Integrand size = 27, antiderivative size = 233 \[ \int \frac {\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac {\log (\sin (c+d x))}{a d}-\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}+\frac {b^6 \log (a+b \sin (c+d x))}{a \left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\sin (c+d x))^2}+\frac {5 a+7 b}{16 (a+b)^2 d (1-\sin (c+d x))}+\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}+\frac {5 a-7 b}{16 (a-b)^2 d (1+\sin (c+d x))} \]
-1/16*(8*a^2+21*a*b+15*b^2)*ln(1-sin(d*x+c))/(a+b)^3/d+ln(sin(d*x+c))/a/d- 1/16*(8*a^2-21*a*b+15*b^2)*ln(1+sin(d*x+c))/(a-b)^3/d+b^6*ln(a+b*sin(d*x+c ))/a/(a^2-b^2)^3/d+1/16/(a+b)/d/(1-sin(d*x+c))^2+1/16*(5*a+7*b)/(a+b)^2/d/ (1-sin(d*x+c))+1/16/(a-b)/d/(1+sin(d*x+c))^2+1/16*(5*a-7*b)/(a-b)^2/d/(1+s in(d*x+c))
Time = 1.98 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.94 \[ \int \frac {\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b^6 \left (-\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\sin (c+d x))}{b^6 (a+b)^3}+\frac {16 \log (\sin (c+d x))}{a b^6}-\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\sin (c+d x))}{(a-b)^3 b^6}+\frac {16 \log (a+b \sin (c+d x))}{a (a-b)^3 (a+b)^3}+\frac {1}{b^6 (a+b) (-1+\sin (c+d x))^2}+\frac {-5 a-7 b}{b^6 (a+b)^2 (-1+\sin (c+d x))}+\frac {1}{(a-b) b^6 (1+\sin (c+d x))^2}+\frac {5 a-7 b}{(a-b)^2 b^6 (1+\sin (c+d x))}\right )}{16 d} \]
(b^6*(-(((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(b^6*(a + b)^3)) + (16*Log[Sin[c + d*x]])/(a*b^6) - ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sin [c + d*x]])/((a - b)^3*b^6) + (16*Log[a + b*Sin[c + d*x]])/(a*(a - b)^3*(a + b)^3) + 1/(b^6*(a + b)*(-1 + Sin[c + d*x])^2) + (-5*a - 7*b)/(b^6*(a + b)^2*(-1 + Sin[c + d*x])) + 1/((a - b)*b^6*(1 + Sin[c + d*x])^2) + (5*a - 7*b)/((a - b)^2*b^6*(1 + Sin[c + d*x]))))/(16*d)
Time = 0.53 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3316, 27, 615, 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.
| \(\displaystyle \int \frac {\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x) \cos (c+d x)^5 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {b^5 \int \frac {\csc (c+d x)}{(a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^6 \int \frac {\csc (c+d x)}{b (a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \frac {b^6 \int \left (\frac {7 b-5 a}{16 (a-b)^2 b^5 (\sin (c+d x) b+b)^2}+\frac {\csc (c+d x)}{a b^7}+\frac {8 a^2+21 b a+15 b^2}{16 b^6 (a+b)^3 (b-b \sin (c+d x))}+\frac {1}{a (a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+\frac {8 a^2-21 b a+15 b^2}{16 b^6 (b-a)^3 (\sin (c+d x) b+b)}+\frac {5 a+7 b}{16 b^5 (a+b)^2 (b-b \sin (c+d x))^2}+\frac {1}{8 b^4 (a+b) (b-b \sin (c+d x))^3}+\frac {1}{8 b^4 (b-a) (\sin (c+d x) b+b)^3}\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^6 \left (\frac {\log (a+b \sin (c+d x))}{a \left (a^2-b^2\right )^3}-\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (b-b \sin (c+d x))}{16 b^6 (a+b)^3}-\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (b \sin (c+d x)+b)}{16 b^6 (a-b)^3}+\frac {\log (b \sin (c+d x))}{a b^6}+\frac {5 a-7 b}{16 b^5 (a-b)^2 (b \sin (c+d x)+b)}+\frac {5 a+7 b}{16 b^5 (a+b)^2 (b-b \sin (c+d x))}+\frac {1}{16 b^4 (a+b) (b-b \sin (c+d x))^2}+\frac {1}{16 b^4 (a-b) (b \sin (c+d x)+b)^2}\right )}{d}\) |
(b^6*(Log[b*Sin[c + d*x]]/(a*b^6) - ((8*a^2 + 21*a*b + 15*b^2)*Log[b - b*S in[c + d*x]])/(16*b^6*(a + b)^3) + Log[a + b*Sin[c + d*x]]/(a*(a^2 - b^2)^ 3) - ((8*a^2 - 21*a*b + 15*b^2)*Log[b + b*Sin[c + d*x]])/(16*(a - b)^3*b^6 ) + 1/(16*b^4*(a + b)*(b - b*Sin[c + d*x])^2) + (5*a + 7*b)/(16*b^5*(a + b )^2*(b - b*Sin[c + d*x])) + 1/(16*(a - b)*b^4*(b + b*Sin[c + d*x])^2) + (5 *a - 7*b)/(16*(a - b)^2*b^5*(b + b*Sin[c + d*x]))))/d
3.14.67.3.1 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[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 1.14 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{6} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a}+\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-5 a +7 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-8 a^{2}+21 a b -15 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}+\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 a +7 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-8 a^{2}-21 a b -15 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a}}{d}\) | \(203\) |
| default | \(\frac {\frac {b^{6} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a}+\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-5 a +7 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-8 a^{2}+21 a b -15 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}+\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 a +7 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-8 a^{2}-21 a b -15 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a}}{d}\) | \(203\) |
| parallelrisch | \(\frac {4 b^{6} \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-4 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}+\frac {21}{8} a b +\frac {15}{8} b^{2}\right ) \left (a -b \right )^{3} a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+4 \left (a +b \right ) \left (-\left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}-\frac {21}{8} a b +\frac {15}{8} b^{2}\right ) \left (a +b \right )^{2} a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{2} \left (a -b \right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (\left (a^{3}-a \,b^{2}\right ) \cos \left (2 d x +2 c \right )+\frac {\left (3 a^{3}-5 a \,b^{2}\right ) \cos \left (4 d x +4 c \right )}{4}+\frac {\left (3 a^{2} b -7 b^{3}\right ) \sin \left (3 d x +3 c \right )}{4}+\frac {\left (11 a^{2} b -15 b^{3}\right ) \sin \left (d x +c \right )}{4}-\frac {7 a^{3}}{4}+\frac {9 a \,b^{2}}{4}\right ) a}{4}\right ) \left (a -b \right )\right )}{d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \left (a +b \right )^{3} \left (a -b \right )^{3} a}\) | \(369\) |
| norman | \(\frac {-\frac {2 \left (-2 a^{3}+3 a \,b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (-2 a^{3}+3 a \,b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {2 \left (-2 a^{3}+4 a \,b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (3 a^{2}+b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {b \left (3 a^{2}+b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {b \left (5 a^{2}-9 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (5 a^{2}-9 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {b^{6} \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{a d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {\left (8 a^{2}-21 a b +15 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {\left (8 a^{2}+21 a b +15 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}\) | \(518\) |
| risch | \(-\frac {21 i a b c}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}+\frac {21 i a b c}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}-\frac {2 i b^{6} c}{a d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 i c}{d a}+\frac {15 i b^{2} x}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {i a^{2} x}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}+\frac {15 i b^{2} x}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {i a^{2} x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}-\frac {i \left (8 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-16 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-3 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+7 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+32 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-48 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-11 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+15 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+8 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-16 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+11 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-15 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-7 b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4} d}+\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 i b^{6} x}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}-\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a b}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}+\frac {i a^{2} c}{\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}-\frac {21 i a b x}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {15 i b^{2} c}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}+\frac {i a^{2} c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {21 i a b x}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 i b^{2} c}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {2 i x}{a}\) | \(1044\) |
1/d*(b^6/(a+b)^3/(a-b)^3/a*ln(a+b*sin(d*x+c))+1/2/(8*a-8*b)/(1+sin(d*x+c)) ^2-1/16*(-5*a+7*b)/(a-b)^2/(1+sin(d*x+c))+1/16/(a-b)^3*(-8*a^2+21*a*b-15*b ^2)*ln(1+sin(d*x+c))+1/2/(8*a+8*b)/(sin(d*x+c)-1)^2-1/16*(5*a+7*b)/(a+b)^2 /(sin(d*x+c)-1)+1/16/(a+b)^3*(-8*a^2-21*a*b-15*b^2)*ln(sin(d*x+c)-1)+1/a*l n(sin(d*x+c)))
Time = 1.57 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.48 \[ \int \frac {\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {16 \, b^{6} \cos \left (d x + c\right )^{4} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \, a^{6} - 8 \, a^{4} b^{2} + 4 \, a^{2} b^{4} + 16 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{4} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left (8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 2 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (2 \, a^{5} b - 4 \, a^{3} b^{3} + 2 \, a b^{5} + {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )^{4}} \]
1/16*(16*b^6*cos(d*x + c)^4*log(b*sin(d*x + c) + a) + 4*a^6 - 8*a^4*b^2 + 4*a^2*b^4 + 16*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(d*x + c)^4*log(-1/2 *sin(d*x + c)) - (8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - (8*a^6 - 3*a^5*b - 24*a^ 4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 8*(a^6 - 3*a^4*b^2 + 2*a^2*b^4)*cos(d*x + c)^2 - 2*(2*a^5*b - 4* a^3*b^3 + 2*a*b^5 + (3*a^5*b - 10*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^2)*sin(d *x + c))/((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c)^4)
Timed out. \[ \int \frac {\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.28 \[ \int \frac {\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {16 \, b^{6} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} - \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left ({\left (3 \, a^{2} b - 7 \, b^{3}\right )} \sin \left (d x + c\right )^{3} + 6 \, a^{3} - 10 \, a b^{2} - 4 \, {\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (d x + c\right )^{2} - {\left (5 \, a^{2} b - 9 \, b^{3}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}} + \frac {16 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{16 \, d} \]
1/16*(16*b^6*log(b*sin(d*x + c) + a)/(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6) - (8*a^2 - 21*a*b + 15*b^2)*log(sin(d*x + c) + 1)/(a^3 - 3*a^2*b + 3*a*b^ 2 - b^3) - (8*a^2 + 21*a*b + 15*b^2)*log(sin(d*x + c) - 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 2*((3*a^2*b - 7*b^3)*sin(d*x + c)^3 + 6*a^3 - 10*a*b^2 - 4*(a^3 - 2*a*b^2)*sin(d*x + c)^2 - (5*a^2*b - 9*b^3)*sin(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^ 2*b^2 + b^4)*sin(d*x + c)^2) + 16*log(sin(d*x + c))/a)/d
Time = 0.42 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.68 \[ \int \frac {\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {16 \, b^{7} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}} - \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {16 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {2 \, {\left (6 \, a^{5} \sin \left (d x + c\right )^{4} - 18 \, a^{3} b^{2} \sin \left (d x + c\right )^{4} + 18 \, a b^{4} \sin \left (d x + c\right )^{4} + 3 \, a^{4} b \sin \left (d x + c\right )^{3} - 10 \, a^{2} b^{3} \sin \left (d x + c\right )^{3} + 7 \, b^{5} \sin \left (d x + c\right )^{3} - 16 \, a^{5} \sin \left (d x + c\right )^{2} + 48 \, a^{3} b^{2} \sin \left (d x + c\right )^{2} - 44 \, a b^{4} \sin \left (d x + c\right )^{2} - 5 \, a^{4} b \sin \left (d x + c\right ) + 14 \, a^{2} b^{3} \sin \left (d x + c\right ) - 9 \, b^{5} \sin \left (d x + c\right ) + 12 \, a^{5} - 34 \, a^{3} b^{2} + 28 \, a b^{4}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
1/16*(16*b^7*log(abs(b*sin(d*x + c) + a))/(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7) - (8*a^2 - 21*a*b + 15*b^2)*log(abs(sin(d*x + c) + 1))/(a^3 - 3*a^ 2*b + 3*a*b^2 - b^3) - (8*a^2 + 21*a*b + 15*b^2)*log(abs(sin(d*x + c) - 1) )/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 16*log(abs(sin(d*x + c)))/a + 2*(6*a^5 *sin(d*x + c)^4 - 18*a^3*b^2*sin(d*x + c)^4 + 18*a*b^4*sin(d*x + c)^4 + 3* a^4*b*sin(d*x + c)^3 - 10*a^2*b^3*sin(d*x + c)^3 + 7*b^5*sin(d*x + c)^3 - 16*a^5*sin(d*x + c)^2 + 48*a^3*b^2*sin(d*x + c)^2 - 44*a*b^4*sin(d*x + c)^ 2 - 5*a^4*b*sin(d*x + c) + 14*a^2*b^3*sin(d*x + c) - 9*b^5*sin(d*x + c) + 12*a^5 - 34*a^3*b^2 + 28*a*b^4)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(sin( d*x + c)^2 - 1)^2))/d
Time = 12.29 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.48 \[ \int \frac {\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {5\,b}{16\,{\left (a+b\right )}^2}+\frac {1}{2\,\left (a+b\right )}+\frac {b^2}{8\,{\left (a+b\right )}^3}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {b^2}{8\,{\left (a-b\right )}^3}-\frac {5\,b}{16\,{\left (a-b\right )}^2}+\frac {1}{2\,\left (a-b\right )}\right )}{d}-\frac {\frac {5\,a\,b^2-3\,a^3}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (2\,a\,b^2-a^3\right )}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (3\,a^2\,b-7\,b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {b\,\sin \left (c+d\,x\right )\,\left (5\,a^2-9\,b^2\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2+{\sin \left (c+d\,x\right )}^4-{\sin \left (c+d\,x\right )}^2\right )}-\frac {b^6\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (-a^7+3\,a^5\,b^2-3\,a^3\,b^4+a\,b^6\right )} \]
log(sin(c + d*x))/(a*d) - (log(sin(c + d*x) - 1)*((5*b)/(16*(a + b)^2) + 1 /(2*(a + b)) + b^2/(8*(a + b)^3)))/d - (log(sin(c + d*x) + 1)*(b^2/(8*(a - b)^3) - (5*b)/(16*(a - b)^2) + 1/(2*(a - b))))/d - ((5*a*b^2 - 3*a^3)/(4* (a^4 + b^4 - 2*a^2*b^2)) - (sin(c + d*x)^2*(2*a*b^2 - a^3))/(2*(a^4 + b^4 - 2*a^2*b^2)) - (sin(c + d*x)^3*(3*a^2*b - 7*b^3))/(8*(a^4 + b^4 - 2*a^2*b ^2)) + (b*sin(c + d*x)*(5*a^2 - 9*b^2))/(8*(a^4 + b^4 - 2*a^2*b^2)))/(d*(c os(c + d*x)^2 - sin(c + d*x)^2 + sin(c + d*x)^4)) - (b^6*log(a + b*sin(c + d*x)))/(d*(a*b^6 - a^7 - 3*a^3*b^4 + 3*a^5*b^2))