Optimal. Leaf size=215 \[ \frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{165 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 d}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{40 d} \]
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Rubi [A] time = 0.807951, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2881, 2762, 21, 2773, 206, 3044, 2975, 2980, 2772} \[ \frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{165 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 d}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{40 d} \]
Antiderivative was successfully verified.
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Rule 2881
Rule 2762
Rule 21
Rule 2773
Rule 206
Rule 3044
Rule 2975
Rule 2980
Rule 2772
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{\int \csc ^5(c+d x) \left (\frac{3 a}{2}-\frac{15}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{5 a}-a \int \frac{\csc (c+d x) \left (-\frac{3 a}{2}-\frac{3}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{\int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \left (-\frac{93 a^2}{4}-\frac{105}{4} a^2 \sin (c+d x)\right ) \, dx}{20 a}+\frac{1}{2} (3 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{1}{32} (73 a) \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{a^2 \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{1}{128} (219 a) \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{1}{256} (219 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{\left (219 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 d}\\ &=-\frac{165 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 d}+\frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}\\ \end{align*}
Mathematica [A] time = 1.70755, size = 404, normalized size = 1.88 \[ -\frac{a \csc ^{16}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-1380 \sin \left (\frac{1}{2} (c+d x)\right )+320 \sin \left (\frac{3}{2} (c+d x)\right )-1296 \sin \left (\frac{5}{2} (c+d x)\right )+2010 \sin \left (\frac{7}{2} (c+d x)\right )+910 \sin \left (\frac{9}{2} (c+d x)\right )+1380 \cos \left (\frac{1}{2} (c+d x)\right )+320 \cos \left (\frac{3}{2} (c+d x)\right )+1296 \cos \left (\frac{5}{2} (c+d x)\right )+2010 \cos \left (\frac{7}{2} (c+d x)\right )-910 \cos \left (\frac{9}{2} (c+d x)\right )+8250 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-8250 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-4125 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+4125 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+825 \sin (5 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-825 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{640 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.161, size = 180, normalized size = 0.8 \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{640\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( -825\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{5}+455\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}\sqrt{a}-2550\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{3/2}+4992\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{5/2}-3850\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{7/2}+825\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{9/2} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \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 \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.27496, size = 1307, normalized size = 6.08 \begin{align*} \frac{825 \,{\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} -{\left (a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} - 9 \, a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \,{\left (455 \, a \cos \left (d x + c\right )^{5} - 275 \, a \cos \left (d x + c\right )^{4} - 982 \, a \cos \left (d x + c\right )^{3} + 174 \, a \cos \left (d x + c\right )^{2} + 399 \, a \cos \left (d x + c\right ) -{\left (455 \, a \cos \left (d x + c\right )^{4} + 730 \, a \cos \left (d x + c\right )^{3} - 252 \, a \cos \left (d x + c\right )^{2} - 426 \, a \cos \left (d x + c\right ) - 27 \, a\right )} \sin \left (d x + c\right ) - 27 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{2560 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} -{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d\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(-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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