Optimal. Leaf size=38 \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3774, 203} \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3774
Rubi steps
\begin {align*} \int \sqrt {a-a \text {sech}(c+d x)} \, dx &=-\frac {(2 i a) \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {i a \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d}\\ &=\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 2.38, size = 70, normalized size = 1.84 \[ \frac {\sqrt {e^{2 (c+d x)}+1} \sqrt {a-a \text {sech}(c+d x)} \left (\sinh ^{-1}\left (e^{c+d x}\right )+\tanh ^{-1}\left (\sqrt {e^{2 (c+d x)}+1}\right )\right )}{d \left (e^{c+d x}-1\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 642, normalized size = 16.89 \[ \frac {\sqrt {a} \log \left (\frac {a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} + 3 \, a \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) + 3 \, a\right )} \sinh \left (d x + c\right )^{3} + 5 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} + 9 \, a \cosh \left (d x + c\right ) + 5 \, a\right )} \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{5} + {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{4} + \sinh \left (d x + c\right )^{5} + 3 \, \cosh \left (d x + c\right )^{4} + {\left (10 \, \cosh \left (d x + c\right )^{2} + 12 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{3} + 5 \, \cosh \left (d x + c\right )^{3} + {\left (10 \, \cosh \left (d x + c\right )^{3} + 18 \, \cosh \left (d x + c\right )^{2} + 15 \, \cosh \left (d x + c\right ) + 7\right )} \sinh \left (d x + c\right )^{2} + 7 \, \cosh \left (d x + c\right )^{2} + {\left (5 \, \cosh \left (d x + c\right )^{4} + 12 \, \cosh \left (d x + c\right )^{3} + 15 \, \cosh \left (d x + c\right )^{2} + 14 \, \cosh \left (d x + c\right ) + 4\right )} \sinh \left (d x + c\right ) + 4 \, \cosh \left (d x + c\right ) + 4\right )} \sqrt {a} \sqrt {\frac {a}{\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}} + 4 \, a \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} + 9 \, a \cosh \left (d x + c\right )^{2} + 10 \, a \cosh \left (d x + c\right ) + 4 \, a\right )} \sinh \left (d x + c\right ) + 4 \, a}{\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}}\right ) + \sqrt {a} \log \left (-\frac {a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} - \cosh \left (d x + c\right )^{2} + {\left (3 \, \cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right ) - 1\right )} \sqrt {a} \sqrt {\frac {a}{\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}} - a \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) - a\right )} \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 101, normalized size = 2.66 \[ -\frac {\frac {2 \, a \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (e^{\left (d x + c\right )} - 1\right )}{\sqrt {-a}} + \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) \mathrm {sgn}\left (e^{\left (d x + c\right )} - 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \sqrt {a -a \,\mathrm {sech}\left (d x +c \right )}\, dx \]
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 \[ \int \sqrt {-a \operatorname {sech}\left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {a-\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \sqrt {- a \operatorname {sech}{\left (c + d x \right )} + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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