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 = 0.10, 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 = {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*}
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Mathematica [A] time = 0.11, 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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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 \[ \int -\frac {5}{21} \, x \sqrt {\operatorname {sech}\relax (x)} + \frac {x}{\operatorname {sech}\relax (x)^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {x}{\mathrm {sech}\relax (x )^{\frac {7}{2}}}-\frac {5 x \sqrt {\mathrm {sech}\relax (x )}}{21}\, dx \]
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 \[ \int -\frac {5}{21} \, x \sqrt {\operatorname {sech}\relax (x)} + \frac {x}{\operatorname {sech}\relax (x)^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {5\,x\,\sqrt {\frac {1}{\mathrm {cosh}\relax (x)}}}{21}-\frac {x}{{\left (\frac {1}{\mathrm {cosh}\relax (x)}\right )}^{7/2}} \,d x \]
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 \[ - \frac {\int \left (- \frac {21 x}{\operatorname {sech}^{\frac {7}{2}}{\relax (x )}}\right )\, dx + \int 5 x \sqrt {\operatorname {sech}{\relax (x )}}\, dx}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
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