Optimal. Leaf size=100 \[ -\frac{\cot ^3(c+d x)}{4 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\cot (c+d x)}{2 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a^2 d \sqrt{-a \tan ^2(c+d x)}} \]
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Rubi [A] time = 0.0505803, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {4121, 3658, 3473, 3475} \[ -\frac{\cot ^3(c+d x)}{4 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\cot (c+d x)}{2 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a^2 d \sqrt{-a \tan ^2(c+d x)}} \]
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 )^{5/2}} \, dx &=\int \frac{1}{\left (-a \tan ^2(c+d x)\right )^{5/2}} \, dx\\ &=\frac{\tan (c+d x) \int \cot ^5(c+d x) \, dx}{a^2 \sqrt{-a \tan ^2(c+d x)}}\\ &=-\frac{\cot ^3(c+d x)}{4 a^2 d \sqrt{-a \tan ^2(c+d x)}}-\frac{\tan (c+d x) \int \cot ^3(c+d x) \, dx}{a^2 \sqrt{-a \tan ^2(c+d x)}}\\ &=\frac{\cot (c+d x)}{2 a^2 d \sqrt{-a \tan ^2(c+d x)}}-\frac{\cot ^3(c+d x)}{4 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \int \cot (c+d x) \, dx}{a^2 \sqrt{-a \tan ^2(c+d x)}}\\ &=\frac{\cot (c+d x)}{2 a^2 d \sqrt{-a \tan ^2(c+d x)}}-\frac{\cot ^3(c+d x)}{4 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\log (\sin (c+d x)) \tan (c+d x)}{a^2 d \sqrt{-a \tan ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.259141, size = 69, normalized size = 0.69 \[ \frac{\tan ^5(c+d x) \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d \left (-a \tan ^2(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.247, size = 203, normalized size = 2. \begin{align*}{\frac{\sin \left ( dx+c \right ) }{32\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}} \left ( 32\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -32\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) -13\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) -6\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+32\,\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -32\,\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) +11 \right ) \left ( -{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47495, size = 107, normalized size = 1.07 \begin{align*} -\frac{\frac{2 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt{-a} a^{2}} - \frac{4 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt{-a} a^{2}} + \frac{2 \, \sqrt{-a} \tan \left (d x + c\right )^{2} - \sqrt{-a}}{a^{3} \tan \left (d x + c\right )^{4}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.512133, size = 315, normalized size = 3.15 \begin{align*} \frac{{\left (4 \, \cos \left (d x + c\right )^{3} - 4 \,{\left (\cos \left (d x + c\right )^{5} - 2 \, \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 ) - 3 \, \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}}}}{4 \,{\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} 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{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.879, size = 369, normalized size = 3.69 \begin{align*} -\frac{\frac{64 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{\sqrt{-a} 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{32 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )}{\sqrt{-a} 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{\sqrt{-a} a^{3} \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 )^{4} - 12 \, \sqrt{-a} a^{3} \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}}{a^{6}} + \frac{48 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}{\sqrt{-a} 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 ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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