Optimal. Leaf size=186 \[ \frac{73 a^2 \cos (c+d x)}{20 d \sqrt{a \sin (c+d x)+a}}+\frac{9 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{4 d}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
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Rubi [A] time = 0.571473, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2881, 2751, 2647, 2646, 3044, 2975, 2981, 2773, 206} \[ \frac{73 a^2 \cos (c+d x)}{20 d \sqrt{a \sin (c+d x)+a}}+\frac{9 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{4 d}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 2881
Rule 2751
Rule 2647
Rule 2646
Rule 3044
Rule 2975
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{3}{5} \int (a+a \sin (c+d x))^{3/2} \, dx+\frac{\int \csc ^2(c+d x) \left (\frac{3 a}{2}-\frac{9}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{2 a}\\ &=-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{3 a \cot (c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \left (-\frac{9 a^2}{4}-\frac{21}{4} a^2 \sin (c+d x)\right ) \, dx}{2 a}+\frac{1}{5} (4 a) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{73 a^2 \cos (c+d x)}{20 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{3 a \cot (c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}-\frac{1}{8} (9 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{73 a^2 \cos (c+d x)}{20 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{3 a \cot (c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{\left (9 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 )}{4 d}\\ &=\frac{9 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 d}+\frac{73 a^2 \cos (c+d x)}{20 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{3 a \cot (c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}\\ \end{align*}
Mathematica [A] time = 1.05447, size = 322, normalized size = 1.73 \[ -\frac{a \csc ^7\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (118 \sin \left (\frac{1}{2} (c+d x)\right )+130 \sin \left (\frac{3}{2} (c+d x)\right )-36 \sin \left (\frac{5}{2} (c+d x)\right )-10 \sin \left (\frac{7}{2} (c+d x)\right )-2 \sin \left (\frac{9}{2} (c+d x)\right )-118 \cos \left (\frac{1}{2} (c+d x)\right )+130 \cos \left (\frac{3}{2} (c+d x)\right )+36 \cos \left (\frac{5}{2} (c+d x)\right )-10 \cos \left (\frac{7}{2} (c+d x)\right )+2 \cos \left (\frac{9}{2} (c+d x)\right )+45 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-45 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-45 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+45 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{20 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.034, size = 178, normalized size = 1. \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{20\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 8\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\sqrt{a}-40\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{3/2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}-45\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{3}-35\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{3/2}+45\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{5/2} \right ){a}^{-{\frac{3}{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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.21174, size = 1061, normalized size = 5.7 \begin{align*} \frac{45 \,{\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right )^{2} - 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 (8 \, a \cos \left (d x + c\right )^{5} - 16 \, a \cos \left (d x + c\right )^{4} + 16 \, a \cos \left (d x + c\right )^{3} + 99 \, a \cos \left (d x + c\right )^{2} - 14 \, a \cos \left (d x + c\right ) -{\left (8 \, a \cos \left (d x + c\right )^{4} + 24 \, a \cos \left (d x + c\right )^{3} + 40 \, a \cos \left (d x + c\right )^{2} - 59 \, a \cos \left (d x + c\right ) - 73 \, a\right )} \sin \left (d x + c\right ) - 73 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{80 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right )^{2} - 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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