Optimal. Leaf size=33 \[ -\frac {\cot (c+d x) \sqrt {-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4121, 3658, 3475} \[ -\frac {\cot (c+d x) \sqrt {-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \sqrt {a-a \sec ^2(c+d x)} \, dx &=\int \sqrt {-a \tan ^2(c+d x)} \, dx\\ &=\left (\cot (c+d x) \sqrt {-a \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=-\frac {\cot (c+d x) \log (\cos (c+d x)) \sqrt {-a \tan ^2(c+d x)}}{d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 33, normalized size = 1.00 \[ -\frac {\cot (c+d x) \sqrt {-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 53, normalized size = 1.61 \[ -\frac {\sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right )}{d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 141, normalized size = 4.27 \[ -\frac {{\left (\log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {-a} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.07, size = 109, normalized size = 3.30 \[ -\frac {\left (\ln \left (-\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+\ln \left (-\frac {-\sin \left (d x +c \right )-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-\ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right )\right ) \cos \left (d x +c \right ) \sqrt {-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{2}}}}{d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 21, normalized size = 0.64 \[ \frac {\sqrt {-a} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
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 \[ \int \sqrt {a-\frac {a}{{\cos \left (c+d\,x\right )}^2}} \,d x \]
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 \[ \int \sqrt {- a \sec ^{2}{\left (c + d x \right )} + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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