Optimal. Leaf size=32 \[ \frac {\tan (c+d x) \log (\sin (c+d x))}{d \sqrt {-a \tan ^2(c+d x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 32, 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} \[ \frac {\tan (c+d x) \log (\sin (c+d x))}{d \sqrt {-a \tan ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx &=\int \frac {1}{\sqrt {-a \tan ^2(c+d x)}} \, dx\\ &=\frac {\tan (c+d x) \int \cot (c+d x) \, dx}{\sqrt {-a \tan ^2(c+d x)}}\\ &=\frac {\log (\sin (c+d x)) \tan (c+d x)}{d \sqrt {-a \tan ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 40, normalized size = 1.25 \[ \frac {\tan (c+d x) (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d \sqrt {-a \tan ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.87, size = 56, normalized size = 1.75 \[ -\frac {\sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}} \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{a d \sin \left (d x + c\right )} \]
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 \[ \int \frac {1}{\sqrt {-a \sec \left (d x + c\right )^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.96, size = 76, normalized size = 2.38 \[ -\frac {\left (-\ln \left (-\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+\ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right )\right ) \sin \left (d x +c \right )}{d \sqrt {-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{2}}}\, \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 37, normalized size = 1.16 \[ -\frac {\frac {\log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt {-a}} - \frac {2 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt {-a}}}{2 \, d} \]
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 \[ \int \frac {1}{\sqrt {a-\frac {a}{{\cos \left (c+d\,x\right )}^2}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- a \sec ^{2}{\left (c + d x \right )} + a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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