Integrand size = 29, antiderivative size = 274 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \csc (c+d x)}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {\left (24 a^2-57 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}+\frac {b^8 \log (a+b \sin (c+d x))}{a^3 \left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\sin (c+d x))^2}+\frac {9 a+11 b}{16 (a+b)^2 d (1-\sin (c+d x))}+\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}+\frac {9 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))} \]
b*csc(d*x+c)/a^2/d-1/2*csc(d*x+c)^2/a/d-1/16*(24*a^2+57*a*b+35*b^2)*ln(1-s in(d*x+c))/(a+b)^3/d+(3*a^2+b^2)*ln(sin(d*x+c))/a^3/d-1/16*(24*a^2-57*a*b+ 35*b^2)*ln(1+sin(d*x+c))/(a-b)^3/d+b^8*ln(a+b*sin(d*x+c))/a^3/(a^2-b^2)^3/ d+1/16/(a+b)/d/(1-sin(d*x+c))^2+1/16*(9*a+11*b)/(a+b)^2/d/(1-sin(d*x+c))+1 /16/(a-b)/d/(1+sin(d*x+c))^2+1/16*(9*a-11*b)/(a-b)^2/d/(1+sin(d*x+c))
Time = 6.17 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b^8 \left (\frac {\csc (c+d x)}{a^2 b^7}-\frac {\csc ^2(c+d x)}{2 a b^8}-\frac {\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 b^8 (a+b)^3}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 b^8}-\frac {\left (24 a^2-57 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 b^8}+\frac {\log (a+b \sin (c+d x))}{a^3 (a-b)^3 (a+b)^3}+\frac {1}{16 b^6 (a+b) (b-b \sin (c+d x))^2}+\frac {9 a+11 b}{16 b^7 (a+b)^2 (b-b \sin (c+d x))}+\frac {1}{16 (a-b) b^6 (b+b \sin (c+d x))^2}+\frac {9 a-11 b}{16 (a-b)^2 b^7 (b+b \sin (c+d x))}\right )}{d} \]
(b^8*(Csc[c + d*x]/(a^2*b^7) - Csc[c + d*x]^2/(2*a*b^8) - ((24*a^2 + 57*a* b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*b^8*(a + b)^3) + ((3*a^2 + b^2)*Log [Sin[c + d*x]])/(a^3*b^8) - ((24*a^2 - 57*a*b + 35*b^2)*Log[1 + Sin[c + d* x]])/(16*(a - b)^3*b^8) + Log[a + b*Sin[c + d*x]]/(a^3*(a - b)^3*(a + b)^3 ) + 1/(16*b^6*(a + b)*(b - b*Sin[c + d*x])^2) + (9*a + 11*b)/(16*b^7*(a + b)^2*(b - b*Sin[c + d*x])) + 1/(16*(a - b)*b^6*(b + b*Sin[c + d*x])^2) + ( 9*a - 11*b)/(16*(a - b)^2*b^7*(b + b*Sin[c + d*x]))))/d
Time = 0.61 (sec) , antiderivative size = 285, 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.
\(\displaystyle \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (c+d x)^3 \cos (c+d x)^5 (a+b \sin (c+d x))}dx\) |
\(\Big \downarrow \) 3316 |
\(\displaystyle \frac {b^5 \int \frac {\csc ^3(c+d x)}{(a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b^8 \int \frac {\csc ^3(c+d x)}{b^3 (a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \frac {b^8 \int \left (\frac {\csc ^3(c+d x)}{a b^9}-\frac {\csc ^2(c+d x)}{a^2 b^8}+\frac {\left (3 a^2+b^2\right ) \csc (c+d x)}{a^3 b^9}+\frac {24 a^2+57 b a+35 b^2}{16 b^8 (a+b)^3 (b-b \sin (c+d x))}+\frac {1}{a^3 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+\frac {24 a^2-57 b a+35 b^2}{16 b^8 (b-a)^3 (\sin (c+d x) b+b)}+\frac {9 a+11 b}{16 b^7 (a+b)^2 (b-b \sin (c+d x))^2}+\frac {11 b-9 a}{16 (a-b)^2 b^7 (\sin (c+d x) b+b)^2}+\frac {1}{8 b^6 (a+b) (b-b \sin (c+d x))^3}+\frac {1}{8 b^6 (b-a) (\sin (c+d x) b+b)^3}\right )d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^8 \left (\frac {\csc (c+d x)}{a^2 b^7}-\frac {\left (24 a^2+57 a b+35 b^2\right ) \log (b-b \sin (c+d x))}{16 b^8 (a+b)^3}-\frac {\left (24 a^2-57 a b+35 b^2\right ) \log (b \sin (c+d x)+b)}{16 b^8 (a-b)^3}+\frac {\log (a+b \sin (c+d x))}{a^3 \left (a^2-b^2\right )^3}+\frac {\left (3 a^2+b^2\right ) \log (b \sin (c+d x))}{a^3 b^8}-\frac {\csc ^2(c+d x)}{2 a b^8}+\frac {9 a+11 b}{16 b^7 (a+b)^2 (b-b \sin (c+d x))}+\frac {9 a-11 b}{16 b^7 (a-b)^2 (b \sin (c+d x)+b)}+\frac {1}{16 b^6 (a+b) (b-b \sin (c+d x))^2}+\frac {1}{16 b^6 (a-b) (b \sin (c+d x)+b)^2}\right )}{d}\) |
(b^8*(Csc[c + d*x]/(a^2*b^7) - Csc[c + d*x]^2/(2*a*b^8) + ((3*a^2 + b^2)*L og[b*Sin[c + d*x]])/(a^3*b^8) - ((24*a^2 + 57*a*b + 35*b^2)*Log[b - b*Sin[ c + d*x]])/(16*b^8*(a + b)^3) + Log[a + b*Sin[c + d*x]]/(a^3*(a^2 - b^2)^3 ) - ((24*a^2 - 57*a*b + 35*b^2)*Log[b + b*Sin[c + d*x]])/(16*(a - b)^3*b^8 ) + 1/(16*b^6*(a + b)*(b - b*Sin[c + d*x])^2) + (9*a + 11*b)/(16*b^7*(a + b)^2*(b - b*Sin[c + d*x])) + 1/(16*(a - b)*b^6*(b + b*Sin[c + d*x])^2) + ( 9*a - 11*b)/(16*(a - b)^2*b^7*(b + b*Sin[c + d*x]))))/d
3.14.69.3.1 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 = 1.52 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {9 a +11 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-24 a^{2}-57 a b -35 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 a \sin \left (d x +c \right )^{2}}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{3}}+\frac {b}{a^{2} \sin \left (d x +c \right )}+\frac {b^{8} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a^{3}}+\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-9 a +11 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-24 a^{2}+57 a b -35 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}}{d}\) | \(238\) |
default | \(\frac {\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {9 a +11 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-24 a^{2}-57 a b -35 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 a \sin \left (d x +c \right )^{2}}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{3}}+\frac {b}{a^{2} \sin \left (d x +c \right )}+\frac {b^{8} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a^{3}}+\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-9 a +11 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-24 a^{2}+57 a b -35 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}}{d}\) | \(238\) |
parallelrisch | \(\frac {256 b^{8} \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 )-768 \left (a^{2}+\frac {19}{8} a b +\frac {35}{24} b^{2}\right ) \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 )^{3} a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-54 \left (\frac {128 \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^{2}-\frac {19}{8} a b +\frac {35}{24} b^{2}\right ) a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{9}+\left (-\frac {128 \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} \left (a^{2}+\frac {b^{2}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (a^{4}-\frac {89}{54} a^{2} b^{2}+\frac {11}{18} b^{4}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {2}{9} a^{4}-\frac {11}{27} a^{2} b^{2}+\frac {1}{9} b^{4}\right ) \cos \left (4 d x +4 c \right )+\left (-\frac {1}{9} a^{4}+\frac {1}{6} a^{2} b^{2}-\frac {1}{54} b^{4}\right ) \cos \left (6 d x +6 c \right )-\frac {13 a^{2} b^{2}}{27}+\frac {13 b^{4}}{27}+\frac {2 a^{4}}{27}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {80 b \left (\left (a^{4}-\frac {9}{5} a^{2} b^{2}+\frac {4}{5} b^{4}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {3}{8} a^{4}-\frac {27}{40} a^{2} b^{2}+\frac {1}{5} b^{4}\right ) \cos \left (4 d x +4 c \right )+\frac {9 a^{4}}{40}-\frac {29 a^{2} b^{2}}{40}+\frac {3 b^{4}}{5}\right )}{27}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right ) a \right ) \left (a -b \right )\right ) \left (a +b \right )}{64 \left (a -b \right )^{3} \left (a +b \right )^{3} a^{3} d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(507\) |
norman | \(\frac {-\frac {1}{8 a d}-\frac {\tan ^{12}\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 ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} d}+\frac {2 \left (-5 a^{4}+8 a^{2} b^{2}-b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (-57 a^{4}+82 a^{2} b^{2}-9 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (-57 a^{4}+82 a^{2} b^{2}-9 b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (15 a^{4}-25 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (15 a^{4}-25 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (5 a^{4}-13 a^{2} b^{2}+4 b^{4}\right ) b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (5 a^{4}-13 a^{2} b^{2}+4 b^{4}\right ) b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\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 )^{4}}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}+\frac {b^{8} \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^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {\left (24 a^{2}-57 a b +35 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 (24 a^{2}+57 a b +35 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}\) | \(683\) |
risch | \(\text {Expression too large to display}\) | \(1418\) |
1/d*(1/2/(8*a+8*b)/(sin(d*x+c)-1)^2-1/16*(9*a+11*b)/(a+b)^2/(sin(d*x+c)-1) +1/16/(a+b)^3*(-24*a^2-57*a*b-35*b^2)*ln(sin(d*x+c)-1)-1/2/a/sin(d*x+c)^2+ (3*a^2+b^2)/a^3*ln(sin(d*x+c))+1/a^2*b/sin(d*x+c)+b^8/(a+b)^3/(a-b)^3/a^3* ln(a+b*sin(d*x+c))+1/2/(8*a-8*b)/(1+sin(d*x+c))^2-1/16*(-9*a+11*b)/(a-b)^2 /(1+sin(d*x+c))+1/16/(a-b)^3*(-24*a^2+57*a*b-35*b^2)*ln(1+sin(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (256) = 512\).
Time = 3.31 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.34 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {4 \, a^{8} - 8 \, a^{6} b^{2} + 4 \, a^{4} b^{4} - 8 \, {\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 5 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{2} - 16 \, {\left (b^{8} \cos \left (d x + c\right )^{6} - b^{8} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 16 \, {\left ({\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - b^{8}\right )} \cos \left (d x + c\right )^{6} - {\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - b^{8}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) + {\left ({\left (24 \, a^{8} + 15 \, a^{7} b - 64 \, a^{6} b^{2} - 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} + 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{6} - {\left (24 \, a^{8} + 15 \, a^{7} b - 64 \, a^{6} b^{2} - 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} + 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (24 \, a^{8} - 15 \, a^{7} b - 64 \, a^{6} b^{2} + 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} - 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{6} - {\left (24 \, a^{8} - 15 \, a^{7} b - 64 \, a^{6} b^{2} + 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} - 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a^{7} b - 4 \, a^{5} b^{3} + 2 \, a^{3} b^{5} - {\left (15 \, a^{7} b - 42 \, a^{5} b^{3} + 35 \, a^{3} b^{5} - 8 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (5 \, a^{7} b - 14 \, a^{5} b^{3} + 9 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{6} - {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
-1/16*(4*a^8 - 8*a^6*b^2 + 4*a^4*b^4 - 8*(3*a^8 - 8*a^6*b^2 + 6*a^4*b^4 - a^2*b^6)*cos(d*x + c)^4 + 4*(3*a^8 - 8*a^6*b^2 + 5*a^4*b^4)*cos(d*x + c)^2 - 16*(b^8*cos(d*x + c)^6 - b^8*cos(d*x + c)^4)*log(b*sin(d*x + c) + a) - 16*((3*a^8 - 8*a^6*b^2 + 6*a^4*b^4 - b^8)*cos(d*x + c)^6 - (3*a^8 - 8*a^6* b^2 + 6*a^4*b^4 - b^8)*cos(d*x + c)^4)*log(-1/2*sin(d*x + c)) + ((24*a^8 + 15*a^7*b - 64*a^6*b^2 - 42*a^5*b^3 + 48*a^4*b^4 + 35*a^3*b^5)*cos(d*x + c )^6 - (24*a^8 + 15*a^7*b - 64*a^6*b^2 - 42*a^5*b^3 + 48*a^4*b^4 + 35*a^3*b ^5)*cos(d*x + c)^4)*log(sin(d*x + c) + 1) + ((24*a^8 - 15*a^7*b - 64*a^6*b ^2 + 42*a^5*b^3 + 48*a^4*b^4 - 35*a^3*b^5)*cos(d*x + c)^6 - (24*a^8 - 15*a ^7*b - 64*a^6*b^2 + 42*a^5*b^3 + 48*a^4*b^4 - 35*a^3*b^5)*cos(d*x + c)^4)* log(-sin(d*x + c) + 1) - 2*(2*a^7*b - 4*a^5*b^3 + 2*a^3*b^5 - (15*a^7*b - 42*a^5*b^3 + 35*a^3*b^5 - 8*a*b^7)*cos(d*x + c)^4 + (5*a^7*b - 14*a^5*b^3 + 9*a^3*b^5)*cos(d*x + c)^2)*sin(d*x + c))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^6 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos (d*x + c)^4)
Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.54 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {16 \, b^{8} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}} - \frac {{\left (24 \, a^{2} - 57 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (24 \, a^{2} + 57 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left ({\left (15 \, a^{4} b - 27 \, a^{2} b^{3} + 8 \, b^{5}\right )} \sin \left (d x + c\right )^{5} - 4 \, a^{5} + 8 \, a^{3} b^{2} - 4 \, a b^{4} - 4 \, {\left (3 \, a^{5} - 5 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{4} - {\left (25 \, a^{4} b - 45 \, a^{2} b^{3} + 16 \, b^{5}\right )} \sin \left (d x + c\right )^{3} + 2 \, {\left (9 \, a^{5} - 15 \, a^{3} b^{2} + 4 \, a b^{4}\right )} \sin \left (d x + c\right )^{2} + 8 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{6} - 2 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{4} + {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{2}} + \frac {16 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{16 \, d} \]
1/16*(16*b^8*log(b*sin(d*x + c) + a)/(a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^ 6) - (24*a^2 - 57*a*b + 35*b^2)*log(sin(d*x + c) + 1)/(a^3 - 3*a^2*b + 3*a *b^2 - b^3) - (24*a^2 + 57*a*b + 35*b^2)*log(sin(d*x + c) - 1)/(a^3 + 3*a^ 2*b + 3*a*b^2 + b^3) + 2*((15*a^4*b - 27*a^2*b^3 + 8*b^5)*sin(d*x + c)^5 - 4*a^5 + 8*a^3*b^2 - 4*a*b^4 - 4*(3*a^5 - 5*a^3*b^2 + a*b^4)*sin(d*x + c)^ 4 - (25*a^4*b - 45*a^2*b^3 + 16*b^5)*sin(d*x + c)^3 + 2*(9*a^5 - 15*a^3*b^ 2 + 4*a*b^4)*sin(d*x + c)^2 + 8*(a^4*b - 2*a^2*b^3 + b^5)*sin(d*x + c))/(( a^6 - 2*a^4*b^2 + a^2*b^4)*sin(d*x + c)^6 - 2*(a^6 - 2*a^4*b^2 + a^2*b^4)* sin(d*x + c)^4 + (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(d*x + c)^2) + 16*(3*a^2 + b^2)*log(sin(d*x + c))/a^3)/d
Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (256) = 512\).
Time = 0.37 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.15 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {16 \, b^{9} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}} - \frac {{\left (24 \, a^{2} - 57 \, a b + 35 \, 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 (24 \, a^{2} + 57 \, a b + 35 \, 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 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (4 \, b^{8} \sin \left (d x + c\right )^{6} + 15 \, a^{7} b \sin \left (d x + c\right )^{5} - 42 \, a^{5} b^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{5} \sin \left (d x + c\right )^{5} - 8 \, a b^{7} \sin \left (d x + c\right )^{5} - 12 \, a^{8} \sin \left (d x + c\right )^{4} + 32 \, a^{6} b^{2} \sin \left (d x + c\right )^{4} - 24 \, a^{4} b^{4} \sin \left (d x + c\right )^{4} + 4 \, a^{2} b^{6} \sin \left (d x + c\right )^{4} - 8 \, b^{8} \sin \left (d x + c\right )^{4} - 25 \, a^{7} b \sin \left (d x + c\right )^{3} + 70 \, a^{5} b^{3} \sin \left (d x + c\right )^{3} - 61 \, a^{3} b^{5} \sin \left (d x + c\right )^{3} + 16 \, a b^{7} \sin \left (d x + c\right )^{3} + 18 \, a^{8} \sin \left (d x + c\right )^{2} - 48 \, a^{6} b^{2} \sin \left (d x + c\right )^{2} + 38 \, a^{4} b^{4} \sin \left (d x + c\right )^{2} - 8 \, a^{2} b^{6} \sin \left (d x + c\right )^{2} + 4 \, b^{8} \sin \left (d x + c\right )^{2} + 8 \, a^{7} b \sin \left (d x + c\right ) - 24 \, a^{5} b^{3} \sin \left (d x + c\right ) + 24 \, a^{3} b^{5} \sin \left (d x + c\right ) - 8 \, a b^{7} \sin \left (d x + c\right ) - 4 \, a^{8} + 12 \, a^{6} b^{2} - 12 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right )}}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} {\left (\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )\right )}^{2}}}{16 \, d} \]
1/16*(16*b^9*log(abs(b*sin(d*x + c) + a))/(a^9*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7) - (24*a^2 - 57*a*b + 35*b^2)*log(abs(sin(d*x + c) + 1))/(a^3 - 3 *a^2*b + 3*a*b^2 - b^3) - (24*a^2 + 57*a*b + 35*b^2)*log(abs(sin(d*x + c) - 1))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 16*(3*a^2 + b^2)*log(abs(sin(d*x + c)))/a^3 + 2*(4*b^8*sin(d*x + c)^6 + 15*a^7*b*sin(d*x + c)^5 - 42*a^5*b^3 *sin(d*x + c)^5 + 35*a^3*b^5*sin(d*x + c)^5 - 8*a*b^7*sin(d*x + c)^5 - 12* a^8*sin(d*x + c)^4 + 32*a^6*b^2*sin(d*x + c)^4 - 24*a^4*b^4*sin(d*x + c)^4 + 4*a^2*b^6*sin(d*x + c)^4 - 8*b^8*sin(d*x + c)^4 - 25*a^7*b*sin(d*x + c) ^3 + 70*a^5*b^3*sin(d*x + c)^3 - 61*a^3*b^5*sin(d*x + c)^3 + 16*a*b^7*sin( d*x + c)^3 + 18*a^8*sin(d*x + c)^2 - 48*a^6*b^2*sin(d*x + c)^2 + 38*a^4*b^ 4*sin(d*x + c)^2 - 8*a^2*b^6*sin(d*x + c)^2 + 4*b^8*sin(d*x + c)^2 + 8*a^7 *b*sin(d*x + c) - 24*a^5*b^3*sin(d*x + c) + 24*a^3*b^5*sin(d*x + c) - 8*a* b^7*sin(d*x + c) - 4*a^8 + 12*a^6*b^2 - 12*a^4*b^4 + 4*a^2*b^6)/((a^9 - 3* a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(sin(d*x + c)^3 - sin(d*x + c))^2))/d
Time = 12.83 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.50 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )\right )\,\left (3\,a^2+b^2\right )}{a^3\,d}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {b^2}{8\,{\left (a-b\right )}^3}-\frac {9\,b}{16\,{\left (a-b\right )}^2}+\frac {3}{2\,\left (a-b\right )}\right )}{d}-\frac {\frac {1}{2\,a}-\frac {b\,\sin \left (c+d\,x\right )}{a^2}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (3\,a^4-5\,a^2\,b^2+b^4\right )}{2\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (9\,a^4-15\,a^2\,b^2+4\,b^4\right )}{4\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^5\,\left (15\,a^4\,b-27\,a^2\,b^3+8\,b^5\right )}{8\,a^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (25\,a^4\,b-45\,a^2\,b^3+16\,b^5\right )}{8\,a^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^6-2\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {9\,b}{16\,{\left (a+b\right )}^2}+\frac {3}{2\,\left (a+b\right )}+\frac {b^2}{8\,{\left (a+b\right )}^3}\right )}{d}+\frac {b^8\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^9-3\,a^7\,b^2+3\,a^5\,b^4-a^3\,b^6\right )} \]
(log(sin(c + d*x))*(3*a^2 + b^2))/(a^3*d) - (log(sin(c + d*x) + 1)*(b^2/(8 *(a - b)^3) - (9*b)/(16*(a - b)^2) + 3/(2*(a - b))))/d - (1/(2*a) - (b*sin (c + d*x))/a^2 + (sin(c + d*x)^4*(3*a^4 + b^4 - 5*a^2*b^2))/(2*a*(a^4 + b^ 4 - 2*a^2*b^2)) - (sin(c + d*x)^2*(9*a^4 + 4*b^4 - 15*a^2*b^2))/(4*a*(a^4 + b^4 - 2*a^2*b^2)) - (sin(c + d*x)^5*(15*a^4*b + 8*b^5 - 27*a^2*b^3))/(8* a^2*(a^4 + b^4 - 2*a^2*b^2)) + (sin(c + d*x)^3*(25*a^4*b + 16*b^5 - 45*a^2 *b^3))/(8*a^2*(a^4 + b^4 - 2*a^2*b^2)))/(d*(sin(c + d*x)^2 - 2*sin(c + d*x )^4 + sin(c + d*x)^6)) - (log(sin(c + d*x) - 1)*((9*b)/(16*(a + b)^2) + 3/ (2*(a + b)) + b^2/(8*(a + b)^3)))/d + (b^8*log(a + b*sin(c + d*x)))/(d*(a^ 9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))