sinh (x)}{21 \sqrt{\text{sech}(x)}} \]
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Rubi [A] time = 0.104468, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4187, 4189} \[ -\frac{20}{63 \text{sech}^{\frac{3}{2}}(x)}-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{10 x \sinh (x)}{21 \sqrt{\text{sech}(x)}} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin{align*} \int \left (\frac{x}{\text{sech}^{\frac{7}{2}}(x)}-\frac{5}{21} x \sqrt{\text{sech}(x)}\right ) \, dx &=-\left (\frac{5}{21} \int x \sqrt{\text{sech}(x)} \, dx\right )+\int \frac{x}{\text{sech}^{\frac{7}{2}}(x)} \, dx\\ &=-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{5}{7} \int \frac{x}{\text{sech}^{\frac{3}{2}}(x)} \, dx-\frac{1}{21} \left (5 \sqrt{\cosh (x)} \sqrt{\text{sech}(x)}\right ) \int \frac{x}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}-\frac{20}{63 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{10 x \sinh (x)}{21 \sqrt{\text{sech}(x)}}+\frac{5}{21} \int x \sqrt{\text{sech}(x)} \, dx-\frac{1}{21} \left (5 \sqrt{\cosh (x)} \sqrt{\text{sech}(x)}\right ) \int \frac{x}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}-\frac{20}{63 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{10 x \sinh (x)}{21 \sqrt{\text{sech}(x)}}\\ \end{align*}
Mathematica [A] time = 0.103575, size = 45, normalized size = 0.96 \[ \sqrt{\text{sech}(x)} \left (\frac{13}{42} x \sinh (2 x)+\frac{1}{28} x \sinh (4 x)-\frac{88}{441} \cosh (2 x)-\frac{1}{98} \cosh (4 x)-\frac{167}{882}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{x \left ({\rm sech} \left (x\right ) \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,x}{21}\sqrt{{\rm sech} \left (x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{5}{21} \, x \sqrt{\operatorname{sech}\left (x\right )} + \frac{x}{\operatorname{sech}\left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{5}{21} \, x \sqrt{\operatorname{sech}\left (x\right )} + \frac{x}{\operatorname{sech}\left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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