Optimal. Leaf size=182 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 a^{3/2} d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 a^2 d}-\frac{3 \cot (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{32 a d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.727433, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2880, 2772, 2773, 206, 3044, 2980} \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 a^{3/2} d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 a^2 d}-\frac{3 \cot (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{32 a d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2880
Rule 2772
Rule 2773
Rule 206
Rule 3044
Rule 2980
Rubi steps
\begin{align*} \int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{\int \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)} \left (1+\sin ^2(c+d x)\right ) \, dx}{a^2}-\frac{2 \int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac{2 \cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 a^2 d}+\frac{\int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{a}{2}+\frac{13}{2} a \sin (c+d x)\right ) \, dx}{4 a^3}-\frac{5 \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{3 a^2}\\ &=\frac{5 \cot (c+d x) \csc (c+d x)}{6 a d \sqrt{a+a \sin (c+d x)}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 a^2 d}-\frac{5 \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{4 a^2}+\frac{83 \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{48 a^2}\\ &=\frac{5 \cot (c+d x)}{4 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{32 a d \sqrt{a+a \sin (c+d x)}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 a^2 d}-\frac{5 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{8 a^2}+\frac{83 \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{64 a^2}\\ &=-\frac{3 \cot (c+d x)}{64 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{32 a d \sqrt{a+a \sin (c+d x)}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 a^2 d}+\frac{83 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{128 a^2}+\frac{5 \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 a d}\\ &=\frac{5 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 a^{3/2} d}-\frac{3 \cot (c+d x)}{64 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{32 a d \sqrt{a+a \sin (c+d x)}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 a^2 d}-\frac{83 \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 )}{64 a d}\\ &=-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{64 a^{3/2} d}-\frac{3 \cot (c+d x)}{64 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{32 a d \sqrt{a+a \sin (c+d x)}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 a^2 d}\\ \end{align*}
Mathematica [B] time = 0.964182, size = 376, normalized size = 2.07 \[ -\frac{\csc ^{12}\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (-446 \sin \left (\frac{1}{2} (c+d x)\right )-182 \sin \left (\frac{3}{2} (c+d x)\right )+2 \sin \left (\frac{5}{2} (c+d x)\right )-6 \sin \left (\frac{7}{2} (c+d x)\right )+446 \cos \left (\frac{1}{2} (c+d x)\right )-182 \cos \left (\frac{3}{2} (c+d x)\right )-2 \cos \left (\frac{5}{2} (c+d x)\right )-6 \cos \left (\frac{7}{2} (c+d x)\right )-12 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+3 \cos (4 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+9 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+12 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-3 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-9 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{64 d (a (\sin (c+d x)+1))^{3/2} \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.987, size = 162, normalized size = 0.9 \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{64\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 3\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{5/2}-11\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{7/2}-3\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{6} \left ( \sin \left ( dx+c \right ) \right ) ^{4}-11\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{9/2}+3\,\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 = 1.18098, size = 1172, normalized size = 6.44 \begin{align*} \frac{3 \,{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + \cos \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 (3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} + 20 \, \cos \left (d x + c\right )^{2} +{\left (3 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 21 \, \cos \left (d x + c\right ) + 39\right )} \sin \left (d x + c\right ) - 18 \, \cos \left (d x + c\right ) - 39\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{256 \,{\left (a^{2} d \cos \left (d x + c\right )^{5} + a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right ) + a^{2} d +{\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sin \left (d x + c\right )\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 [B] time = 2.26825, size = 995, normalized size = 5.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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