Optimal. Leaf size=245 \[ -\frac {55 a \cot (c+d x)}{512 d \sqrt {a \sin (c+d x)+a}}-\frac {55 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{512 d}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a \sin (c+d x)+a}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a \sin (c+d x)+a}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a \sin (c+d x)+a}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.81, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ -\frac {55 a \cot (c+d x)}{512 d \sqrt {a \sin (c+d x)+a}}-\frac {55 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{512 d}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a \sin (c+d x)+a}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a \sin (c+d x)+a}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a \sin (c+d x)+a}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a \sin (c+d x)+a}} \]
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 ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\int \csc ^7(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 (c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d}+\frac {3}{4} \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\frac {\int \csc ^6(c+d x) \left (\frac {a}{2}-\frac {15}{2} a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{6 a}\\ &=-\frac {3 a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d}+\frac {3}{8} \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {47}{40} \int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {3 a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d}-\frac {329}{320} \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {(3 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 )}{4 d}\\ &=-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {3 a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a+a \sin (c+d x)}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d}-\frac {329}{384} \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {3 a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a+a \sin (c+d x)}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a+a \sin (c+d x)}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d}-\frac {329}{512} \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {55 a \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a+a \sin (c+d x)}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a+a \sin (c+d x)}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d}-\frac {329 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{1024}\\ &=-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {55 a \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a+a \sin (c+d x)}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a+a \sin (c+d x)}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d}+\frac {(329 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 )}{512 d}\\ &=-\frac {55 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{512 d}-\frac {55 a \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a+a \sin (c+d x)}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a+a \sin (c+d x)}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d}\\ \end {align*}
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Mathematica [A] time = 7.69, size = 485, normalized size = 1.98 \[ \frac {\csc ^{19}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sin (c+d x)+1)} \left (24540 \sin \left (\frac {1}{2} (c+d x)\right )-25684 \sin \left (\frac {3}{2} (c+d x)\right )+14490 \sin \left (\frac {5}{2} (c+d x)\right )-15006 \sin \left (\frac {7}{2} (c+d x)\right )+550 \sin \left (\frac {9}{2} (c+d x)\right )-1650 \sin \left (\frac {11}{2} (c+d x)\right )-24540 \cos \left (\frac {1}{2} (c+d x)\right )-25684 \cos \left (\frac {3}{2} (c+d x)\right )-14490 \cos \left (\frac {5}{2} (c+d x)\right )-15006 \cos \left (\frac {7}{2} (c+d x)\right )-550 \cos \left (\frac {9}{2} (c+d x)\right )-1650 \cos \left (\frac {11}{2} (c+d x)\right )+12375 \cos (2 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-4950 \cos (4 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+825 \cos (6 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-8250 \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-12375 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+4950 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-825 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+8250 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{7680 d \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 525, normalized size = 2.14 \[ \frac {825 \, {\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} + {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \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 (825 \, \cos \left (d x + c\right )^{6} + 550 \, \cos \left (d x + c\right )^{5} + 707 \, \cos \left (d x + c\right )^{4} + 1156 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + {\left (825 \, \cos \left (d x + c\right )^{5} + 275 \, \cos \left (d x + c\right )^{4} + 982 \, \cos \left (d x + c\right )^{3} - 174 \, \cos \left (d x + c\right )^{2} - 399 \, \cos \left (d x + c\right ) + 27\right )} \sin \left (d x + c\right ) - 426 \, \cos \left (d x + c\right ) - 27\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{30720 \, {\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 ) + {\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} - 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 = 1.32, size = 198, normalized size = 0.81 \[ -\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-825 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {15}{2}}+4675 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {13}{2}}+1398 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {11}{2}}-7818 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {9}{2}}+4675 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {7}{2}}-825 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {5}{2}}+825 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{8} \left (\sin ^{6}\left (d x +c \right )\right )\right )}{7680 a^{\frac {15}{2}} \sin \left (d x +c \right )^{6} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{7}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^7} \,d x \]
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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