Optimal. Leaf size=178 \[ \frac{171 a^2 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}+\frac{69 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]
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Rubi [A] time = 0.649182, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {2881, 2759, 2751, 2647, 2646, 3044, 2976, 2981, 2773, 206} \[ \frac{171 a^2 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}+\frac{69 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]
Antiderivative was successfully verified.
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Rule 2881
Rule 2759
Rule 2751
Rule 2647
Rule 2646
Rule 3044
Rule 2976
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \sin ^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} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}+\frac{2 \int \left (\frac{5 a}{2}-a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{7 a}+\frac{\int \csc (c+d x) \left (\frac{3 a}{2}-\frac{7}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{a}\\ &=\frac{7 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+\frac{4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}+\frac{19}{35} \int (a+a \sin (c+d x))^{3/2} \, dx+\frac{2 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{9 a^2}{4}-\frac{19}{4} a^2 \sin (c+d x)\right ) \, dx}{3 a}\\ &=\frac{19 a^2 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{69 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{35 d}+\frac{4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}+\frac{1}{105} (76 a) \int \sqrt{a+a \sin (c+d x)} \, dx+\frac{1}{2} (3 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{171 a^2 \cos (c+d x)}{35 d \sqrt{a+a \sin (c+d x)}}+\frac{69 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{35 d}+\frac{4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}-\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{171 a^2 \cos (c+d x)}{35 d \sqrt{a+a \sin (c+d x)}}+\frac{69 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{35 d}+\frac{4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}\\ \end{align*}
Mathematica [A] time = 1.30973, size = 283, normalized size = 1.59 \[ -\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-840 \sin \left (\frac{1}{2} (c+d x)\right )-574 \sin \left (\frac{3}{2} (c+d x)\right )-30 \sin \left (\frac{5}{2} (c+d x)\right )-21 \sin \left (\frac{7}{2} (c+d x)\right )-5 \sin \left (\frac{9}{2} (c+d x)\right )+840 \cos \left (\frac{1}{2} (c+d x)\right )-574 \cos \left (\frac{3}{2} (c+d x)\right )+30 \cos \left (\frac{5}{2} (c+d x)\right )-21 \cos \left (\frac{7}{2} (c+d x)\right )+5 \cos \left (\frac{9}{2} (c+d x)\right )+420 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-420 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{140 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc \left (\frac{1}{4} (c+d x)\right )-\sec \left (\frac{1}{4} (c+d x)\right )\right ) \left (\csc \left (\frac{1}{4} (c+d x)\right )+\sec \left (\frac{1}{4} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.948, size = 180, normalized size = 1. \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{35\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( \sin \left ( dx+c \right ) \left ( 140\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{7/2}+70\,{a}^{5/2} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}-56\,{a}^{3/2} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}+10\,\sqrt{a} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{7/2}-105\,{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( dx+c \right ) }}{\sqrt{a}}} \right ){a}^{4} \right ) -35\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{7/2} \right ){a}^{-{\frac{5}{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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.26272, size = 968, normalized size = 5.44 \begin{align*} \frac{105 \,{\left (a \cos \left (d x + c\right )^{2} -{\left (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 (10 \, a \cos \left (d x + c\right )^{5} - 16 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{3} - 120 \, a \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right ) -{\left (10 \, a \cos \left (d x + c\right )^{4} + 26 \, a \cos \left (d x + c\right )^{3} + 18 \, a \cos \left (d x + c\right )^{2} + 138 \, a \cos \left (d x + c\right ) + 171 \, a\right )} \sin \left (d x + c\right ) + 171 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{140 \,{\left (d \cos \left (d x + c\right )^{2} -{\left (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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