Optimal. Leaf size=291 \[ -\frac {171 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{1024 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a \sin (c+d x)+a}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a \sin (c+d x)+a}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a \sin (c+d x)+a}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a \sin (c+d x)+a}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{28 d} \]
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Rubi [A] time = 1.06, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2881, 2762, 21, 2772, 2773, 206, 3044, 2975, 2980} \[ -\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a \sin (c+d x)+a}}-\frac {171 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{1024 d}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a \sin (c+d x)+a}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a \sin (c+d x)+a}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a \sin (c+d x)+a}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{28 d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 206
Rule 2762
Rule 2772
Rule 2773
Rule 2881
Rule 2975
Rule 2980
Rule 3044
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^8(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {a^2 \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) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {\int \csc ^7(c+d x) \left (\frac {3 a}{2}-\frac {19}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{7 a}-\frac {1}{3} a \int \frac {\csc ^3(c+d x) \left (-\frac {11 a}{2}-\frac {11}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {a^2 \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) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {\int \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {189 a^2}{4}-\frac {201}{4} a^2 \sin (c+d x)\right ) \, dx}{42 a}+\frac {1}{6} (11 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {1}{8} (11 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {1}{560} (1237 a) \int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {1}{16} (11 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {1}{640} (1237 a) \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {1}{768} (1237 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {\left (11 a^2\right ) \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 {11 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {(1237 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{1024}\\ &=-\frac {11 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {(1237 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{2048}\\ &=-\frac {11 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {\left (1237 a^2\right ) \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 {171 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{1024 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}\\ \end {align*}
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Mathematica [A] time = 4.94, size = 522, normalized size = 1.79 \[ \frac {a \csc ^{22}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sin (c+d x)+1)} \left (306488 \sin \left (\frac {1}{2} (c+d x)\right )-177170 \sin \left (\frac {3}{2} (c+d x)\right )-6566 \sin \left (\frac {5}{2} (c+d x)\right )-219540 \sin \left (\frac {7}{2} (c+d x)\right )-33292 \sin \left (\frac {9}{2} (c+d x)\right )-3990 \sin \left (\frac {11}{2} (c+d x)\right )-11970 \sin \left (\frac {13}{2} (c+d x)\right )-306488 \cos \left (\frac {1}{2} (c+d x)\right )-177170 \cos \left (\frac {3}{2} (c+d x)\right )+6566 \cos \left (\frac {5}{2} (c+d x)\right )-219540 \cos \left (\frac {7}{2} (c+d x)\right )+33292 \cos \left (\frac {9}{2} (c+d x)\right )-3990 \cos \left (\frac {11}{2} (c+d x)\right )+11970 \cos \left (\frac {13}{2} (c+d x)\right )-209475 \sin (c+d x) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+209475 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+125685 \sin (3 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-125685 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-41895 \sin (5 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+41895 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+5985 \sin (7 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-5985 \sin (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{35840 d \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 600, normalized size = 2.06 \[ \frac {5985 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{7} + a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{5} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\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 (5985 \, a \cos \left (d x + c\right )^{7} + 1995 \, a \cos \left (d x + c\right )^{6} - 6811 \, a \cos \left (d x + c\right )^{5} - 14633 \, a \cos \left (d x + c\right )^{4} - 5997 \, a \cos \left (d x + c\right )^{3} + 10097 \, a \cos \left (d x + c\right )^{2} + 1703 \, a \cos \left (d x + c\right ) - {\left (5985 \, a \cos \left (d x + c\right )^{6} + 3990 \, a \cos \left (d x + c\right )^{5} - 2821 \, a \cos \left (d x + c\right )^{4} + 11812 \, a \cos \left (d x + c\right )^{3} + 5815 \, a \cos \left (d x + c\right )^{2} - 4282 \, a \cos \left (d x + c\right ) - 2579 \, a\right )} \sin \left (d x + c\right ) - 2579 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{143360 \, {\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 = 1.66, size = 216, normalized size = 0.74 \[ \frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (5985 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {17}{2}}-39900 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {15}{2}}-1771 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {13}{2}}+95232 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {11}{2}}-98581 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {9}{2}}+39900 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {7}{2}}-5985 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {13}{2}} a^{\frac {5}{2}}-5985 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{9} \left (\sin ^{7}\left (d x +c \right )\right )\right )}{35840 a^{\frac {15}{2}} \sin \left (d x +c \right )^{7} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{8}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^8} \,d x \]
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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