Optimal. Leaf size=152 \[ -\frac{a}{12 d (a \sec (c+d x)+a)^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}+\frac{7}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{\sqrt{a} d}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{8 \sqrt{2} \sqrt{a} d} \]
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Rubi [A] time = 0.135295, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3880, 103, 152, 156, 63, 207} \[ -\frac{a}{12 d (a \sec (c+d x)+a)^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}+\frac{7}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{\sqrt{a} d}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{8 \sqrt{2} \sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 3880
Rule 103
Rule 152
Rule 156
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx &=\frac{a^4 \operatorname{Subst}\left (\int \frac{1}{x (-a+a x)^2 (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac{a \operatorname{Subst}\left (\int \frac{2 a^2+\frac{5 a^2 x}{2}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-6 a^4-\frac{3 a^4 x}{4}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{6 a^2 d}\\ &=-\frac{a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac{7}{8 d \sqrt{a+a \sec (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{6 a^6-\frac{21 a^6 x}{8}}{x (-a+a x) \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{6 a^5 d}\\ &=-\frac{a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac{7}{8 d \sqrt{a+a \sec (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{d}-\frac{(9 a) \operatorname{Subst}\left (\int \frac{1}{(-a+a x) \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{16 d}\\ &=-\frac{a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac{7}{8 d \sqrt{a+a \sec (c+d x)}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{-2 a+x^2} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{8 d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{a d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{8 \sqrt{2} \sqrt{a} d}-\frac{a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac{7}{8 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.166507, size = 90, normalized size = 0.59 \[ \frac{a \left (-9 (\sec (c+d x)-1) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\frac{1}{2} (\sec (c+d x)+1)\right )+8 (\sec (c+d x)-1) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\sec (c+d x)+1\right )-6\right )}{12 d (\sec (c+d x)-1) (a (\sec (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.274, size = 504, normalized size = 3.3 \begin{align*} \text{result too large to display} \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 \frac{\cot \left (d x + c\right )^{3}}{\sqrt{a \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\cot ^{3}{\left (c + d x \right )}}{\sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 9.8231, size = 309, normalized size = 2.03 \begin{align*} -\frac{\sqrt{2}{\left (\frac{48 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{27 \, \arctan \left (\frac{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{3 \, \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + \frac{2 \,{\left ({\left (-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{3}{2}} a^{4} + 12 \, \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a^{5}\right )}}{a^{6} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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