Optimal. Leaf size=281 \[ -\frac {61 a \cot (c+d x)}{1024 d \sqrt {a \sin (c+d x)+a}}-\frac {61 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{1024 d}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a \sin (c+d x)+a}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a \sin (c+d x)+a}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a \sin (c+d x)+a}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a \sin (c+d x)+a}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.95, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ -\frac {61 a \cot (c+d x)}{1024 d \sqrt {a \sin (c+d x)+a}}-\frac {61 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{1024 d}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a \sin (c+d x)+a}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a \sin (c+d x)+a}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a \sin (c+d x)+a}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a \sin (c+d x)+a}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2772
Rule 2773
Rule 2881
Rule 2980
Rule 3044
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\int \csc ^8(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) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {5}{6} \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\frac {\int \csc ^7(c+d x) \left (\frac {a}{2}-\frac {17}{2} a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{7 a}\\ &=-\frac {5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {5}{8} \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {193}{168} \int \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {5 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {5}{16} \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {579}{560} \int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {5 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}-\frac {579}{640} \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {(5 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 {5 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {5 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}-\frac {193}{256} \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {5 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {5 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}-\frac {579 \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{1024}\\ &=-\frac {5 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}-\frac {579 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{2048}\\ &=-\frac {5 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {(579 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 )}{1024 d}\\ &=-\frac {61 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{1024 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}\\ \end {align*}
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Mathematica [A] time = 2.03, size = 191, normalized size = 0.68 \[ \frac {\sqrt {a (\sin (c+d x)+1)} \left (-102480 \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+102480 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+\csc ^7(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (49128 \sin (c+d x)-179636 \sin (3 (c+d x))-8540 \sin (5 (c+d x))-244533 \cos (2 (c+d x))-52094 \cos (4 (c+d x))+6405 \cos (6 (c+d x))-201298)\right )}{3440640 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 567, normalized size = 2.02 \[ \frac {6405 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{7} + \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} + 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 (6405 \, \cos \left (d x + c\right )^{7} + 2135 \, \cos \left (d x + c\right )^{6} - 22631 \, \cos \left (d x + c\right )^{5} - 37613 \, \cos \left (d x + c\right )^{4} + 1343 \, \cos \left (d x + c\right )^{3} + 27477 \, \cos \left (d x + c\right )^{2} - {\left (6405 \, \cos \left (d x + c\right )^{6} + 4270 \, \cos \left (d x + c\right )^{5} - 18361 \, \cos \left (d x + c\right )^{4} + 19252 \, \cos \left (d x + c\right )^{3} + 20595 \, \cos \left (d x + c\right )^{2} - 6882 \, \cos \left (d x + c\right ) - 7359\right )} \sin \left (d x + c\right ) - 477 \, \cos \left (d x + c\right ) - 7359\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{430080 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} + 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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.39, size = 216, normalized size = 0.77 \[ -\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (6405 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {13}{2}} a^{\frac {7}{2}}-42700 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {9}{2}}+120841 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {11}{2}}+6405 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{10} \left (\sin ^{7}\left (d x +c \right )\right )-156672 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {13}{2}}+51191 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {15}{2}}+42700 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {17}{2}}-6405 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {19}{2}}\right )}{107520 a^{\frac {19}{2}} \sin \left (d x +c \right )^{7} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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