Integrand size = 18, antiderivative size = 64 \[ \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 x}{5 b \cosh ^{\frac {5}{2}}(a+b x)}-\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{15 b^2}+\frac {4 \sinh (a+b x)}{15 b^2 \cosh ^{\frac {3}{2}}(a+b x)} \]
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Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5481, 2716, 2720} \[ \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{15 b^2}+\frac {4 \sinh (a+b x)}{15 b^2 \cosh ^{\frac {3}{2}}(a+b x)}-\frac {2 x}{5 b \cosh ^{\frac {5}{2}}(a+b x)} \]
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Rule 2716
Rule 2720
Rule 5481
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {2 \int \frac {1}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx}{5 b} \\ & = -\frac {2 x}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {4 \sinh (a+b x)}{15 b^2 \cosh ^{\frac {3}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{15 b} \\ & = -\frac {2 x}{5 b \cosh ^{\frac {5}{2}}(a+b x)}-\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{15 b^2}+\frac {4 \sinh (a+b x)}{15 b^2 \cosh ^{\frac {3}{2}}(a+b x)} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\frac {2 \left (-3 b x-2 i \cosh ^{\frac {5}{2}}(a+b x) \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )+\sinh (2 (a+b x))\right )}{15 b^2 \cosh ^{\frac {5}{2}}(a+b x)} \]
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\[\int \frac {x \sinh \left (b x +a \right )}{\cosh \left (b x +a \right )^{\frac {7}{2}}}d x\]
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Exception generated. \[ \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\text {Timed out} \]
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\[ \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (a+b\,x\right )}^{7/2}} \,d x \]
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