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 = 0.0349387, antiderivative size = 32, normalized size of antiderivative = 1., 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 4121
Rule 3658
Rule 3475
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*}
Mathematica [A] time = 0.0451237, 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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Maple [B] time = 0.33, size = 75, normalized size = 2.3 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d\cos \left ( dx+c \right ) } \left ( -\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) +\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \right ){\frac{1}{\sqrt{-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51489, size = 50, normalized size = 1.56 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.495317, size = 135, normalized size = 4.22 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- a \sec ^{2}{\left (c + d x \right )} + a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a \sec \left (d x + c\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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