Optimal. Leaf size=137 \[ \frac{3 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{8 d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.479941, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2874, 2975, 2980, 2772, 2773, 206} \[ \frac{3 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{8 d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2874
Rule 2975
Rule 2980
Rule 2772
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\frac{\int \csc ^4(c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{3/2} \, dx}{a^2}\\ &=-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+\frac{\int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{a^2}{2}-\frac{3}{2} a^2 \sin (c+d x)\right ) \, dx}{3 a^2}\\ &=-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{3}{8} \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{3 a \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{3}{16} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{3 a \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+\frac{(3 a) \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 )}{8 d}\\ &=\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}+\frac{3 a \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}\\ \end{align*}
Mathematica [B] time = 1.35948, size = 285, normalized size = 2.08 \[ -\frac{\csc ^{10}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (12 \sin \left (\frac{1}{2} (c+d x)\right )+58 \sin \left (\frac{3}{2} (c+d x)\right )-18 \sin \left (\frac{5}{2} (c+d x)\right )-12 \cos \left (\frac{1}{2} (c+d x)\right )+58 \cos \left (\frac{3}{2} (c+d x)\right )+18 \cos \left (\frac{5}{2} (c+d x)\right )-27 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+27 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+9 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-9 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{24 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.122, size = 144, normalized size = 1.1 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{24\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 9\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{7/2}+8\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{5/2}-9\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{3/2}-9\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3} \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 \sqrt{a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \csc \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73819, size = 961, normalized size = 7.01 \begin{align*} \frac{9 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\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 (9 \, \cos \left (d x + c\right )^{3} + 19 \, \cos \left (d x + c\right )^{2} -{\left (9 \, \cos \left (d x + c\right )^{2} - 10 \, \cos \left (d x + c\right ) - 11\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 11\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{96 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} -{\left (d \cos \left (d x + c\right )^{3} + 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(-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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