Optimal. Leaf size=64 \[ -\frac{a \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0407729, antiderivative size = 64, 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} \[ -\frac{a \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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 )^{3/2} \, dx &=\int \left (-a \tan ^2(c+d x)\right )^{3/2} \, dx\\ &=-\left (\left (a \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan ^3(c+d x) \, dx\right )\\ &=-\frac{a \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}+\left (a \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=-\frac{a \cot (c+d x) \log (\cos (c+d x)) \sqrt{-a \tan ^2(c+d x)}}{d}-\frac{a \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}\\ \end{align*}
Mathematica [A] time = 0.107481, size = 48, normalized size = 0.75 \[ \frac{\cot ^3(c+d x) \left (-a \tan ^2(c+d x)\right )^{3/2} \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.264, size = 145, normalized size = 2.3 \begin{align*} -{\frac{\cos \left ( dx+c \right ) }{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ( 2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) -2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ({\frac{1-\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) + \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{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48758, size = 54, normalized size = 0.84 \begin{align*} -\frac{\sqrt{-a} a \tan \left (d x + c\right )^{2} - \sqrt{-a} a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.528953, size = 167, normalized size = 2.61 \begin{align*} -\frac{{\left (2 \, a \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + a\right )} \sqrt{\frac{a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{2 \, d \cos \left (d x + c\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 \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 = 1.40452, size = 185, normalized size = 2.89 \begin{align*} -\frac{\sqrt{-a} a \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 ) - \sqrt{-a} a \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{{\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 - 6 \, \sqrt{-a} a}{\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}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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