Integrand size = 29, antiderivative size = 264 \[ \int \frac {\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {843 \log (1-\sin (c+d x))}{512 a d}-\frac {2229 \log (1+\sin (c+d x))}{512 a d}+\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {3 a^2}{64 d (a-a \sin (c+d x))^3}+\frac {141 a}{512 d (a-a \sin (c+d x))^2}-\frac {39}{32 d (a-a \sin (c+d x))}-\frac {a^4}{160 d (a+a \sin (c+d x))^5}+\frac {19 a^3}{256 d (a+a \sin (c+d x))^4}-\frac {53 a^2}{128 d (a+a \sin (c+d x))^3}+\frac {765 a}{512 d (a+a \sin (c+d x))^2}-\frac {1155}{256 d (a+a \sin (c+d x))} \]
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Time = 0.21 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90} \[ \int \frac {\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {19 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac {3 a^2}{64 d (a-a \sin (c+d x))^3}-\frac {53 a^2}{128 d (a \sin (c+d x)+a)^3}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {141 a}{512 d (a-a \sin (c+d x))^2}+\frac {765 a}{512 d (a \sin (c+d x)+a)^2}-\frac {39}{32 d (a-a \sin (c+d x))}-\frac {1155}{256 d (a \sin (c+d x)+a)}+\frac {\sin (c+d x)}{a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}-\frac {2229 \log (\sin (c+d x)+1)}{512 a d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {a^9 \text {Subst}\left (\int \frac {x^{12}}{a^{12} (a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {x^{12}}{(a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \left (a+\frac {a^6}{64 (a-x)^5}-\frac {9 a^5}{64 (a-x)^4}+\frac {141 a^4}{256 (a-x)^3}-\frac {39 a^3}{32 (a-x)^2}+\frac {843 a^2}{512 (a-x)}-x+\frac {a^7}{32 (a+x)^6}-\frac {19 a^6}{64 (a+x)^5}+\frac {159 a^5}{128 (a+x)^4}-\frac {765 a^4}{256 (a+x)^3}+\frac {1155 a^3}{256 (a+x)^2}-\frac {2229 a^2}{512 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = -\frac {843 \log (1-\sin (c+d x))}{512 a d}-\frac {2229 \log (1+\sin (c+d x))}{512 a d}+\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {3 a^2}{64 d (a-a \sin (c+d x))^3}+\frac {141 a}{512 d (a-a \sin (c+d x))^2}-\frac {39}{32 d (a-a \sin (c+d x))}-\frac {a^4}{160 d (a+a \sin (c+d x))^5}+\frac {19 a^3}{256 d (a+a \sin (c+d x))^4}-\frac {53 a^2}{128 d (a+a \sin (c+d x))^3}+\frac {765 a}{512 d (a+a \sin (c+d x))^2}-\frac {1155}{256 d (a+a \sin (c+d x))} \\ \end{align*}
Time = 6.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.64 \[ \int \frac {\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {4215 \log (1-\sin (c+d x))+11145 \log (1+\sin (c+d x))-\frac {10}{(1-\sin (c+d x))^4}+\frac {120}{(1-\sin (c+d x))^3}-\frac {705}{(1-\sin (c+d x))^2}+\frac {3120}{1-\sin (c+d x)}-2560 \sin (c+d x)+1280 \sin ^2(c+d x)+\frac {16}{(1+\sin (c+d x))^5}-\frac {190}{(1+\sin (c+d x))^4}+\frac {1060}{(1+\sin (c+d x))^3}-\frac {3825}{(1+\sin (c+d x))^2}+\frac {11550}{1+\sin (c+d x)}}{2560 a d} \]
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Time = 4.70 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.59
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {3}{64 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {141}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {39}{32 \left (\sin \left (d x +c \right )-1\right )}-\frac {843 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {19}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {53}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {765}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1155}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {2229 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(155\) |
default | \(\frac {-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {3}{64 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {141}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {39}{32 \left (\sin \left (d x +c \right )-1\right )}-\frac {843 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {19}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {53}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {765}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1155}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {2229 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(155\) |
risch | \(\frac {6 i x}{a}+\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 a d}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a d}+\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a d}+\frac {12 i c}{d a}-\frac {i \left (-31370 i {\mathrm e}^{14 i \left (d x +c \right )}+4215 \,{\mathrm e}^{17 i \left (d x +c \right )}-10770 i {\mathrm e}^{16 i \left (d x +c \right )}+40900 \,{\mathrm e}^{15 i \left (d x +c \right )}+20922 i {\mathrm e}^{8 i \left (d x +c \right )}+152108 \,{\mathrm e}^{13 i \left (d x +c \right )}+59954 i {\mathrm e}^{6 i \left (d x +c \right )}+316476 \,{\mathrm e}^{11 i \left (d x +c \right )}-59954 i {\mathrm e}^{12 i \left (d x +c \right )}+398010 \,{\mathrm e}^{9 i \left (d x +c \right )}-20922 i {\mathrm e}^{10 i \left (d x +c \right )}+316476 \,{\mathrm e}^{7 i \left (d x +c \right )}+31370 i {\mathrm e}^{4 i \left (d x +c \right )}+152108 \,{\mathrm e}^{5 i \left (d x +c \right )}+10770 i {\mathrm e}^{2 i \left (d x +c \right )}+40900 \,{\mathrm e}^{3 i \left (d x +c \right )}+4215 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{640 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}-\frac {2229 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}-\frac {843 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}\) | \(364\) |
parallelrisch | \(\frac {\left (215040 \sin \left (3 d x +3 c \right )+153600 \sin \left (5 d x +5 c \right )+53760 \sin \left (7 d x +7 c \right )+7680 \sin \left (9 d x +9 c \right )+860160 \cos \left (2 d x +2 c \right )+430080 \cos \left (4 d x +4 c \right )+122880 \cos \left (6 d x +6 c \right )+15360 \cos \left (8 d x +8 c \right )+107520 \sin \left (d x +c \right )+537600\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-118020 \sin \left (3 d x +3 c \right )-84300 \sin \left (5 d x +5 c \right )-29505 \sin \left (7 d x +7 c \right )-4215 \sin \left (9 d x +9 c \right )-472080 \cos \left (2 d x +2 c \right )-236040 \cos \left (4 d x +4 c \right )-67440 \cos \left (6 d x +6 c \right )-8430 \cos \left (8 d x +8 c \right )-59010 \sin \left (d x +c \right )-295050\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-312060 \sin \left (3 d x +3 c \right )-222900 \sin \left (5 d x +5 c \right )-78015 \sin \left (7 d x +7 c \right )-11145 \sin \left (9 d x +9 c \right )-1248240 \cos \left (2 d x +2 c \right )-624120 \cos \left (4 d x +4 c \right )-178320 \cos \left (6 d x +6 c \right )-22290 \cos \left (8 d x +8 c \right )-156030 \sin \left (d x +c \right )-780150\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+68316 \sin \left (3 d x +3 c \right )+73420 \sin \left (5 d x +5 c \right )+27516 \sin \left (7 d x +7 c \right )+7648 \sin \left (9 d x +9 c \right )-320 \cos \left (10 d x +10 c \right )+160 \sin \left (11 d x +11 c \right )-4856 \cos \left (2 d x +2 c \right )+5032 \cos \left (4 d x +4 c \right )+3128 \cos \left (6 d x +6 c \right )+786 \cos \left (8 d x +8 c \right )+51788 \sin \left (d x +c \right )-3770}{1280 a d \left (70+\sin \left (9 d x +9 c \right )+7 \sin \left (7 d x +7 c \right )+20 \sin \left (5 d x +5 c \right )+28 \sin \left (3 d x +3 c \right )+14 \sin \left (d x +c \right )+2 \cos \left (8 d x +8 c \right )+16 \cos \left (6 d x +6 c \right )+56 \cos \left (4 d x +4 c \right )+112 \cos \left (2 d x +2 c \right )\right )}\) | \(560\) |
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Time = 0.31 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {1280 \, \cos \left (d x + c\right )^{10} + 6510 \, \cos \left (d x + c\right )^{8} + 3590 \, \cos \left (d x + c\right )^{6} - 1124 \, \cos \left (d x + c\right )^{4} + 272 \, \cos \left (d x + c\right )^{2} + 11145 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 4215 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (640 \, \cos \left (d x + c\right )^{10} + 960 \, \cos \left (d x + c\right )^{8} - 5385 \, \cos \left (d x + c\right )^{6} + 2810 \, \cos \left (d x + c\right )^{4} - 952 \, \cos \left (d x + c\right )^{2} + 144\right )} \sin \left (d x + c\right ) - 32}{2560 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
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Timed out. \[ \int \frac {\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (4215 \, \sin \left (d x + c\right )^{8} - 5385 \, \sin \left (d x + c\right )^{7} - 18655 \, \sin \left (d x + c\right )^{6} + 13345 \, \sin \left (d x + c\right )^{5} + 30113 \, \sin \left (d x + c\right )^{4} - 11487 \, \sin \left (d x + c\right )^{3} - 21257 \, \sin \left (d x + c\right )^{2} + 3383 \, \sin \left (d x + c\right ) + 5568\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} + \frac {1280 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )}}{a} + \frac {11145 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {4215 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{2560 \, d} \]
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Time = 0.45 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {44580 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {16860 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5120 \, {\left (a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{a^{2}} - \frac {5 \, {\left (7025 \, \sin \left (d x + c\right )^{4} - 25604 \, \sin \left (d x + c\right )^{3} + 35226 \, \sin \left (d x + c\right )^{2} - 21644 \, \sin \left (d x + c\right ) + 5005\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {101791 \, \sin \left (d x + c\right )^{5} + 462755 \, \sin \left (d x + c\right )^{4} + 848410 \, \sin \left (d x + c\right )^{3} + 782370 \, \sin \left (d x + c\right )^{2} + 362335 \, \sin \left (d x + c\right ) + 67347}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \]
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Time = 13.56 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.45 \[ \int \frac {\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
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