Optimal. Leaf size=191 \[ \frac{23 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{8 a^{3/2} d}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{3/2} d}-\frac{9 \cot (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}+\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.714488, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2874, 2984, 2985, 2649, 206, 2773} \[ \frac{23 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{8 a^{3/2} d}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{3/2} d}-\frac{9 \cot (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}+\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2874
Rule 2984
Rule 2985
Rule 2649
Rule 206
Rule 2773
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{\int \frac{\csc ^4(c+d x) (a-a \sin (c+d x))}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^3(c+d x) \left (-\frac{7 a^2}{2}+\frac{5}{2} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{3 a^3}\\ &=\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^2(c+d x) \left (\frac{27 a^3}{4}-\frac{21}{4} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{6 a^4}\\ &=-\frac{9 \cot (c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}+\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc (c+d x) \left (-\frac{69 a^4}{8}+\frac{27}{8} a^4 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{6 a^5}\\ &=-\frac{9 \cot (c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}+\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+a \sin (c+d x)}}-\frac{23 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{16 a^2}+\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a}\\ &=-\frac{9 \cot (c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}+\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+a \sin (c+d x)}}+\frac{23 \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 a d}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{a d}\\ &=\frac{23 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 a^{3/2} d}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{a^{3/2} d}-\frac{9 \cot (c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}+\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 2.23112, size = 332, normalized size = 1.74 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (-\frac{8 \csc ^9\left (\frac{1}{2} (c+d x)\right ) \left (-228 \sin \left (\frac{1}{2} (c+d x)\right )-110 \sin \left (\frac{3}{2} (c+d x)\right )+54 \sin \left (\frac{5}{2} (c+d x)\right )+228 \cos \left (\frac{1}{2} (c+d x)\right )-110 \cos \left (\frac{3}{2} (c+d x)\right )-54 \cos \left (\frac{5}{2} (c+d x)\right )-207 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+207 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+69 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-69 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{\left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^3}+(768+768 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{192 d (a (\sin (c+d x)+1))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.032, size = 182, normalized size = 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 ( -69\,{a}^{6}{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}+27\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{7/2}-40\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{9/2}+48\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{6} \left ( \sin \left ( dx+c \right ) \right ) ^{3}+21\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{11/2} \right ){a}^{-{\frac{15}{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(-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 [B] time = 1.93417, size = 1501, normalized size = 7.86 \begin{align*} \text{result too large to display} \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 [B] time = 2.58764, size = 938, normalized size = 4.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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