Optimal. Leaf size=173 \[ \frac{61 a \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{67 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.538, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2881, 2773, 206, 3044, 2980, 2772} \[ \frac{61 a \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{67 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2881
Rule 2773
Rule 206
Rule 3044
Rule 2980
Rule 2772
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\int \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}+\frac{\int \csc ^4(c+d x) \left (\frac{a}{2}-\frac{11}{2} a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx}{4 a}-\frac{(2 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 )}{d}\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 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 d}-\frac{61}{48} \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 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 d}-\frac{61}{64} \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{61 a \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 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 d}-\frac{61}{128} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{61 a \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 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 d}+\frac{(61 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 )}{64 d}\\ &=-\frac{67 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{64 d}+\frac{61 a \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 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 d}\\ \end{align*}
Mathematica [B] time = 2.66687, size = 367, normalized size = 2.12 \[ -\frac{\csc ^{13}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-442 \sin \left (\frac{1}{2} (c+d x)\right )-162 \sin \left (\frac{3}{2} (c+d x)\right )-122 \sin \left (\frac{5}{2} (c+d x)\right )+366 \sin \left (\frac{7}{2} (c+d x)\right )+442 \cos \left (\frac{1}{2} (c+d x)\right )-162 \cos \left (\frac{3}{2} (c+d x)\right )+122 \cos \left (\frac{5}{2} (c+d x)\right )+366 \cos \left (\frac{7}{2} (c+d x)\right )-804 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+201 \cos (4 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+603 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+804 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-201 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-603 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{192 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.003, size = 162, normalized size = 0.9 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{192\, \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 ( 201\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}{a}^{4}+183\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}\sqrt{a}-671\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{3/2}+737\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{5/2}-201\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{7/2} \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 )^{4} \csc \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.26237, size = 1141, normalized size = 6.6 \begin{align*} \frac{201 \,{\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 (183 \, \cos \left (d x + c\right )^{4} + 122 \, \cos \left (d x + c\right )^{3} - 188 \, \cos \left (d x + c\right )^{2} +{\left (183 \, \cos \left (d x + c\right )^{3} + 61 \, \cos \left (d x + c\right )^{2} - 127 \, \cos \left (d x + c\right ) - 53\right )} \sin \left (d x + c\right ) - 74 \, \cos \left (d x + c\right ) + 53\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{768 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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(-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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