Integrand size = 27, antiderivative size = 247 \[ \int \frac {\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {193 \log (1-\sin (c+d x))}{512 a d}+\frac {\log (\sin (c+d x))}{a d}-\frac {319 \log (1+\sin (c+d x))}{512 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {a^2}{48 d (a-a \sin (c+d x))^3}+\frac {37 a}{512 d (a-a \sin (c+d x))^2}+\frac {65}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}+\frac {7 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {29 a^2}{384 d (a+a \sin (c+d x))^3}+\frac {93 a}{512 d (a+a \sin (c+d x))^2}+\frac {1}{2 d (a+a \sin (c+d x))} \]
-193/512*ln(1-sin(d*x+c))/a/d+ln(sin(d*x+c))/a/d-319/512*ln(1+sin(d*x+c))/ a/d+1/256*a^3/d/(a-a*sin(d*x+c))^4+1/48*a^2/d/(a-a*sin(d*x+c))^3+37/512*a/ d/(a-a*sin(d*x+c))^2+65/256/d/(a-a*sin(d*x+c))+1/160*a^4/d/(a+a*sin(d*x+c) )^5+7/256*a^3/d/(a+a*sin(d*x+c))^4+29/384*a^2/d/(a+a*sin(d*x+c))^3+93/512* a/d/(a+a*sin(d*x+c))^2+1/2/d/(a+a*sin(d*x+c))
Time = 6.13 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.92 \[ \int \frac {\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^9 \left (-\frac {193 \log (1-\sin (c+d x))}{512 a^{10}}+\frac {\log (\sin (c+d x))}{a^{10}}-\frac {319 \log (1+\sin (c+d x))}{512 a^{10}}+\frac {1}{256 a^6 (a-a \sin (c+d x))^4}+\frac {1}{48 a^7 (a-a \sin (c+d x))^3}+\frac {37}{512 a^8 (a-a \sin (c+d x))^2}+\frac {65}{256 a^9 (a-a \sin (c+d x))}+\frac {1}{160 a^5 (a+a \sin (c+d x))^5}+\frac {7}{256 a^6 (a+a \sin (c+d x))^4}+\frac {29}{384 a^7 (a+a \sin (c+d x))^3}+\frac {93}{512 a^8 (a+a \sin (c+d x))^2}+\frac {1}{2 a^9 (a+a \sin (c+d x))}\right )}{d} \]
(a^9*((-193*Log[1 - Sin[c + d*x]])/(512*a^10) + Log[Sin[c + d*x]]/a^10 - ( 319*Log[1 + Sin[c + d*x]])/(512*a^10) + 1/(256*a^6*(a - a*Sin[c + d*x])^4) + 1/(48*a^7*(a - a*Sin[c + d*x])^3) + 37/(512*a^8*(a - a*Sin[c + d*x])^2) + 65/(256*a^9*(a - a*Sin[c + d*x])) + 1/(160*a^5*(a + a*Sin[c + d*x])^5) + 7/(256*a^6*(a + a*Sin[c + d*x])^4) + 29/(384*a^7*(a + a*Sin[c + d*x])^3) + 93/(512*a^8*(a + a*Sin[c + d*x])^2) + 1/(2*a^9*(a + a*Sin[c + d*x]))))/ d
Time = 0.43 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3315, 27, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\csc (c+d x) \sec ^9(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x) \cos (c+d x)^9 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^9 \int \frac {\csc (c+d x)}{(a-a \sin (c+d x))^5 (\sin (c+d x) a+a)^6}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^{10} \int \frac {\csc (c+d x)}{a (a-a \sin (c+d x))^5 (\sin (c+d x) a+a)^6}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^{10} \int \left (\frac {\csc (c+d x)}{a^{12}}+\frac {193}{512 a^{11} (a-a \sin (c+d x))}-\frac {319}{512 a^{11} (\sin (c+d x) a+a)}+\frac {65}{256 a^{10} (a-a \sin (c+d x))^2}-\frac {1}{2 a^{10} (\sin (c+d x) a+a)^2}+\frac {37}{256 a^9 (a-a \sin (c+d x))^3}-\frac {93}{256 a^9 (\sin (c+d x) a+a)^3}+\frac {1}{16 a^8 (a-a \sin (c+d x))^4}-\frac {29}{128 a^8 (\sin (c+d x) a+a)^4}+\frac {1}{64 a^7 (a-a \sin (c+d x))^5}-\frac {7}{64 a^7 (\sin (c+d x) a+a)^5}-\frac {1}{32 a^6 (\sin (c+d x) a+a)^6}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{10} \left (\frac {\log (a \sin (c+d x))}{a^{11}}-\frac {193 \log (a-a \sin (c+d x))}{512 a^{11}}-\frac {319 \log (a \sin (c+d x)+a)}{512 a^{11}}+\frac {65}{256 a^{10} (a-a \sin (c+d x))}+\frac {1}{2 a^{10} (a \sin (c+d x)+a)}+\frac {37}{512 a^9 (a-a \sin (c+d x))^2}+\frac {93}{512 a^9 (a \sin (c+d x)+a)^2}+\frac {1}{48 a^8 (a-a \sin (c+d x))^3}+\frac {29}{384 a^8 (a \sin (c+d x)+a)^3}+\frac {1}{256 a^7 (a-a \sin (c+d x))^4}+\frac {7}{256 a^7 (a \sin (c+d x)+a)^4}+\frac {1}{160 a^6 (a \sin (c+d x)+a)^5}\right )}{d}\) |
(a^10*(Log[a*Sin[c + d*x]]/a^11 - (193*Log[a - a*Sin[c + d*x]])/(512*a^11) - (319*Log[a + a*Sin[c + d*x]])/(512*a^11) + 1/(256*a^7*(a - a*Sin[c + d* x])^4) + 1/(48*a^8*(a - a*Sin[c + d*x])^3) + 37/(512*a^9*(a - a*Sin[c + d* x])^2) + 65/(256*a^10*(a - a*Sin[c + d*x])) + 1/(160*a^6*(a + a*Sin[c + d* x])^5) + 7/(256*a^7*(a + a*Sin[c + d*x])^4) + 29/(384*a^8*(a + a*Sin[c + d *x])^3) + 93/(512*a^9*(a + a*Sin[c + d*x])^2) + 1/(2*a^10*(a + a*Sin[c + d *x]))))/d
3.10.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 3.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.59
| method | result | size |
| derivativedivides | \(\frac {\ln \left (\sin \left (d x +c \right )\right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{48 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {37}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {65}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {193 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {7}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {29}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {93}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 \sin \left (d x +c \right )+2}-\frac {319 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(146\) |
| default | \(\frac {\ln \left (\sin \left (d x +c \right )\right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{48 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {37}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {65}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {193 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {7}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {29}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {93}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 \sin \left (d x +c \right )+2}-\frac {319 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(146\) |
| risch | \(\frac {i \left (78324 \,{\mathrm e}^{5 i \left (d x +c \right )}-12390 i {\mathrm e}^{14 i \left (d x +c \right )}+13980 \,{\mathrm e}^{15 i \left (d x +c \right )}-1950 i {\mathrm e}^{16 i \left (d x +c \right )}+25526 i {\mathrm e}^{8 i \left (d x +c \right )}+29822 i {\mathrm e}^{6 i \left (d x +c \right )}-29822 i {\mathrm e}^{12 i \left (d x +c \right )}+945 \,{\mathrm e}^{i \left (d x +c \right )}+238948 \,{\mathrm e}^{11 i \left (d x +c \right )}+78324 \,{\mathrm e}^{13 i \left (d x +c \right )}+945 \,{\mathrm e}^{17 i \left (d x +c \right )}+457910 \,{\mathrm e}^{9 i \left (d x +c \right )}-25526 i {\mathrm e}^{10 i \left (d x +c \right )}+12390 i {\mathrm e}^{4 i \left (d x +c \right )}+1950 i {\mathrm e}^{2 i \left (d x +c \right )}+13980 \,{\mathrm e}^{3 i \left (d x +c \right )}+238948 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}-\frac {319 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}-\frac {193 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(296\) |
| parallelrisch | \(\frac {\left (-81060 \sin \left (3 d x +3 c \right )-57900 \sin \left (5 d x +5 c \right )-20265 \sin \left (7 d x +7 c \right )-2895 \sin \left (9 d x +9 c \right )-324240 \cos \left (2 d x +2 c \right )-162120 \cos \left (4 d x +4 c \right )-46320 \cos \left (6 d x +6 c \right )-5790 \cos \left (8 d x +8 c \right )-40530 \sin \left (d x +c \right )-202650\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-133980 \sin \left (3 d x +3 c \right )-95700 \sin \left (5 d x +5 c \right )-33495 \sin \left (7 d x +7 c \right )-4785 \sin \left (9 d x +9 c \right )-535920 \cos \left (2 d x +2 c \right )-267960 \cos \left (4 d x +4 c \right )-76560 \cos \left (6 d x +6 c \right )-9570 \cos \left (8 d x +8 c \right )-66990 \sin \left (d x +c \right )-334950\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (107520 \sin \left (3 d x +3 c \right )+76800 \sin \left (5 d x +5 c \right )+26880 \sin \left (7 d x +7 c \right )+3840 \sin \left (9 d x +9 c \right )+430080 \cos \left (2 d x +2 c \right )+215040 \cos \left (4 d x +4 c \right )+61440 \cos \left (6 d x +6 c \right )+7680 \cos \left (8 d x +8 c \right )+53760 \sin \left (d x +c \right )+268800\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-63108 \sin \left (3 d x +3 c \right )-62900 \sin \left (5 d x +5 c \right )-26788 \sin \left (7 d x +7 c \right )-4384 \sin \left (9 d x +9 c \right )-13112 \cos \left (2 d x +2 c \right )-88856 \cos \left (4 d x +4 c \right )-42184 \cos \left (6 d x +6 c \right )-6878 \cos \left (8 d x +8 c \right )-10324 \sin \left (d x +c \right )+151030}{3840 a d \left (70+\sin \left (9 d x +9 c \right )+7 \sin \left (7 d x +7 c \right )+20 \sin \left (5 d x +5 c \right )+28 \sin \left (3 d x +3 c \right )+14 \sin \left (d x +c \right )+2 \cos \left (8 d x +8 c \right )+16 \cos \left (6 d x +6 c \right )+56 \cos \left (4 d x +4 c \right )+112 \cos \left (2 d x +2 c \right )\right )}\) | \(536\) |
1/d/a*(ln(sin(d*x+c))+1/256/(sin(d*x+c)-1)^4-1/48/(sin(d*x+c)-1)^3+37/512/ (sin(d*x+c)-1)^2-65/256/(sin(d*x+c)-1)-193/512*ln(sin(d*x+c)-1)+1/160/(1+s in(d*x+c))^5+7/256/(1+sin(d*x+c))^4+29/384/(1+sin(d*x+c))^3+93/512/(1+sin( d*x+c))^2+1/2/(1+sin(d*x+c))-319/512*ln(1+sin(d*x+c)))
Time = 0.33 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.90 \[ \int \frac {\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1890 \, \cos \left (d x + c\right )^{8} + 3210 \, \cos \left (d x + c\right )^{6} + 1668 \, \cos \left (d x + c\right )^{4} + 1136 \, \cos \left (d x + c\right )^{2} + 7680 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4785 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2895 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (975 \, \cos \left (d x + c\right )^{6} + 330 \, \cos \left (d x + c\right )^{4} + 136 \, \cos \left (d x + c\right )^{2} + 48\right )} \sin \left (d x + c\right ) + 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
1/7680*(1890*cos(d*x + c)^8 + 3210*cos(d*x + c)^6 + 1668*cos(d*x + c)^4 + 1136*cos(d*x + c)^2 + 7680*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8)* log(1/2*sin(d*x + c)) - 4785*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8 )*log(sin(d*x + c) + 1) - 2895*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c) ^8)*log(-sin(d*x + c) + 1) + 2*(975*cos(d*x + c)^6 + 330*cos(d*x + c)^4 + 136*cos(d*x + c)^2 + 48)*sin(d*x + c) + 864)/(a*d*cos(d*x + c)^8*sin(d*x + c) + a*d*cos(d*x + c)^8)
Timed out. \[ \int \frac {\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.91 \[ \int \frac {\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (945 \, \sin \left (d x + c\right )^{8} - 975 \, \sin \left (d x + c\right )^{7} - 5385 \, \sin \left (d x + c\right )^{6} + 3255 \, \sin \left (d x + c\right )^{5} + 11319 \, \sin \left (d x + c\right )^{4} - 3721 \, \sin \left (d x + c\right )^{3} - 10831 \, \sin \left (d x + c\right )^{2} + 1489 \, \sin \left (d x + c\right ) + 4384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {4785 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {2895 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {7680 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{7680 \, d} \]
1/7680*(2*(945*sin(d*x + c)^8 - 975*sin(d*x + c)^7 - 5385*sin(d*x + c)^6 + 3255*sin(d*x + c)^5 + 11319*sin(d*x + c)^4 - 3721*sin(d*x + c)^3 - 10831* sin(d*x + c)^2 + 1489*sin(d*x + c) + 4384)/(a*sin(d*x + c)^9 + a*sin(d*x + c)^8 - 4*a*sin(d*x + c)^7 - 4*a*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 + 6*a *sin(d*x + c)^4 - 4*a*sin(d*x + c)^3 - 4*a*sin(d*x + c)^2 + a*sin(d*x + c) + a) - 4785*log(sin(d*x + c) + 1)/a - 2895*log(sin(d*x + c) - 1)/a + 7680 *log(sin(d*x + c))/a)/d
Time = 0.36 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.68 \[ \int \frac {\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {19140 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {11580 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {30720 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {5 \, {\left (4825 \, \sin \left (d x + c\right )^{4} - 20860 \, \sin \left (d x + c\right )^{3} + 34074 \, \sin \left (d x + c\right )^{2} - 24996 \, \sin \left (d x + c\right ) + 6981\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {43703 \, \sin \left (d x + c\right )^{5} + 233875 \, \sin \left (d x + c\right )^{4} + 504050 \, \sin \left (d x + c\right )^{3} + 548250 \, \sin \left (d x + c\right )^{2} + 302175 \, \sin \left (d x + c\right ) + 67995}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
-1/30720*(19140*log(abs(sin(d*x + c) + 1))/a + 11580*log(abs(sin(d*x + c) - 1))/a - 30720*log(abs(sin(d*x + c)))/a - 5*(4825*sin(d*x + c)^4 - 20860* sin(d*x + c)^3 + 34074*sin(d*x + c)^2 - 24996*sin(d*x + c) + 6981)/(a*(sin (d*x + c) - 1)^4) - (43703*sin(d*x + c)^5 + 233875*sin(d*x + c)^4 + 504050 *sin(d*x + c)^3 + 548250*sin(d*x + c)^2 + 302175*sin(d*x + c) + 67995)/(a* (sin(d*x + c) + 1)^5))/d
Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.94 \[ \int \frac {\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {63\,{\sin \left (c+d\,x\right )}^8}{256}-\frac {65\,{\sin \left (c+d\,x\right )}^7}{256}-\frac {359\,{\sin \left (c+d\,x\right )}^6}{256}+\frac {217\,{\sin \left (c+d\,x\right )}^5}{256}+\frac {3773\,{\sin \left (c+d\,x\right )}^4}{1280}-\frac {3721\,{\sin \left (c+d\,x\right )}^3}{3840}-\frac {10831\,{\sin \left (c+d\,x\right )}^2}{3840}+\frac {1489\,\sin \left (c+d\,x\right )}{3840}+\frac {137}{120}}{d\,\left (a\,{\sin \left (c+d\,x\right )}^9+a\,{\sin \left (c+d\,x\right )}^8-4\,a\,{\sin \left (c+d\,x\right )}^7-4\,a\,{\sin \left (c+d\,x\right )}^6+6\,a\,{\sin \left (c+d\,x\right )}^5+6\,a\,{\sin \left (c+d\,x\right )}^4-4\,a\,{\sin \left (c+d\,x\right )}^3-4\,a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )+a\right )}-\frac {319\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{512\,a\,d}-\frac {193\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{512\,a\,d}+\frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \]
((1489*sin(c + d*x))/3840 - (10831*sin(c + d*x)^2)/3840 - (3721*sin(c + d* x)^3)/3840 + (3773*sin(c + d*x)^4)/1280 + (217*sin(c + d*x)^5)/256 - (359* sin(c + d*x)^6)/256 - (65*sin(c + d*x)^7)/256 + (63*sin(c + d*x)^8)/256 + 137/120)/(d*(a + a*sin(c + d*x) - 4*a*sin(c + d*x)^2 - 4*a*sin(c + d*x)^3 + 6*a*sin(c + d*x)^4 + 6*a*sin(c + d*x)^5 - 4*a*sin(c + d*x)^6 - 4*a*sin(c + d*x)^7 + a*sin(c + d*x)^8 + a*sin(c + d*x)^9)) - (319*log(sin(c + d*x) + 1))/(512*a*d) - (193*log(sin(c + d*x) - 1))/(512*a*d) + log(sin(c + d*x) )/(a*d)