Optimal. Leaf size=193 \[ \frac{43 a^2}{96 d (a \sec (c+d x)+a)^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}-\frac{21 a}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}-\frac{107 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{64 \sqrt{2} d} \]
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Rubi [A] time = 0.157803, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3880, 103, 151, 152, 156, 63, 207} \[ \frac{43 a^2}{96 d (a \sec (c+d x)+a)^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}-\frac{21 a}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}-\frac{107 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{64 \sqrt{2} d} \]
Antiderivative was successfully verified.
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Rule 3880
Rule 103
Rule 151
Rule 152
Rule 156
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \cot ^5(c+d x) \sqrt{a+a \sec (c+d x)} \, dx &=\frac{a^6 \operatorname{Subst}\left (\int \frac{1}{x (-a+a x)^3 (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac{a^3 \operatorname{Subst}\left (\int \frac{4 a^2+\frac{7 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{4 d}\\ &=-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{8 a^4+\frac{75 a^4 x}{4}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{8 d}\\ &=\frac{43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{-24 a^6-\frac{129 a^6 x}{8}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{24 a^3 d}\\ &=\frac{43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac{21 a}{64 d \sqrt{a+a \sec (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{24 a^8-\frac{63 a^8 x}{16}}{x (-a+a x) \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{24 a^6 d}\\ &=\frac{43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac{21 a}{64 d \sqrt{a+a \sec (c+d x)}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{d}+\frac{\left (107 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{(-a+a x) \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{128 d}\\ &=\frac{43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac{21 a}{64 d \sqrt{a+a \sec (c+d x)}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{d}+\frac{(107 a) \operatorname{Subst}\left (\int \frac{1}{-2 a+x^2} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{64 d}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{d}-\frac{107 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{64 \sqrt{2} d}+\frac{43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac{21 a}{64 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.301505, size = 102, normalized size = 0.53 \[ \frac{\cot ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (107 (\sec (c+d x)-1)^2 \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\frac{1}{2} (\sec (c+d x)+1)\right )-2 \left (32 (\sec (c+d x)-1)^2 \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\sec (c+d x)+1\right )-45 \sec (c+d x)+57\right )\right )}{96 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.335, size = 407, normalized size = 2.1 \begin{align*} -{\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}{384\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 384\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +321\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}} \right ) -768\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +410\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-642\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}} \right ) +384\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) -142\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-298\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+321\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}} \right ) +126\,\cos \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 2.63913, size = 1438, normalized size = 7.45 \begin{align*} \left [\frac{384 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sqrt{a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 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 )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 321 \,{\left (\sqrt{2} \cos \left (d x + c\right )^{4} - 2 \, \sqrt{2} \cos \left (d x + c\right )^{2} + \sqrt{2}\right )} \sqrt{a} \log \left (-\frac{2 \, \sqrt{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 ) - 3 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) - 1}\right ) - 4 \,{\left (205 \, \cos \left (d x + c\right )^{4} - 71 \, \cos \left (d x + c\right )^{3} - 149 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{768 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}}, \frac{321 \,{\left (\sqrt{2} \cos \left (d x + c\right )^{4} - 2 \, \sqrt{2} \cos \left (d x + c\right )^{2} + \sqrt{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{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 )}{a \cos \left (d x + c\right ) + a}\right ) - 384 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sqrt{-a} \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 )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \,{\left (205 \, \cos \left (d x + c\right )^{4} - 71 \, \cos \left (d x + c\right )^{3} - 149 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{384 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\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 [A] time = 4.74632, size = 271, normalized size = 1.4 \begin{align*} -\frac{\sqrt{2}{\left (\frac{384 \, \sqrt{2} a \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}} - \frac{321 \, a \arctan \left (\frac{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{8 \,{\left ({\left (-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{3}{2}} a^{2} + 15 \, \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a^{3}\right )}}{a^{3}} + \frac{3 \,{\left (21 \,{\left (-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{3}{2}} a - 19 \, \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a^{2}\right )}}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}\right )} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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