Optimal. Leaf size=104 \[ -\frac {10 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{21 b^4 d}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 104, 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 = {3769, 3771, 2641} \[ \frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}}-\frac {10 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{21 b^4 d}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx &=\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {5 \int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx}{7 b^2}\\ &=\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}}+\frac {5 \int \sqrt {b \text {sech}(c+d x)} \, dx}{21 b^4}\\ &=\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}}+\frac {\left (5 \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}\right ) \int \frac {1}{\sqrt {\cosh (c+d x)}} \, dx}{21 b^4}\\ &=-\frac {10 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{21 b^4 d}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 70, normalized size = 0.67 \[ \frac {\sqrt {b \text {sech}(c+d x)} \left (26 \sinh (2 (c+d x))+3 \sinh (4 (c+d x))-40 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right )\right )}{84 b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \operatorname {sech}\left (d x + c\right )}}{b^{4} \operatorname {sech}\left (d x + c\right )^{4}}, x\right ) \]
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 \[ \int \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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