Optimal. Leaf size=183 \[ \frac {\cos ^{11}(c+d x)}{11 a d}-\frac {2 \cos ^9(c+d x)}{9 a d}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}-\frac {3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{160 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{128 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{256 a d}+\frac {3 x}{256 a} \]
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Rubi [A] time = 0.24, antiderivative size = 183, 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, 2568, 2635, 8, 2565, 270} \[ \frac {\cos ^{11}(c+d x)}{11 a d}-\frac {2 \cos ^9(c+d x)}{9 a d}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}-\frac {3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{160 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{128 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{256 a d}+\frac {3 x}{256 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2839
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^6(c+d x) \sin ^4(c+d x) \, dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^5(c+d x) \, dx}{a}\\ &=-\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac {3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{10 a}+\frac {\operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac {3 \int \cos ^6(c+d x) \, dx}{80 a}+\frac {\operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \cos ^9(c+d x)}{9 a d}+\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac {\int \cos ^4(c+d x) \, dx}{32 a}\\ &=\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \cos ^9(c+d x)}{9 a d}+\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac {3 \int \cos ^2(c+d x) \, dx}{128 a}\\ &=\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \cos ^9(c+d x)}{9 a d}+\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac {3 \int 1 \, dx}{256 a}\\ &=\frac {3 x}{256 a}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \cos ^9(c+d x)}{9 a d}+\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}\\ \end {align*}
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Mathematica [B] time = 12.16, size = 573, normalized size = 3.13 \[ \frac {\frac {97020 \sin ^2\left (\frac {1}{2} (c+d x)\right )}{d (a \sin (c+d x)+a)}+\frac {103950 \sin (c) \sin (d x)}{a d}-\frac {66990 \sin (3 c) \sin (3 d x)}{a d}+\frac {24948 \sin (5 c) \sin (5 d x)}{a d}-\frac {1980 \sin (7 c) \sin (7 d x)}{a d}-\frac {76230 \sin (2 (c+d x))}{a d}+\frac {27720 \sin (4 (c+d x))}{a d}-\frac {11550 \sin (6 (c+d x))}{a d}+\frac {3465 \sin (8 (c+d x))}{a d}+\frac {1386 \sin (10 (c+d x))}{a d}+\frac {48510 \sin (c+d x)}{a d (\sin (c+d x)+1)}-\frac {103950 \cos (c) \cos (d x)}{a d}+\frac {66990 \cos (3 c) \cos (3 d x)}{a d}-\frac {24948 \cos (5 c) \cos (5 d x)}{a d}+\frac {1980 \cos (7 c) \cos (7 d x)}{a d}+\frac {173250 \cos (c+d x)}{a d}-\frac {43890 \cos (3 (c+d x))}{a d}+\frac {18018 \cos (5 (c+d x))}{a d}-\frac {6930 \cos (7 (c+d x))}{a d}+\frac {770 \cos (9 (c+d x))}{a d}+\frac {630 \cos (11 (c+d x))}{a d}-\frac {20790 \sin \left (\frac {1}{2} (c+d x)\right )}{a d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {90090 \sin (2 c) \cos (2 d x)}{a d}-\frac {55440 \sin (4 c) \cos (4 d x)}{a d}+\frac {4620 \sin (6 c) \cos (6 d x)}{a d}+\frac {90090 \cos (2 c) \sin (2 d x)}{a d}-\frac {55440 \cos (4 c) \sin (4 d x)}{a d}+\frac {4620 \cos (6 c) \sin (6 d x)}{a d}-\frac {76230 \sin \left (\frac {d x}{2}\right )}{a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {97020 c}{a d}+\frac {83160 x}{a}}{7096320} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 100, normalized size = 0.55 \[ \frac {80640 \, \cos \left (d x + c\right )^{11} - 197120 \, \cos \left (d x + c\right )^{9} + 126720 \, \cos \left (d x + c\right )^{7} + 10395 \, d x + 693 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 176 \, \cos \left (d x + c\right )^{7} + 8 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 270, normalized size = 1.48 \[ \frac {\frac {10395 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{21} + 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} - 2302839 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 4730880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 4790016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 11827200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 5828130 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 26019840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 21288960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 5828130 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 15206400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 4790016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3041280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2302839 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 563200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10240\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{11} a}}{887040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 653, normalized size = 3.57 \[ \frac {16}{693 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {80 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {3323 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {48 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {54 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {240 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {841 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {48 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {176 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {841 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {80 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {54 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {32 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {3323 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {3 \left (\tan ^{21}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 624, normalized size = 3.41 \[ -\frac {\frac {\frac {10395 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {112640 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {110880 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {563200 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2302839 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3041280 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {4790016 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {15206400 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {5828130 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {21288960 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {26019840 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {5828130 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} + \frac {11827200 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {4790016 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {4730880 \, \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {2302839 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {110880 \, \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} - \frac {10395 \, \sin \left (d x + c\right )^{21}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{21}} - 10240}{a + \frac {11 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {55 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {165 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {330 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {462 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {462 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {330 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {165 \, a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {55 \, a \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {11 \, a \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}} + \frac {a \sin \left (d x + c\right )^{22}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{22}}} - \frac {10395 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{443520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.60, size = 263, normalized size = 1.44 \[ \frac {3\,x}{256\,a}+\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{4}-\frac {3323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{640}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{3}+\frac {54\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}-\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}-\frac {841\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {176\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {841\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {240\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}-\frac {54\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {3323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640}+\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{63}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {16}{693}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]
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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