Optimal. Leaf size=102 \[ \frac {6 i b^4 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {6 b^3 \sinh (c+d x) \sqrt {b \text {sech}(c+d x)}}{5 d}+\frac {2 b \sinh (c+d x) (b \text {sech}(c+d x))^{5/2}}{5 d} \]
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Rubi [A] time = 0.06, 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 = {3768, 3771, 2639} \[ \frac {6 b^3 \sinh (c+d x) \sqrt {b \text {sech}(c+d x)}}{5 d}+\frac {6 i b^4 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {2 b \sinh (c+d x) (b \text {sech}(c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3768
Rule 3771
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 = 0.20, size = 68, normalized size = 0.67 \[ \frac {b^2 (b \text {sech}(c+d x))^{3/2} \left (3 \sinh (2 (c+d x))+2 \tanh (c+d x)+6 i \cosh ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )\right )}{5 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \operatorname {sech}\left (d x + c\right )} b^{3} \operatorname {sech}\left (d x + c\right )^{3}, x\right ) \]
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 \[ \int \left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, 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 \[ \int \left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
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 \[ \int {\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{7/2} \,d x \]
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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