Optimal. Leaf size=70 \[ \frac {2 b \sinh (c+d x) \sqrt {b \text {sech}(c+d x)}}{d}+\frac {2 i b^2 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \]
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Rubi [A] time = 0.04, 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 = {3768, 3771, 2639} \[ \frac {2 b \sinh (c+d x) \sqrt {b \text {sech}(c+d x)}}{d}+\frac {2 i b^2 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \]
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))^{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 = 0.04, size = 52, normalized size = 0.74 \[ \frac {2 b \sqrt {b \text {sech}(c+d x)} \left (\sinh (c+d x)+i \sqrt {\cosh (c+d x)} E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \operatorname {sech}\left (d x + c\right )} b \operatorname {sech}\left (d x + c\right ), x\right ) \]
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 \[ \int \left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, 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 \[ \int \left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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 \[ \int \left (b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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