Optimal. Leaf size=152 \[ -\frac {\cot ^9(c+d x)}{9 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{128 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d} \]
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Rubi [A] time = 0.24, antiderivative size = 152, normalized size of antiderivative = 1.00, 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 = {2839, 2607, 14, 2611, 3768, 3770} \[ -\frac {\cot ^9(c+d x)}{9 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{128 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rule 2611
Rule 2839
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^6(c+d x) \csc ^3(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc ^4(c+d x) \, dx}{a}\\ &=\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac {5 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx}{8 a}+\frac {\operatorname {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{16 a}+\frac {\operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {\cot ^7(c+d x)}{7 a d}-\frac {\cot ^9(c+d x)}{9 a d}+\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac {5 \int \csc ^3(c+d x) \, dx}{64 a}\\ &=-\frac {\cot ^7(c+d x)}{7 a d}-\frac {\cot ^9(c+d x)}{9 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}+\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac {5 \int \csc (c+d x) \, dx}{128 a}\\ &=-\frac {5 \tanh ^{-1}(\cos (c+d x))}{128 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {\cot ^9(c+d x)}{9 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}+\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}\\ \end {align*}
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Mathematica [B] time = 1.37, size = 313, normalized size = 2.06 \[ -\frac {\csc ^9(c+d x) \left (-36540 \sin (2 (c+d x))-20916 \sin (4 (c+d x))-16044 \sin (6 (c+d x))-630 \sin (8 (c+d x))+129024 \cos (c+d x)+75264 \cos (3 (c+d x))+23040 \cos (5 (c+d x))+2304 \cos (7 (c+d x))-256 \cos (9 (c+d x))-39690 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+26460 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-11340 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2835 \sin (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-315 \sin (9 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+39690 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-26460 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+11340 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2835 \sin (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+315 \sin (9 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2064384 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 249, normalized size = 1.64 \[ \frac {512 \, \cos \left (d x + c\right )^{9} - 2304 \, \cos \left (d x + c\right )^{7} - 315 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 315 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 42 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 273, normalized size = 1.80 \[ \frac {\frac {5040 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {28 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 63 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 108 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 336 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1008 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1512 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {14258 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1512 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 108 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 28}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{129024 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.60, size = 322, normalized size = 2.12 \[ \frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4608 a d}-\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2048 a d}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3584 a d}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 a d}-\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{256 a d}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{192 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256 a d}-\frac {1}{384 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {3}{256 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d}+\frac {3}{3584 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{4608 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {1}{2048 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {1}{256 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{192 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 355, normalized size = 2.34 \[ -\frac {\frac {\frac {1512 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1008 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {672 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {504 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {108 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {28 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a} - \frac {5040 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {108 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {336 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {504 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {672 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1008 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {1512 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 28\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{9}}{a \sin \left (d x + c\right )^{9}}}{129024 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.55, size = 435, normalized size = 2.86 \[ \frac {28\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-28\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-63\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-108\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-504\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-1512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+504\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+108\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5040\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{129024\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
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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