Integrand size = 29, antiderivative size = 247 \[ \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {437 \log (1-\sin (c+d x))}{512 a d}+\frac {949 \log (1+\sin (c+d x))}{512 a d}-\frac {\sin (c+d x)}{a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {a^2}{24 d (a-a \sin (c+d x))^3}+\frac {109 a}{512 d (a-a \sin (c+d x))^2}-\frac {203}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}-\frac {17 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {125 a^2}{384 d (a+a \sin (c+d x))^3}-\frac {515 a}{512 d (a+a \sin (c+d x))^2}+\frac {5}{2 d (a+a \sin (c+d x))} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 247, 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} \[ \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {17 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac {a^2}{24 d (a-a \sin (c+d x))^3}+\frac {125 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac {109 a}{512 d (a-a \sin (c+d x))^2}-\frac {515 a}{512 d (a \sin (c+d x)+a)^2}-\frac {203}{256 d (a-a \sin (c+d x))}+\frac {5}{2 d (a \sin (c+d x)+a)}-\frac {\sin (c+d x)}{a d}-\frac {437 \log (1-\sin (c+d x))}{512 a d}+\frac {949 \log (\sin (c+d x)+1)}{512 a d} \]
[In]
[Out]
Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {a^9 \text {Subst}\left (\int \frac {x^{11}}{a^{11} (a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {x^{11}}{(a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (-1+\frac {a^5}{64 (a-x)^5}-\frac {a^4}{8 (a-x)^4}+\frac {109 a^3}{256 (a-x)^3}-\frac {203 a^2}{256 (a-x)^2}+\frac {437 a}{512 (a-x)}-\frac {a^6}{32 (a+x)^6}+\frac {17 a^5}{64 (a+x)^5}-\frac {125 a^4}{128 (a+x)^4}+\frac {515 a^3}{256 (a+x)^3}-\frac {5 a^2}{2 (a+x)^2}+\frac {949 a}{512 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = -\frac {437 \log (1-\sin (c+d x))}{512 a d}+\frac {949 \log (1+\sin (c+d x))}{512 a d}-\frac {\sin (c+d x)}{a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {a^2}{24 d (a-a \sin (c+d x))^3}+\frac {109 a}{512 d (a-a \sin (c+d x))^2}-\frac {203}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}-\frac {17 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {125 a^2}{384 d (a+a \sin (c+d x))^3}-\frac {515 a}{512 d (a+a \sin (c+d x))^2}+\frac {5}{2 d (a+a \sin (c+d x))} \\ \end{align*}
Time = 6.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.64 \[ \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {6555 \log (1-\sin (c+d x))-14235 \log (1+\sin (c+d x))-\frac {30}{(1-\sin (c+d x))^4}+\frac {320}{(1-\sin (c+d x))^3}-\frac {1635}{(1-\sin (c+d x))^2}+\frac {6090}{1-\sin (c+d x)}+7680 \sin (c+d x)-\frac {48}{(1+\sin (c+d x))^5}+\frac {510}{(1+\sin (c+d x))^4}-\frac {2500}{(1+\sin (c+d x))^3}+\frac {7725}{(1+\sin (c+d x))^2}-\frac {19200}{1+\sin (c+d x)}}{7680 a d} \]
[In]
[Out]
Time = 3.64 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.60
method | result | size |
derivativedivides | \(\frac {-\sin \left (d x +c \right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{24 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {109}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {203}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {437 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {17}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {125}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {515}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{2 \left (1+\sin \left (d x +c \right )\right )}+\frac {949 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(147\) |
default | \(\frac {-\sin \left (d x +c \right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{24 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {109}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {203}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {437 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {17}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {125}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {515}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{2 \left (1+\sin \left (d x +c \right )\right )}+\frac {949 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(147\) |
risch | \(-\frac {i x}{a}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a d}-\frac {2 i c}{d a}+\frac {i \left (149124 \,{\mathrm e}^{5 i \left (d x +c \right )}+12645 \,{\mathrm e}^{i \left (d x +c \right )}+12645 \,{\mathrm e}^{17 i \left (d x +c \right )}+45900 \,{\mathrm e}^{15 i \left (d x +c \right )}+149124 \,{\mathrm e}^{13 i \left (d x +c \right )}+207028 \,{\mathrm e}^{11 i \left (d x +c \right )}+292910 \,{\mathrm e}^{9 i \left (d x +c \right )}+21090 i {\mathrm e}^{14 i \left (d x +c \right )}+6090 i {\mathrm e}^{16 i \left (d x +c \right )}-16594 i {\mathrm e}^{8 i \left (d x +c \right )}-37738 i {\mathrm e}^{6 i \left (d x +c \right )}+37738 i {\mathrm e}^{12 i \left (d x +c \right )}+16594 i {\mathrm e}^{10 i \left (d x +c \right )}-21090 i {\mathrm e}^{4 i \left (d x +c \right )}-6090 i {\mathrm e}^{2 i \left (d x +c \right )}+45900 \,{\mathrm e}^{3 i \left (d x +c \right )}+207028 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}-\frac {437 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}+\frac {949 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}\) | \(330\) |
parallelrisch | \(\frac {\left (-107520 \sin \left (3 d x +3 c \right )-76800 \sin \left (5 d x +5 c \right )-26880 \sin \left (7 d x +7 c \right )-3840 \sin \left (9 d x +9 c \right )-430080 \cos \left (2 d x +2 c \right )-215040 \cos \left (4 d x +4 c \right )-61440 \cos \left (6 d x +6 c \right )-7680 \cos \left (8 d x +8 c \right )-53760 \sin \left (d x +c \right )-268800\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-183540 \sin \left (3 d x +3 c \right )-131100 \sin \left (5 d x +5 c \right )-45885 \sin \left (7 d x +7 c \right )-6555 \sin \left (9 d x +9 c \right )-734160 \cos \left (2 d x +2 c \right )-367080 \cos \left (4 d x +4 c \right )-104880 \cos \left (6 d x +6 c \right )-13110 \cos \left (8 d x +8 c \right )-91770 \sin \left (d x +c \right )-458850\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (398580 \sin \left (3 d x +3 c \right )+284700 \sin \left (5 d x +5 c \right )+99645 \sin \left (7 d x +7 c \right )+14235 \sin \left (9 d x +9 c \right )+1594320 \cos \left (2 d x +2 c \right )+797160 \cos \left (4 d x +4 c \right )+227760 \cos \left (6 d x +6 c \right )+28470 \cos \left (8 d x +8 c \right )+199290 \sin \left (d x +c \right )+996450\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-305748 \sin \left (3 d x +3 c \right )-206660 \sin \left (5 d x +5 c \right )-69748 \sin \left (7 d x +7 c \right )-8224 \sin \left (9 d x +9 c \right )+1920 \cos \left (10 d x +10 c \right )-103832 \cos \left (2 d x +2 c \right )+68104 \cos \left (4 d x +4 c \right )+46616 \cos \left (6 d x +6 c \right )+28042 \cos \left (8 d x +8 c \right )-148324 \sin \left (d x +c \right )-40850}{3840 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 )}\) | \(549\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {7680 \, \cos \left (d x + c\right )^{10} + 17610 \, \cos \left (d x + c\right )^{8} - 27630 \, \cos \left (d x + c\right )^{6} + 15828 \, \cos \left (d x + c\right )^{4} - 5584 \, \cos \left (d x + c\right )^{2} + 14235 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6555 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (3840 \, \cos \left (d x + c\right )^{8} + 3045 \, \cos \left (d x + c\right )^{6} - 1170 \, \cos \left (d x + c\right )^{4} + 344 \, \cos \left (d x + c\right )^{2} - 48\right )} \sin \left (d x + c\right ) + 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.91 \[ \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (12645 \, \sin \left (d x + c\right )^{8} + 3045 \, \sin \left (d x + c\right )^{7} - 36765 \, \sin \left (d x + c\right )^{6} - 7965 \, \sin \left (d x + c\right )^{5} + 42339 \, \sin \left (d x + c\right )^{4} + 7139 \, \sin \left (d x + c\right )^{3} - 22171 \, \sin \left (d x + c\right )^{2} - 2171 \, \sin \left (d x + c\right ) + 4384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} + \frac {14235 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {6555 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} - \frac {7680 \, \sin \left (d x + c\right )}{a}}{7680 \, d} \]
[In]
[Out]
none
Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.68 \[ \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {56940 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {26220 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {30720 \, \sin \left (d x + c\right )}{a} + \frac {5 \, {\left (10925 \, \sin \left (d x + c\right )^{4} - 38828 \, \sin \left (d x + c\right )^{3} + 52242 \, \sin \left (d x + c\right )^{2} - 31444 \, \sin \left (d x + c\right ) + 7129\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {130013 \, \sin \left (d x + c\right )^{5} + 573265 \, \sin \left (d x + c\right )^{4} + 1023830 \, \sin \left (d x + c\right )^{3} + 922030 \, \sin \left (d x + c\right )^{2} + 417605 \, \sin \left (d x + c\right ) + 75961}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
[In]
[Out]
Time = 11.78 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.41 \[ \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
[In]
[Out]