Integrand size = 29, antiderivative size = 236 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {1795 \log (1+\sin (c+d x))}{256 a d}+\frac {5 \sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin ^3(c+d x)}{3 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {17 a}{128 d (a-a \sin (c+d x))^2}+\frac {125}{128 d (a-a \sin (c+d x))}+\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {3 a^2}{16 d (a+a \sin (c+d x))^3}+\frac {71 a}{64 d (a+a \sin (c+d x))^2}-\frac {5}{d (a+a \sin (c+d x))} \]
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Time = 0.18 (sec) , antiderivative size = 236, 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} \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {3 a^2}{16 d (a \sin (c+d x)+a)^3}+\frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^2(c+d x)}{2 a d}-\frac {17 a}{128 d (a-a \sin (c+d x))^2}+\frac {71 a}{64 d (a \sin (c+d x)+a)^2}+\frac {125}{128 d (a-a \sin (c+d x))}-\frac {5}{d (a \sin (c+d x)+a)}+\frac {5 \sin (c+d x)}{a d}+\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {1795 \log (\sin (c+d x)+1)}{256 a d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {a^7 \text {Subst}\left (\int \frac {x^{11}}{a^{11} (a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {x^{11}}{(a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = \frac {\text {Subst}\left (\int \left (5 a^2+\frac {a^6}{32 (a-x)^4}-\frac {17 a^5}{64 (a-x)^3}+\frac {125 a^4}{128 (a-x)^2}-\frac {515 a^3}{256 (a-x)}-a x+x^2-\frac {a^7}{16 (a+x)^5}+\frac {9 a^6}{16 (a+x)^4}-\frac {71 a^5}{32 (a+x)^3}+\frac {5 a^4}{(a+x)^2}-\frac {1795 a^3}{256 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = \frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {1795 \log (1+\sin (c+d x))}{256 a d}+\frac {5 \sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin ^3(c+d x)}{3 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {17 a}{128 d (a-a \sin (c+d x))^2}+\frac {125}{128 d (a-a \sin (c+d x))}+\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {3 a^2}{16 d (a+a \sin (c+d x))^3}+\frac {71 a}{64 d (a+a \sin (c+d x))^2}-\frac {5}{d (a+a \sin (c+d x))} \\ \end{align*}
Time = 6.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1545 \log (1-\sin (c+d x))-5385 \log (1+\sin (c+d x))+\frac {8}{(1-\sin (c+d x))^3}-\frac {102}{(1-\sin (c+d x))^2}+\frac {750}{1-\sin (c+d x)}+3840 \sin (c+d x)-384 \sin ^2(c+d x)+256 \sin ^3(c+d x)+\frac {12}{(1+\sin (c+d x))^4}-\frac {144}{(1+\sin (c+d x))^3}+\frac {852}{(1+\sin (c+d x))^2}-\frac {3840}{1+\sin (c+d x)}}{768 a d} \]
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Time = 2.93 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+5 \sin \left (d x +c \right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {17}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {125}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {515 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {3}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {71}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5}{1+\sin \left (d x +c \right )}-\frac {1795 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(143\) |
default | \(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+5 \sin \left (d x +c \right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {17}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {125}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {515 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {3}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {71}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5}{1+\sin \left (d x +c \right )}-\frac {1795 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(143\) |
risch | \(\frac {5 i x}{a}+\frac {i {\mathrm e}^{3 i \left (d x +c \right )}}{24 d a}+\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 a d}-\frac {21 i {\mathrm e}^{i \left (d x +c \right )}}{8 a d}+\frac {21 i {\mathrm e}^{-i \left (d x +c \right )}}{8 a d}+\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a d}-\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d a}+\frac {10 i c}{d a}-\frac {i \left (18313 \,{\mathrm e}^{5 i \left (d x +c \right )}+750 i {\mathrm e}^{12 i \left (d x +c \right )}+1690 i {\mathrm e}^{10 i \left (d x +c \right )}+1196 i {\mathrm e}^{8 i \left (d x +c \right )}-1196 i {\mathrm e}^{6 i \left (d x +c \right )}-1690 i {\mathrm e}^{4 i \left (d x +c \right )}-750 i {\mathrm e}^{2 i \left (d x +c \right )}+2295 \,{\mathrm e}^{i \left (d x +c \right )}+2295 \,{\mathrm e}^{13 i \left (d x +c \right )}+18313 \,{\mathrm e}^{9 i \left (d x +c \right )}+8150 \,{\mathrm e}^{11 i \left (d x +c \right )}+8150 \,{\mathrm e}^{3 i \left (d x +c \right )}+21332 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}+\frac {515 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}-\frac {1795 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}\) | \(354\) |
parallelrisch | \(\frac {7724+1920 \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1545 \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+5385 \left (-20-\sin \left (7 d x +7 c \right )-5 \sin \left (5 d x +5 c \right )-9 \sin \left (3 d x +3 c \right )-5 \sin \left (d x +c \right )-2 \cos \left (6 d x +6 c \right )-12 \cos \left (4 d x +4 c \right )-30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+31620 \sin \left (3 d x +3 c \right )+17260 \sin \left (5 d x +5 c \right )+3200 \sin \left (7 d x +7 c \right )+16 \sin \left (9 d x +9 c \right )+16 \cos \left (10 d x +10 c \right )+3822 \cos \left (2 d x +2 c \right )-5100 \cos \left (4 d x +4 c \right )-5630 \cos \left (6 d x +6 c \right )-832 \cos \left (8 d x +8 c \right )+18056 \sin \left (d x +c \right )}{384 a d \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right )}\) | \(460\) |
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Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.88 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {256 \, \cos \left (d x + c\right )^{10} - 3968 \, \cos \left (d x + c\right )^{8} - 686 \, \cos \left (d x + c\right )^{6} + 2810 \, \cos \left (d x + c\right )^{4} - 796 \, \cos \left (d x + c\right )^{2} - 5385 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 1545 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (64 \, \cos \left (d x + c\right )^{8} + 1952 \, \cos \left (d x + c\right )^{6} + 375 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
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Timed out. \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (2295 \, \sin \left (d x + c\right )^{6} + 375 \, \sin \left (d x + c\right )^{5} - 5480 \, \sin \left (d x + c\right )^{4} - 680 \, \sin \left (d x + c\right )^{3} + 4473 \, \sin \left (d x + c\right )^{2} + 313 \, \sin \left (d x + c\right ) - 1232\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {128 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 30 \, \sin \left (d x + c\right )\right )}}{a} + \frac {5385 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {1545 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.76 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {21540 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {6180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {512 \, {\left (2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} \sin \left (d x + c\right )^{2} + 30 \, a^{2} \sin \left (d x + c\right )\right )}}{a^{3}} + \frac {2 \, {\left (5665 \, \sin \left (d x + c\right )^{3} - 15495 \, \sin \left (d x + c\right )^{2} + 14199 \, \sin \left (d x + c\right ) - 4353\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {44875 \, \sin \left (d x + c\right )^{4} + 164140 \, \sin \left (d x + c\right )^{3} + 226578 \, \sin \left (d x + c\right )^{2} + 139660 \, \sin \left (d x + c\right ) + 32395}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
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Time = 11.89 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.40 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {515\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{128\,a\,d}-\frac {1795\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{128\,a\,d}-\frac {-\frac {1155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{64}-\frac {835\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{32}+\frac {3205\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {305\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{4}+\frac {41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{16}+\frac {53\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{24}-\frac {5521\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{48}-\frac {2387\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{12}+\frac {6697\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{96}+\frac {6901\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{48}+\frac {6697\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{96}-\frac {2387\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{12}-\frac {5521\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{48}+\frac {53\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{24}+\frac {41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {305\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {3205\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}-\frac {835\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {1155\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}+\frac {5\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]
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