Integrand size = 29, antiderivative size = 197 \[ \int \frac {\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \csc (c+d x)}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {(4 a+5 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {\left (2 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {(4 a-5 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}-\frac {b^6 \log (a+b \sin (c+d x))}{a^3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}+\frac {1}{4 (a-b) d (1+\sin (c+d x))} \]
b*csc(d*x+c)/a^2/d-1/2*csc(d*x+c)^2/a/d-1/4*(4*a+5*b)*ln(1-sin(d*x+c))/(a+ b)^2/d+(2*a^2+b^2)*ln(sin(d*x+c))/a^3/d-1/4*(4*a-5*b)*ln(1+sin(d*x+c))/(a- b)^2/d-b^6*ln(a+b*sin(d*x+c))/a^3/(a^2-b^2)^2/d+1/4/(a+b)/d/(1-sin(d*x+c)) +1/4/(a-b)/d/(1+sin(d*x+c))
Time = 0.97 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85 \[ \int \frac {\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {-\frac {4 b \csc (c+d x)}{a^2}+\frac {2 \csc ^2(c+d x)}{a}+\frac {(4 a+5 b) \log (1-\sin (c+d x))}{(a+b)^2}-\frac {4 \left (2 a^2+b^2\right ) \log (\sin (c+d x))}{a^3}+\frac {(4 a-5 b) \log (1+\sin (c+d x))}{(a-b)^2}+\frac {4 b^6 \log (a+b \sin (c+d x))}{a^3 (a-b)^2 (a+b)^2}+\frac {1}{(a+b) (-1+\sin (c+d x))}-\frac {1}{(a-b) (1+\sin (c+d x))}}{4 d} \]
-1/4*((-4*b*Csc[c + d*x])/a^2 + (2*Csc[c + d*x]^2)/a + ((4*a + 5*b)*Log[1 - Sin[c + d*x]])/(a + b)^2 - (4*(2*a^2 + b^2)*Log[Sin[c + d*x]])/a^3 + ((4 *a - 5*b)*Log[1 + Sin[c + d*x]])/(a - b)^2 + (4*b^6*Log[a + b*Sin[c + d*x] ])/(a^3*(a - b)^2*(a + b)^2) + 1/((a + b)*(-1 + Sin[c + d*x])) - 1/((a - b )*(1 + Sin[c + d*x])))/d
Time = 0.48 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, 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.
| \(\displaystyle \int \frac {\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^3 \cos (c+d x)^3 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {b^3 \int \frac {\csc ^3(c+d x)}{(a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^6 \int \frac {\csc ^3(c+d x)}{b^3 (a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \frac {b^6 \int \left (\frac {\csc ^3(c+d x)}{a b^7}-\frac {\csc ^2(c+d x)}{a^2 b^6}+\frac {\left (2 a^2+b^2\right ) \csc (c+d x)}{a^3 b^7}+\frac {4 a+5 b}{4 b^6 (a+b)^2 (b-b \sin (c+d x))}-\frac {1}{a^3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac {5 b-4 a}{4 (a-b)^2 b^6 (\sin (c+d x) b+b)}+\frac {1}{4 b^5 (a+b) (b-b \sin (c+d x))^2}-\frac {1}{4 (a-b) b^5 (\sin (c+d x) b+b)^2}\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^6 \left (\frac {\csc (c+d x)}{a^2 b^5}-\frac {\log (a+b \sin (c+d x))}{a^3 \left (a^2-b^2\right )^2}+\frac {\left (2 a^2+b^2\right ) \log (b \sin (c+d x))}{a^3 b^6}-\frac {\csc ^2(c+d x)}{2 a b^6}-\frac {(4 a+5 b) \log (b-b \sin (c+d x))}{4 b^6 (a+b)^2}-\frac {(4 a-5 b) \log (b \sin (c+d x)+b)}{4 b^6 (a-b)^2}+\frac {1}{4 b^5 (a+b) (b-b \sin (c+d x))}+\frac {1}{4 b^5 (a-b) (b \sin (c+d x)+b)}\right )}{d}\) |
(b^6*(Csc[c + d*x]/(a^2*b^5) - Csc[c + d*x]^2/(2*a*b^6) + ((2*a^2 + b^2)*L og[b*Sin[c + d*x]])/(a^3*b^6) - ((4*a + 5*b)*Log[b - b*Sin[c + d*x]])/(4*b ^6*(a + b)^2) - Log[a + b*Sin[c + d*x]]/(a^3*(a^2 - b^2)^2) - ((4*a - 5*b) *Log[b + b*Sin[c + d*x]])/(4*(a - b)^2*b^6) + 1/(4*b^5*(a + b)*(b - b*Sin[ c + d*x])) + 1/(4*(a - b)*b^5*(b + b*Sin[c + d*x]))))/d
3.14.51.3.1 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 = 1.05 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-4 a +5 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-4 a -5 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {b^{6} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {1}{2 a \sin \left (d x +c \right )^{2}}+\frac {\left (2 a^{2}+b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{3}}+\frac {b}{a^{2} \sin \left (d x +c \right )}}{d}\) | \(172\) |
| default | \(\frac {\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-4 a +5 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-4 a -5 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {b^{6} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {1}{2 a \sin \left (d x +c \right )^{2}}+\frac {\left (2 a^{2}+b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{3}}+\frac {b}{a^{2} \sin \left (d x +c \right )}}{d}\) | \(172\) |
| norman | \(\frac {-\frac {1}{8 a d}-\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2} d}+\frac {b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} d}-\frac {\left (-9 a^{2}+b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (a^{2}-b^{2}\right )}-\frac {b \left (3 a^{2}-b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} d \left (a^{2}-b^{2}\right )}-\frac {b \left (3 a^{2}-b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} d \left (a^{2}-b^{2}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (2 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}-\frac {\left (4 a -5 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (4 a +5 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {b^{6} \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{a^{3} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}\) | \(374\) |
| parallelrisch | \(\frac {-4 b^{6} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-8 \left (a +\frac {5 b}{4}\right ) \left (a -b \right )^{2} \left (1+\cos \left (2 d x +2 c \right )\right ) a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-8 \left (a -\frac {5 b}{4}\right ) \left (a +b \right )^{2} \left (1+\cos \left (2 d x +2 c \right )\right ) a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+8 \left (a^{2}+\frac {b^{2}}{2}\right ) \left (a +b \right )^{2} \left (a -b \right )^{2} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\left (\left (\left (a^{4}-\frac {3}{2} a^{2} b^{2}\right ) \cos \left (2 d x +2 c \right )+\frac {\left (-a^{4}+a^{2} b^{2}\right ) \cos \left (4 d x +4 c \right )}{8}+\frac {a^{4}}{8}-\frac {5 a^{2} b^{2}}{8}+b^{4}\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 b^{4}\right ) a \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b \left (\left (a^{2}-\frac {2 b^{2}}{3}\right ) \cos \left (2 d x +2 c \right )+\frac {a^{2}}{3}-\frac {2 b^{2}}{3}\right ) \left (a +b \right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+4 a \,b^{4} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a}{4 \left (a -b \right )^{2} \left (a +b \right )^{2} a^{3} d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(374\) |
| risch | \(\frac {2 i a x}{a^{2}-2 a b +b^{2}}+\frac {2 i a c}{\left (a^{2}-2 a b +b^{2}\right ) d}-\frac {5 i b c}{2 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 i a x}{a^{2}+2 a b +b^{2}}+\frac {2 i b^{6} x}{a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i b^{2} c}{a^{3} d}-\frac {5 i b x}{2 \left (a^{2}-2 a b +b^{2}\right )}-\frac {4 i x}{a}+\frac {5 i b c}{2 \left (a^{2}+2 a b +b^{2}\right ) d}+\frac {5 i b x}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {i \left (4 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-2 i b^{2} a \,{\mathrm e}^{6 i \left (d x +c \right )}-3 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-4 i b^{2} a \,{\mathrm e}^{4 i \left (d x +c \right )}+a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+4 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-2 i b^{2} a \,{\mathrm e}^{2 i \left (d x +c \right )}-a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-2 b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (-a^{2}+b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {2 i b^{2} x}{a^{3}}+\frac {2 i b^{6} c}{a^{3} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {4 i c}{d a}+\frac {2 i a c}{d \left (a^{2}+2 a b +b^{2}\right )}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a}{\left (a^{2}+2 a b +b^{2}\right ) d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b}{2 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a}{\left (a^{2}-2 a b +b^{2}\right ) d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b}{2 \left (a^{2}-2 a b +b^{2}\right ) 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^{3} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{3} d}\) | \(721\) |
1/d*(1/(4*a-4*b)/(1+sin(d*x+c))+1/4/(a-b)^2*(-4*a+5*b)*ln(1+sin(d*x+c))-1/ (4*a+4*b)/(sin(d*x+c)-1)+1/4/(a+b)^2*(-4*a-5*b)*ln(sin(d*x+c)-1)-b^6/(a+b) ^2/(a-b)^2/a^3*ln(a+b*sin(d*x+c))-1/2/a/sin(d*x+c)^2+(2*a^2+b^2)/a^3*ln(si n(d*x+c))+1/a^2*b/sin(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (185) = 370\).
Time = 1.24 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.23 \[ \int \frac {\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 \, a^{6} - 2 \, a^{4} b^{2} - 2 \, {\left (2 \, a^{6} - 3 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (b^{6} \cos \left (d x + c\right )^{4} - b^{6} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \, {\left ({\left (2 \, a^{6} - 3 \, a^{4} b^{2} + b^{6}\right )} \cos \left (d x + c\right )^{4} - {\left (2 \, a^{6} - 3 \, a^{4} b^{2} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) + {\left ({\left (4 \, a^{6} + 3 \, a^{5} b - 6 \, a^{4} b^{2} - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} - {\left (4 \, a^{6} + 3 \, a^{5} b - 6 \, a^{4} b^{2} - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (4 \, a^{6} - 3 \, a^{5} b - 6 \, a^{4} b^{2} + 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} - {\left (4 \, a^{6} - 3 \, a^{5} b - 6 \, a^{4} b^{2} + 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a^{5} b - a^{3} b^{3} - {\left (3 \, a^{5} b - 5 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d x + c\right )^{4} - {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d x + c\right )^{2}\right )}} \]
-1/4*(2*a^6 - 2*a^4*b^2 - 2*(2*a^6 - 3*a^4*b^2 + a^2*b^4)*cos(d*x + c)^2 + 4*(b^6*cos(d*x + c)^4 - b^6*cos(d*x + c)^2)*log(b*sin(d*x + c) + a) - 4*( (2*a^6 - 3*a^4*b^2 + b^6)*cos(d*x + c)^4 - (2*a^6 - 3*a^4*b^2 + b^6)*cos(d *x + c)^2)*log(-1/2*sin(d*x + c)) + ((4*a^6 + 3*a^5*b - 6*a^4*b^2 - 5*a^3* b^3)*cos(d*x + c)^4 - (4*a^6 + 3*a^5*b - 6*a^4*b^2 - 5*a^3*b^3)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + ((4*a^6 - 3*a^5*b - 6*a^4*b^2 + 5*a^3*b^3)*c os(d*x + c)^4 - (4*a^6 - 3*a^5*b - 6*a^4*b^2 + 5*a^3*b^3)*cos(d*x + c)^2)* log(-sin(d*x + c) + 1) - 2*(a^5*b - a^3*b^3 - (3*a^5*b - 5*a^3*b^3 + 2*a*b ^5)*cos(d*x + c)^2)*sin(d*x + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c)^4 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c)^2)
Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.24 \[ \int \frac {\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {4 \, b^{6} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}} + \frac {{\left (4 \, a - 5 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {{\left (4 \, a + 5 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {2 \, {\left ({\left (3 \, a^{2} b - 2 \, b^{3}\right )} \sin \left (d x + c\right )^{3} + a^{3} - a b^{2} - {\left (2 \, a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{2} - 2 \, {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{4} - a^{2} b^{2}\right )} \sin \left (d x + c\right )^{4} - {\left (a^{4} - a^{2} b^{2}\right )} \sin \left (d x + c\right )^{2}} - \frac {4 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{4 \, d} \]
-1/4*(4*b^6*log(b*sin(d*x + c) + a)/(a^7 - 2*a^5*b^2 + a^3*b^4) + (4*a - 5 *b)*log(sin(d*x + c) + 1)/(a^2 - 2*a*b + b^2) + (4*a + 5*b)*log(sin(d*x + c) - 1)/(a^2 + 2*a*b + b^2) - 2*((3*a^2*b - 2*b^3)*sin(d*x + c)^3 + a^3 - a*b^2 - (2*a^3 - a*b^2)*sin(d*x + c)^2 - 2*(a^2*b - b^3)*sin(d*x + c))/((a ^4 - a^2*b^2)*sin(d*x + c)^4 - (a^4 - a^2*b^2)*sin(d*x + c)^2) - 4*(2*a^2 + b^2)*log(sin(d*x + c))/a^3)/d
Time = 0.39 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.40 \[ \int \frac {\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {4 \, b^{7} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}} + \frac {{\left (4 \, a - 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {{\left (4 \, a + 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {2 \, {\left (2 \, a^{3} \sin \left (d x + c\right )^{2} - 3 \, a b^{2} \sin \left (d x + c\right )^{2} + a^{2} b \sin \left (d x + c\right ) - b^{3} \sin \left (d x + c\right ) - 3 \, a^{3} + 4 \, a b^{2}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}} - \frac {4 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (6 \, a^{2} \sin \left (d x + c\right )^{2} + 3 \, b^{2} \sin \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) + a^{2}\right )}}{a^{3} \sin \left (d x + c\right )^{2}}}{4 \, d} \]
-1/4*(4*b^7*log(abs(b*sin(d*x + c) + a))/(a^7*b - 2*a^5*b^3 + a^3*b^5) + ( 4*a - 5*b)*log(abs(sin(d*x + c) + 1))/(a^2 - 2*a*b + b^2) + (4*a + 5*b)*lo g(abs(sin(d*x + c) - 1))/(a^2 + 2*a*b + b^2) - 2*(2*a^3*sin(d*x + c)^2 - 3 *a*b^2*sin(d*x + c)^2 + a^2*b*sin(d*x + c) - b^3*sin(d*x + c) - 3*a^3 + 4* a*b^2)/((a^4 - 2*a^2*b^2 + b^4)*(sin(d*x + c)^2 - 1)) - 4*(2*a^2 + b^2)*lo g(abs(sin(d*x + c)))/a^3 + 2*(6*a^2*sin(d*x + c)^2 + 3*b^2*sin(d*x + c)^2 - 2*a*b*sin(d*x + c) + a^2)/(a^3*sin(d*x + c)^2))/d
Time = 12.06 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.22 \[ \int \frac {\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {b}{4\,{\left (a-b\right )}^2}-\frac {1}{a-b}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {b}{4\,{\left (a+b\right )}^2}+\frac {1}{a+b}\right )}{d}-\frac {\frac {1}{2\,a}-\frac {b\,\sin \left (c+d\,x\right )}{a^2}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (2\,a^2-b^2\right )}{2\,a\,\left (a^2-b^2\right )}+\frac {b\,{\sin \left (c+d\,x\right )}^3\,\left (3\,a^2-2\,b^2\right )}{2\,a^2\,\left (a^2-b^2\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^2-{\sin \left (c+d\,x\right )}^4\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )\right )\,\left (2\,a^2+b^2\right )}{a^3\,d}-\frac {b^6\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )} \]
(log(sin(c + d*x) + 1)*(b/(4*(a - b)^2) - 1/(a - b)))/d - (log(sin(c + d*x ) - 1)*(b/(4*(a + b)^2) + 1/(a + b)))/d - (1/(2*a) - (b*sin(c + d*x))/a^2 - (sin(c + d*x)^2*(2*a^2 - b^2))/(2*a*(a^2 - b^2)) + (b*sin(c + d*x)^3*(3* a^2 - 2*b^2))/(2*a^2*(a^2 - b^2)))/(d*(sin(c + d*x)^2 - sin(c + d*x)^4)) + (log(sin(c + d*x))*(2*a^2 + b^2))/(a^3*d) - (b^6*log(a + b*sin(c + d*x))) /(d*(a^7 + a^3*b^4 - 2*a^5*b^2))