Optimal. Leaf size=194 \[ -\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{256 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d} \]
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Rubi [A] time = 0.25, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2839, 2607, 270, 2611, 3768, 3770} \[ -\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{256 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2607
Rule 2611
Rule 2839
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^6(c+d x) \csc ^5(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc ^6(c+d x) \, dx}{a}\\ &=\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {\int \cot ^4(c+d x) \csc ^5(c+d x) \, dx}{2 a}+\frac {\operatorname {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{16 a}+\frac {\operatorname {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {\int \csc ^5(c+d x) \, dx}{32 a}\\ &=-\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {3 \int \csc ^3(c+d x) \, dx}{128 a}\\ &=-\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {3 \int \csc (c+d x) \, dx}{256 a}\\ &=-\frac {3 \tanh ^{-1}(\cos (c+d x))}{256 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}\\ \end {align*}
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Mathematica [A] time = 2.94, size = 187, normalized size = 0.96 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-2661120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (-3219678 \sin (c+d x)-2608452 \sin (3 (c+d x))-2181564 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x))+9973760 \cos (2 (c+d x))+3543040 \cos (4 (c+d x))+343040 \cos (6 (c+d x))-61440 \cos (8 (c+d x))+5120 \cos (10 (c+d x))+6840320)\right )}{227082240 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 302, normalized size = 1.56 \[ \frac {20480 \, \cos \left (d x + c\right )^{11} - 112640 \, \cos \left (d x + c\right )^{9} + 253440 \, \cos \left (d x + c\right )^{7} - 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 1386 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1774080 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 360, normalized size = 1.86 \[ \frac {\frac {166320 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {630 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1386 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 770 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3465 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 4950 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6930 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27720 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 23100 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 13860 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 69300 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{11}} - \frac {502266 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 69300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 23100 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 27720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6930 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4950 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1386 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 630}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{14192640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.63, size = 436, normalized size = 2.25 \[ \frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{22528 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{10240 a d}-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{18432 a d}+\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4096 a d}-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14336 a d}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2048 a d}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2048 a d}-\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{512 a d}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 a d}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 a d}-\frac {1}{2048 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {5}{1024 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 a d}-\frac {1}{2048 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {5}{14336 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{1024 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{18432 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-\frac {1}{4096 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {1}{512 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{22528 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}-\frac {5}{3072 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{10240 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 475, normalized size = 2.45 \[ -\frac {\frac {\frac {69300 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {13860 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {23100 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {27720 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6930 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6930 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {4950 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3465 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {1386 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {630 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a} - \frac {166320 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {1386 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {770 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3465 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {4950 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6930 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6930 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {27720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {23100 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {13860 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {69300 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - 630\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{11}}{a \sin \left (d x + c\right )^{11}}}{14192640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.27, size = 579, normalized size = 2.98 \[ \frac {630\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}-630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}-1386\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}+1386\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+3465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-4950\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}+6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-27720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+23100\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-13860\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-69300\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+69300\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+13860\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-23100\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+27720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4950\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+166320\,\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 )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{14192640\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]
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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