Optimal. Leaf size=64 \[ -\frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}-\frac{2 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d} \]
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Rubi [A] time = 0.0709612, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3887, 325, 203} \[ -\frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}-\frac{2 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
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Rule 3887
Rule 325
Rule 203
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx &=-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a x^2\right )} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{2 a \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}-\frac{2 a \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.362542, size = 102, normalized size = 1.59 \[ -\frac{2 \cot (c+d x) \sqrt{\frac{1}{\sec (c+d x)+1}} (a (\sec (c+d x)+1))^{3/2} \left (\sqrt{\cos (c+d x)} \sqrt{\frac{1}{\cos (c+d x)+1}}+\tan \left (\frac{1}{2} (c+d x)\right ) \sin ^{-1}\left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\frac{1}{\cos (c+d x)+1}}}\right )\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.197, size = 115, normalized size = 1.8 \begin{align*} -{\frac{a}{d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( -\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}\sin \left ( dx+c \right ) }{2\,\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \sin \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cot \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96566, size = 684, normalized size = 10.69 \begin{align*} \left [\frac{\sqrt{-a} a \log \left (-\frac{8 \, a \cos \left (d x + c\right )^{3} + 4 \,{\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) - 4 \, a \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, d \sin \left (d x + c\right )}, -\frac{a^{\frac{3}{2}} \arctan \left (\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \, a \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{d \sin \left (d x + c\right )}\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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