Optimal. Leaf size=101 \[ \frac{a^2 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^2 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a^2 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0513671, antiderivative size = 101, 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} \[ \frac{a^2 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^2 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a^2 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (a-a \sec ^2(c+d x)\right )^{5/2} \, dx &=\int \left (-a \tan ^2(c+d x)\right )^{5/2} \, dx\\ &=\left (a^2 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan ^5(c+d x) \, dx\\ &=\frac{a^2 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\left (a^2 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan ^3(c+d x) \, dx\\ &=-\frac{a^2 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}+\frac{a^2 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}+\left (a^2 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=-\frac{a^2 \cot (c+d x) \log (\cos (c+d x)) \sqrt{-a \tan ^2(c+d x)}}{d}-\frac{a^2 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}+\frac{a^2 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}\\ \end{align*}
Mathematica [A] time = 0.554348, size = 60, normalized size = 0.59 \[ -\frac{\cot ^5(c+d x) \left (-a \tan ^2(c+d x)\right )^{5/2} \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.304, size = 157, normalized size = 1.6 \begin{align*} -{\frac{\cos \left ( dx+c \right ) }{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( 4\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}\ln \left ({\frac{1-\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +4\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -4\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1 \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 = 1.51641, size = 84, normalized size = 0.83 \begin{align*} \frac{\sqrt{-a} a^{2} \tan \left (d x + c\right )^{4} - 2 \, \sqrt{-a} a^{2} \tan \left (d x + c\right )^{2} + 2 \, \sqrt{-a} a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.510939, size = 207, normalized size = 2.05 \begin{align*} -\frac{{\left (4 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (-\cos \left (d x + c\right )\right ) + 4 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{4 \, d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.63818, size = 246, normalized size = 2.44 \begin{align*} -\frac{2 \, \sqrt{-a} a^{2} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + 2\right ) - 2 \, \sqrt{-a} a^{2} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} - 2\right ) + \frac{3 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}\right )}^{2} \sqrt{-a} a^{2} - 20 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}\right )} \sqrt{-a} a^{2} + 44 \, \sqrt{-a} a^{2}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} - 2\right )}^{2}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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