Optimal. Leaf size=102 \[ \frac {6 i b^4 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {6 b^3 \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{5 d}+\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3856,
2719} \begin {gather*} \frac {6 i b^4 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {6 b^3 \sinh (c+d x) \sqrt {b \text {sech}(c+d x)}}{5 d}+\frac {2 b \sinh (c+d x) (b \text {sech}(c+d x))^{5/2}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int (b \text {sech}(c+d x))^{7/2} \, dx &=\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d}+\frac {1}{5} \left (3 b^2\right ) \int (b \text {sech}(c+d x))^{3/2} \, dx\\ &=\frac {6 b^3 \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{5 d}+\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d}-\frac {1}{5} \left (3 b^4\right ) \int \frac {1}{\sqrt {b \text {sech}(c+d x)}} \, dx\\ &=\frac {6 b^3 \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{5 d}+\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d}-\frac {\left (3 b^4\right ) \int \sqrt {\cosh (c+d x)} \, dx}{5 \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}\\ &=\frac {6 i b^4 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {6 b^3 \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{5 d}+\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 68, normalized size = 0.67 \begin {gather*} \frac {b^2 (b \text {sech}(c+d x))^{3/2} \left (6 i \cosh ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )+3 \sinh (2 (c+d x))+2 \tanh (c+d x)\right )}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.53, size = 0, normalized size = 0.00 \[\int \left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {7}{2}}\, dx\]
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 478, normalized size = 4.69 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {2} {\left (b^{3} \cosh \left (d x + c\right )^{4} + 4 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{3} \sinh \left (d x + c\right )^{4} + 2 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b^{3} \cosh \left (d x + c\right )^{3} + b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (3 \, b^{3} \cosh \left (d x + c\right )^{5} + 15 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 3 \, b^{3} \sinh \left (d x + c\right )^{5} + 8 \, b^{3} \cosh \left (d x + c\right )^{3} + b^{3} \cosh \left (d x + c\right ) + 2 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{2} + 4 \, b^{3}\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{3} + 4 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (15 \, b^{3} \cosh \left (d x + c\right )^{4} + 24 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3}\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}\right )}}{5 \, {\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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