Optimal. Leaf size=107 \[ \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}+\frac {20 i \sqrt {\cosh (a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{147 b^2}-\frac {4 \sinh (a+b x)}{49 b^2 \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {20 \sinh (a+b x)}{147 b^2 \sqrt {\text {sech}(a+b x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5552, 3854,
3856, 2720} \begin {gather*} -\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)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{147 b^2}+\frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3854
Rule 3856
Rule 5552
Rubi steps
\begin {align*} \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 {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)} F\left (\left .\frac {1}{2} i (a+b x)\right |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*}
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Mathematica [A]
time = 0.18, size = 93, normalized size = 0.87 \begin {gather*} \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)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )-52 \sinh (2 (a+b x))-6 \sinh (4 (a+b x))\right )}{588 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.48, size = 0, normalized size = 0.00 \[\int \frac {x \sinh \left (b x +a \right )}{\mathrm {sech}\left (b x +a \right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x \sinh {\left (a + b x \right )}}{\operatorname {sech}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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