Optimal. Leaf size=76 \[ -\frac {2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{3 b^2 d}+\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 76, 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 = {3854, 3856,
2720} \begin {gather*} \frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}-\frac {2 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)}}{3 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx &=\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}+\frac {\int \sqrt {b \text {sech}(c+d x)} \, dx}{3 b^2}\\ &=\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}+\frac {\left (\sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}\right ) \int \frac {1}{\sqrt {\cosh (c+d x)}} \, dx}{3 b^2}\\ &=-\frac {2 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)}}{3 b^2 d}+\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 63, normalized size = 0.83 \begin {gather*} \frac {\text {sech}^2(c+d x) \left (-2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right )+\sinh (2 (c+d x))\right )}{3 d (b \text {sech}(c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.39, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
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 = 0.13, size = 231, normalized size = 3.04 \begin {gather*} \frac {4 \, \sqrt {2} {\left (\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}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4} - 1\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}}}{6 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2}\right )}} \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 {1}{\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 \frac {1}{{\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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