Integrand size = 18, antiderivative size = 107 \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}+\frac {20 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{147 b^2}-\frac {4 \sinh (a+b x)}{49 b^2 \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {20 \sinh (a+b x)}{147 b^2 \sqrt {\text {sech}(a+b x)}} \]
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Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5552, 3854, 3856, 2720} \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=-\frac {4 \sinh (a+b x)}{49 b^2 \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {20 \sinh (a+b x)}{147 b^2 \sqrt {\text {sech}(a+b x)}}+\frac {20 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{147 b^2}+\frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)} \]
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Rule 2720
Rule 3854
Rule 3856
Rule 5552
Rubi steps \begin{align*} \text {integral}& = \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\text {sech}^{\frac {7}{2}}(a+b x)} \, dx}{7 b} \\ & = \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {4 \sinh (a+b x)}{49 b^2 \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {10 \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx}{49 b} \\ & = \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {4 \sinh (a+b x)}{49 b^2 \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {20 \sinh (a+b x)}{147 b^2 \sqrt {\text {sech}(a+b x)}}-\frac {10 \int \sqrt {\text {sech}(a+b x)} \, dx}{147 b} \\ & = \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {4 \sinh (a+b x)}{49 b^2 \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {20 \sinh (a+b x)}{147 b^2 \sqrt {\text {sech}(a+b x)}}-\frac {\left (10 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{147 b} \\ & = \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}+\frac {20 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{147 b^2}-\frac {4 \sinh (a+b x)}{49 b^2 \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {20 \sinh (a+b x)}{147 b^2 \sqrt {\text {sech}(a+b x)}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {\text {sech}(a+b x)} \left (63 b x+84 b x \cosh (2 (a+b x))+21 b x \cosh (4 (a+b x))+80 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )-52 \sinh (2 (a+b x))-6 \sinh (4 (a+b x))\right )}{588 b^2} \]
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\[\int \frac {x \sinh \left (b x +a \right )}{\operatorname {sech}\left (b x +a \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x \sinh {\left (a + b x \right )}}{\operatorname {sech}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \sinh \left (b x + a\right )}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \sinh \left (b x + a\right )}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
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