Optimal. Leaf size=209 \[ -\frac{31 a \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{31 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.690556, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2881, 2772, 2773, 206, 3044, 2980} \[ -\frac{31 a \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{31 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2881
Rule 2772
Rule 2773
Rule 206
Rule 3044
Rule 2980
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\int \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}+\frac{1}{2} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\frac{\int \csc ^5(c+d x) \left (\frac{a}{2}-\frac{13}{2} a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx}{5 a}\\ &=-\frac{a \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{97}{80} \int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{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{\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)}{d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{97}{96} \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{\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)}{d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{97}{128} \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{31 a \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{97}{256} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{31 a \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}+\frac{(97 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 )}{128 d}\\ &=-\frac{31 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 d}-\frac{31 a \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}\\ \end{align*}
Mathematica [A] time = 4.37091, size = 403, normalized size = 1.93 \[ -\frac{\csc ^{16}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-10180 \sin \left (\frac{1}{2} (c+d x)\right )-2240 \sin \left (\frac{3}{2} (c+d x)\right )+1392 \sin \left (\frac{5}{2} (c+d x)\right )+4810 \sin \left (\frac{7}{2} (c+d x)\right )-930 \sin \left (\frac{9}{2} (c+d x)\right )+10180 \cos \left (\frac{1}{2} (c+d x)\right )-2240 \cos \left (\frac{3}{2} (c+d x)\right )-1392 \cos \left (\frac{5}{2} (c+d x)\right )+4810 \cos \left (\frac{7}{2} (c+d x)\right )+930 \cos \left (\frac{9}{2} (c+d x)\right )+4650 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-4650 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-2325 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+2325 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+465 \sin (5 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-465 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{1920 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.2, size = 180, normalized size = 0.9 \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{1920\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 465\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{11/2}-2170\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{9/2}+896\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{7/2}+890\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{5/2}-465\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}{a}^{3/2}-465\,{\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 ) ^{5} \right ){a}^{-{\frac{11}{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 )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.30777, size = 1261, normalized size = 6.03 \begin{align*} \frac{465 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} -{\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} + \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 (465 \, \cos \left (d x + c\right )^{5} + 1435 \, \cos \left (d x + c\right )^{4} - 154 \, \cos \left (d x + c\right )^{3} - 1662 \, \cos \left (d x + c\right )^{2} -{\left (465 \, \cos \left (d x + c\right )^{4} - 970 \, \cos \left (d x + c\right )^{3} - 1124 \, \cos \left (d x + c\right )^{2} + 538 \, \cos \left (d x + c\right ) + 611\right )} \sin \left (d x + c\right ) + 73 \, \cos \left (d x + c\right ) + 611\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{7680 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} -{\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 ) + 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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