Optimal. Leaf size=84 \[ \frac {4 \sinh (a+b x) \text {sech}^{\frac {3}{2}}(a+b x)}{15 b^2}-\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{15 b^2}-\frac {2 x \text {sech}^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rubi [A] time = 0.05, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5444, 3768, 3771, 2641} \[ \frac {4 \sinh (a+b x) \text {sech}^{\frac {3}{2}}(a+b x)}{15 b^2}-\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{15 b^2}-\frac {2 x \text {sech}^{\frac {5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3768
Rule 3771
Rule 5444
Rubi steps
\begin {align*} \int x \text {sech}^{\frac {7}{2}}(a+b x) \sinh (a+b x) \, dx &=-\frac {2 x \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}+\frac {2 \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx}{5 b}\\ &=-\frac {2 x \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}+\frac {4 \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{15 b^2}+\frac {2 \int \sqrt {\text {sech}(a+b x)} \, dx}{15 b}\\ &=-\frac {2 x \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}+\frac {4 \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{15 b^2}+\frac {\left (2 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{15 b}\\ &=-\frac {4 i \sqrt {\cosh (a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{15 b^2}-\frac {2 x \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}+\frac {4 \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{15 b^2}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 65, normalized size = 0.77 \[ -\frac {2 \sqrt {\text {sech}(a+b x)} \left (-2 \tanh (a+b x)+3 b x \text {sech}^2(a+b x)+2 i \sqrt {\cosh (a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )\right )}{15 b^2} \]
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 \[ \int x \operatorname {sech}\left (b x + a\right )^{\frac {7}{2}} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int x \mathrm {sech}\left (b x +a \right )^{\frac {7}{2}} \sinh \left (b x +a \right )\, dx \]
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 \[ \int x \operatorname {sech}\left (b x + a\right )^{\frac {7}{2}} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {sinh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{7/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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