Optimal. Leaf size=70 \[ \frac {2 i b^2 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {2 b \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{d} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {2 b \sinh (c+d x) \sqrt {b \text {sech}(c+d x)}}{d}+\frac {2 i b^2 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \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))^{3/2} \, dx &=\frac {2 b \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{d}-b^2 \int \frac {1}{\sqrt {b \text {sech}(c+d x)}} \, dx\\ &=\frac {2 b \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{d}-\frac {b^2 \int \sqrt {\cosh (c+d x)} \, dx}{\sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}\\ &=\frac {2 i b^2 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {2 b \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 52, normalized size = 0.74 \begin {gather*} \frac {2 b \sqrt {b \text {sech}(c+d x)} \left (i \sqrt {\cosh (c+d x)} E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )+\sinh (c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.31, size = 0, normalized size = 0.00 \[\int \left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}\, dx\]
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 = 107, normalized size = 1.53 \begin {gather*} \frac {2 \, {\left (\sqrt {2} b^{\frac {3}{2}} {\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 (b \cosh \left (d x + c\right ) + b \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 )}}{d} \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 \left (b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, 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 {\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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