Optimal. Leaf size=67 \[ \frac{\cot (c+d x)}{2 a d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a d \sqrt{-a \tan ^2(c+d x)}} \]
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Rubi [A] time = 0.0415855, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {4121, 3658, 3473, 3475} \[ \frac{\cot (c+d x)}{2 a d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a d \sqrt{-a \tan ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\left (a-a \sec ^2(c+d x)\right )^{3/2}} \, dx &=\int \frac{1}{\left (-a \tan ^2(c+d x)\right )^{3/2}} \, dx\\ &=-\frac{\tan (c+d x) \int \cot ^3(c+d x) \, dx}{a \sqrt{-a \tan ^2(c+d x)}}\\ &=\frac{\cot (c+d x)}{2 a d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \int \cot (c+d x) \, dx}{a \sqrt{-a \tan ^2(c+d x)}}\\ &=\frac{\cot (c+d x)}{2 a d \sqrt{-a \tan ^2(c+d x)}}+\frac{\log (\sin (c+d x)) \tan (c+d x)}{a d \sqrt{-a \tan ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.135554, size = 57, normalized size = 0.85 \[ -\frac{\tan ^3(c+d x) \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d \left (-a \tan ^2(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.24, size = 141, normalized size = 2.1 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}} \left ( 4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) - \left ( \cos \left ( dx+c \right ) \right ) ^{2}-4\,\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +4\,\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) -1 \right ) \left ( -{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48285, size = 81, normalized size = 1.21 \begin{align*} -\frac{\frac{\log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt{-a} a} - \frac{2 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt{-a} a} + \frac{\sqrt{-a}}{a^{2} \tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.503698, size = 228, normalized size = 3.4 \begin{align*} -\frac{{\left (2 \,{\left (\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{2 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \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}{\left (- a \sec ^{2}{\left (c + d x \right )} + a\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.81844, size = 285, normalized size = 4.25 \begin{align*} -\frac{\frac{\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{2} \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 )} + \frac{8 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{\sqrt{-a} a \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 )} - \frac{4 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )}{\sqrt{-a} a \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 )} + \frac{4 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1}{\sqrt{-a} a \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 ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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