Integrand size = 27, antiderivative size = 202 \[ \int \frac {\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {93 \log (1-\sin (c+d x))}{256 a d}+\frac {\log (\sin (c+d x))}{a d}-\frac {163 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {7 a}{128 d (a-a \sin (c+d x))^2}+\frac {29}{128 d (a-a \sin (c+d x))}+\frac {a^3}{64 d (a+a \sin (c+d x))^4}+\frac {a^2}{16 d (a+a \sin (c+d x))^3}+\frac {11 a}{64 d (a+a \sin (c+d x))^2}+\frac {1}{2 d (a+a \sin (c+d x))} \]
-93/256*ln(1-sin(d*x+c))/a/d+ln(sin(d*x+c))/a/d-163/256*ln(1+sin(d*x+c))/a /d+1/96*a^2/d/(a-a*sin(d*x+c))^3+7/128*a/d/(a-a*sin(d*x+c))^2+29/128/d/(a- a*sin(d*x+c))+1/64*a^3/d/(a+a*sin(d*x+c))^4+1/16*a^2/d/(a+a*sin(d*x+c))^3+ 11/64*a/d/(a+a*sin(d*x+c))^2+1/2/d/(a+a*sin(d*x+c))
Time = 6.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94 \[ \int \frac {\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^7 \left (-\frac {93 \log (1-\sin (c+d x))}{256 a^8}+\frac {\log (\sin (c+d x))}{a^8}-\frac {163 \log (1+\sin (c+d x))}{256 a^8}+\frac {1}{96 a^5 (a-a \sin (c+d x))^3}+\frac {7}{128 a^6 (a-a \sin (c+d x))^2}+\frac {29}{128 a^7 (a-a \sin (c+d x))}+\frac {1}{64 a^4 (a+a \sin (c+d x))^4}+\frac {1}{16 a^5 (a+a \sin (c+d x))^3}+\frac {11}{64 a^6 (a+a \sin (c+d x))^2}+\frac {1}{2 a^7 (a+a \sin (c+d x))}\right )}{d} \]
(a^7*((-93*Log[1 - Sin[c + d*x]])/(256*a^8) + Log[Sin[c + d*x]]/a^8 - (163 *Log[1 + Sin[c + d*x]])/(256*a^8) + 1/(96*a^5*(a - a*Sin[c + d*x])^3) + 7/ (128*a^6*(a - a*Sin[c + d*x])^2) + 29/(128*a^7*(a - a*Sin[c + d*x])) + 1/( 64*a^4*(a + a*Sin[c + d*x])^4) + 1/(16*a^5*(a + a*Sin[c + d*x])^3) + 11/(6 4*a^6*(a + a*Sin[c + d*x])^2) + 1/(2*a^7*(a + a*Sin[c + d*x]))))/d
Time = 0.40 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.96, 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.
| \(\displaystyle \int \frac {\csc (c+d x) \sec ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x) \cos (c+d x)^7 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^7 \int \frac {\csc (c+d x)}{(a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^8 \int \frac {\csc (c+d x)}{a (a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^8 \int \left (\frac {\csc (c+d x)}{a^{10}}+\frac {93}{256 a^9 (a-a \sin (c+d x))}-\frac {163}{256 a^9 (\sin (c+d x) a+a)}+\frac {29}{128 a^8 (a-a \sin (c+d x))^2}-\frac {1}{2 a^8 (\sin (c+d x) a+a)^2}+\frac {7}{64 a^7 (a-a \sin (c+d x))^3}-\frac {11}{32 a^7 (\sin (c+d x) a+a)^3}+\frac {1}{32 a^6 (a-a \sin (c+d x))^4}-\frac {3}{16 a^6 (\sin (c+d x) a+a)^4}-\frac {1}{16 a^5 (\sin (c+d x) a+a)^5}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^8 \left (\frac {\log (a \sin (c+d x))}{a^9}-\frac {93 \log (a-a \sin (c+d x))}{256 a^9}-\frac {163 \log (a \sin (c+d x)+a)}{256 a^9}+\frac {29}{128 a^8 (a-a \sin (c+d x))}+\frac {1}{2 a^8 (a \sin (c+d x)+a)}+\frac {7}{128 a^7 (a-a \sin (c+d x))^2}+\frac {11}{64 a^7 (a \sin (c+d x)+a)^2}+\frac {1}{96 a^6 (a-a \sin (c+d x))^3}+\frac {1}{16 a^6 (a \sin (c+d x)+a)^3}+\frac {1}{64 a^5 (a \sin (c+d x)+a)^4}\right )}{d}\) |
(a^8*(Log[a*Sin[c + d*x]]/a^9 - (93*Log[a - a*Sin[c + d*x]])/(256*a^9) - ( 163*Log[a + a*Sin[c + d*x]])/(256*a^9) + 1/(96*a^6*(a - a*Sin[c + d*x])^3) + 7/(128*a^7*(a - a*Sin[c + d*x])^2) + 29/(128*a^8*(a - a*Sin[c + d*x])) + 1/(64*a^5*(a + a*Sin[c + d*x])^4) + 1/(16*a^6*(a + a*Sin[c + d*x])^3) + 11/(64*a^7*(a + a*Sin[c + d*x])^2) + 1/(2*a^8*(a + a*Sin[c + d*x]))))/d
3.9.90.3.1 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 = 1.72 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.60
| method | result | size |
| derivativedivides | \(\frac {\ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {7}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {29}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {93 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {11}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 \sin \left (d x +c \right )+2}-\frac {163 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(122\) |
| default | \(\frac {\ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {7}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {29}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {93 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {11}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 \sin \left (d x +c \right )+2}-\frac {163 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(122\) |
| risch | \(\frac {i \left (5399 \,{\mathrm e}^{5 i \left (d x +c \right )}+1258 \,{\mathrm e}^{11 i \left (d x +c \right )}+105 \,{\mathrm e}^{13 i \left (d x +c \right )}+730 i {\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}+1258 \,{\mathrm e}^{3 i \left (d x +c \right )}+174 i {\mathrm e}^{2 i \left (d x +c \right )}-812 i {\mathrm e}^{8 i \left (d x +c \right )}+812 i {\mathrm e}^{6 i \left (d x +c \right )}-730 i {\mathrm e}^{10 i \left (d x +c \right )}+12076 \,{\mathrm e}^{7 i \left (d x +c \right )}+5399 \,{\mathrm e}^{9 i \left (d x +c \right )}-174 i {\mathrm e}^{12 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}-\frac {163 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}-\frac {93 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(250\) |
| norman | \(\frac {-\frac {93 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {93 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {163 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {163 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {437 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}+\frac {437 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}-\frac {467 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {467 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {1163 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {1163 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {2101 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}-\frac {5993 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {5993 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {93 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128 a d}-\frac {163 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128 a d}\) | \(332\) |
| parallelrisch | \(\frac {\left (-1395 \sin \left (5 d x +5 c \right )-279 \sin \left (7 d x +7 c \right )-8370 \cos \left (2 d x +2 c \right )-3348 \cos \left (4 d x +4 c \right )-558 \cos \left (6 d x +6 c \right )-1395 \sin \left (d x +c \right )-2511 \sin \left (3 d x +3 c \right )-5580\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-2445 \sin \left (5 d x +5 c \right )-489 \sin \left (7 d x +7 c \right )-14670 \cos \left (2 d x +2 c \right )-5868 \cos \left (4 d x +4 c \right )-978 \cos \left (6 d x +6 c \right )-2445 \sin \left (d x +c \right )-4401 \sin \left (3 d x +3 c \right )-9780\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1920 \sin \left (5 d x +5 c \right )+384 \sin \left (7 d x +7 c \right )+11520 \cos \left (2 d x +2 c \right )+4608 \cos \left (4 d x +4 c \right )+768 \cos \left (6 d x +6 c \right )+1920 \sin \left (d x +c \right )+3456 \sin \left (3 d x +3 c \right )+7680\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1652 \sin \left (5 d x +5 c \right )-400 \sin \left (7 d x +7 c \right )-1202 \cos \left (2 d x +2 c \right )-2284 \cos \left (4 d x +4 c \right )-590 \cos \left (6 d x +6 c \right )-376 \sin \left (d x +c \right )-2140 \sin \left (3 d x +3 c \right )+4076}{384 a d \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right )}\) | \(426\) |
1/d/a*(ln(sin(d*x+c))-1/96/(sin(d*x+c)-1)^3+7/128/(sin(d*x+c)-1)^2-29/128/ (sin(d*x+c)-1)-93/256*ln(sin(d*x+c)-1)+1/64/(1+sin(d*x+c))^4+1/16/(1+sin(d *x+c))^3+11/64/(1+sin(d*x+c))^2+1/2/(1+sin(d*x+c))-163/256*ln(1+sin(d*x+c) ))
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {210 \, \cos \left (d x + c\right )^{6} + 314 \, \cos \left (d x + c\right )^{4} + 164 \, \cos \left (d x + c\right )^{2} + 768 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 489 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 279 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (87 \, \cos \left (d x + c\right )^{4} + 26 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
1/768*(210*cos(d*x + c)^6 + 314*cos(d*x + c)^4 + 164*cos(d*x + c)^2 + 768* (cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(1/2*sin(d*x + c)) - 489 *(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) - 27 9*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(-sin(d*x + c) + 1) + 2*(87*cos(d*x + c)^4 + 26*cos(d*x + c)^2 + 8)*sin(d*x + c) + 112)/(a*d*cos (d*x + c)^6*sin(d*x + c) + a*d*cos(d*x + c)^6)
Timed out. \[ \int \frac {\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.93 \[ \int \frac {\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} - 87 \, \sin \left (d x + c\right )^{5} - 472 \, \sin \left (d x + c\right )^{4} + 200 \, \sin \left (d x + c\right )^{3} + 711 \, \sin \left (d x + c\right )^{2} - 121 \, \sin \left (d x + c\right ) - 400\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {489 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {279 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {768 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \]
1/768*(2*(105*sin(d*x + c)^6 - 87*sin(d*x + c)^5 - 472*sin(d*x + c)^4 + 20 0*sin(d*x + c)^3 + 711*sin(d*x + c)^2 - 121*sin(d*x + c) - 400)/(a*sin(d*x + c)^7 + a*sin(d*x + c)^6 - 3*a*sin(d*x + c)^5 - 3*a*sin(d*x + c)^4 + 3*a *sin(d*x + c)^3 + 3*a*sin(d*x + c)^2 - a*sin(d*x + c) - a) - 489*log(sin(d *x + c) + 1)/a - 279*log(sin(d*x + c) - 1)/a + 768*log(sin(d*x + c))/a)/d
Time = 0.42 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.74 \[ \int \frac {\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {1956 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {1116 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {3072 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {2 \, {\left (1023 \, \sin \left (d x + c\right )^{3} - 3417 \, \sin \left (d x + c\right )^{2} + 3849 \, \sin \left (d x + c\right ) - 1471\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {4075 \, \sin \left (d x + c\right )^{4} + 17836 \, \sin \left (d x + c\right )^{3} + 29586 \, \sin \left (d x + c\right )^{2} + 22156 \, \sin \left (d x + c\right ) + 6379}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
-1/3072*(1956*log(abs(sin(d*x + c) + 1))/a + 1116*log(abs(sin(d*x + c) - 1 ))/a - 3072*log(abs(sin(d*x + c)))/a - 2*(1023*sin(d*x + c)^3 - 3417*sin(d *x + c)^2 + 3849*sin(d*x + c) - 1471)/(a*(sin(d*x + c) - 1)^3) - (4075*sin (d*x + c)^4 + 17836*sin(d*x + c)^3 + 29586*sin(d*x + c)^2 + 22156*sin(d*x + c) + 6379)/(a*(sin(d*x + c) + 1)^4))/d
Time = 0.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.95 \[ \int \frac {\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d}-\frac {163\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {93\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}+\frac {-\frac {35\,{\sin \left (c+d\,x\right )}^6}{128}+\frac {29\,{\sin \left (c+d\,x\right )}^5}{128}+\frac {59\,{\sin \left (c+d\,x\right )}^4}{48}-\frac {25\,{\sin \left (c+d\,x\right )}^3}{48}-\frac {237\,{\sin \left (c+d\,x\right )}^2}{128}+\frac {121\,\sin \left (c+d\,x\right )}{384}+\frac {25}{24}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^7-a\,{\sin \left (c+d\,x\right )}^6+3\,a\,{\sin \left (c+d\,x\right )}^5+3\,a\,{\sin \left (c+d\,x\right )}^4-3\,a\,{\sin \left (c+d\,x\right )}^3-3\,a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )+a\right )} \]
log(sin(c + d*x))/(a*d) - (163*log(sin(c + d*x) + 1))/(256*a*d) - (93*log( sin(c + d*x) - 1))/(256*a*d) + ((121*sin(c + d*x))/384 - (237*sin(c + d*x) ^2)/128 - (25*sin(c + d*x)^3)/48 + (59*sin(c + d*x)^4)/48 + (29*sin(c + d* x)^5)/128 - (35*sin(c + d*x)^6)/128 + 25/24)/(d*(a + a*sin(c + d*x) - 3*a* sin(c + d*x)^2 - 3*a*sin(c + d*x)^3 + 3*a*sin(c + d*x)^4 + 3*a*sin(c + d*x )^5 - a*sin(c + d*x)^6 - a*sin(c + d*x)^7))