Optimal. Leaf size=125 \[ \frac {2 \coth (c+d x) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (\text {sech}(c+d x)+1)}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x)) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right )|\frac {a-b}{a+b}\right )}{d \sqrt {a+b}} \]
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Rubi [A] time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3780} \[ \frac {2 \coth (c+d x) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (\text {sech}(c+d x)+1)}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x)) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right )|\frac {a-b}{a+b}\right )}{d \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 3780
Rubi steps
\begin {align*} \int \sqrt {a+b \text {sech}(c+d x)} \, dx &=\frac {2 \coth (c+d x) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right )|\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d}\\ \end {align*}
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Mathematica [F] time = 7.86, size = 0, normalized size = 0.00 \[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 2.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \sqrt {b \operatorname {sech}\left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}\, dx \]
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 \[ \int \sqrt {b \operatorname {sech}\left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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