Optimal. Leaf size=26 \[ \frac {2 a \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3792} \[ \frac {2 a \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3792
Rubi steps
\begin {align*} \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx &=\frac {2 a \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 29, normalized size = 1.12 \[ \frac {2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sec (c+d x)+1)}}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 41, normalized size = 1.58 \[ \frac {2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.08, size = 62, normalized size = 2.38 \[ -\frac {2 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 42, normalized size = 1.62 \[ -\frac {2 \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (-1+\cos \left (d x +c \right )\right )}{d \sin \left (d x +c \right )} \]
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 \[ \int \sqrt {a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 41, normalized size = 1.58 \[ \frac {2\,\sin \left (c+d\,x\right )\,\sqrt {\frac {a\,\left (\cos \left (c+d\,x\right )+1\right )}{\cos \left (c+d\,x\right )}}}{d\,\left (\cos \left (c+d\,x\right )+1\right )} \]
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 \[ \int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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