Integrand size = 27, antiderivative size = 195 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {x}{b}+\frac {2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b d}-\frac {\left (15 a^4-20 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^5 d}+\frac {b \left (-2 a^2+b^2\right ) \cot (c+d x)}{a^4 d}+\frac {b \cot ^3(c+d x)}{3 a^2 d}+\frac {\left (7 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d} \]
-x/b+2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^ 5/b/d-1/8*(15*a^4-20*a^2*b^2+8*b^4)*arctanh(cos(d*x+c))/a^5/d+b*(-2*a^2+b^ 2)*cot(d*x+c)/a^4/d+1/3*b*cot(d*x+c)^3/a^2/d+1/8*(7*a^2-4*b^2)*cot(d*x+c)* csc(d*x+c)/a^3/d-1/4*cot(d*x+c)^3*csc(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(448\) vs. \(2(195)=390\).
Time = 6.36 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.30 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {c+d x}{b d}+\frac {2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b d}+\frac {\left (-7 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )+3 b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^4 d}+\frac {\left (9 a^2-4 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\left (-15 a^4+20 a^2 b^2-8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (15 a^4-20 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (-9 a^2+4 b^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (7 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^2 d} \]
-((c + d*x)/(b*d)) + (2*(a^2 - b^2)^(5/2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[ (c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*b*d) + ((-7*a^2 *b*Cos[(c + d*x)/2] + 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + ((9*a^2 - 4*b^2)*Csc[(c + d*x)/2]^2)/(32*a^3*d) + (b*Cot[(c + d*x)/2]*Cs c[(c + d*x)/2]^2)/(24*a^2*d) - Csc[(c + d*x)/2]^4/(64*a*d) + ((-15*a^4 + 2 0*a^2*b^2 - 8*b^4)*Log[Cos[(c + d*x)/2]])/(8*a^5*d) + ((15*a^4 - 20*a^2*b^ 2 + 8*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^5*d) + ((-9*a^2 + 4*b^2)*Sec[(c + d *x)/2]^2)/(32*a^3*d) + Sec[(c + d*x)/2]^4/(64*a*d) + (Sec[(c + d*x)/2]*(7* a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[(c + d*x)/2]))/(6*a^4*d) - (b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)
Time = 0.54 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3376, 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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3376 |
| \(\displaystyle \int \left (-\frac {b \csc ^4(c+d x)}{a^2}+\frac {\left (a^2-b^2\right )^3}{a^5 b (a+b \sin (c+d x))}+\frac {\left (3 a^2 b-b^3\right ) \csc ^2(c+d x)}{a^4}+\frac {\left (b^2-3 a^2\right ) \csc ^3(c+d x)}{a^3}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \csc (c+d x)}{a^5}+\frac {\csc ^5(c+d x)}{a}-\frac {1}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \cot ^3(c+d x)}{3 a^2 d}+\frac {b \cot (c+d x)}{a^2 d}+\frac {2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b d}-\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac {\left (3 a^2-b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {x}{b}\) |
-(x/b) + (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b ^2]])/(a^5*b*d) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) + ((3*a^2 - b^2)*ArcTa nh[Cos[c + d*x]])/(2*a^3*d) - ((3*a^4 - 3*a^2*b^2 + b^4)*ArcTanh[Cos[c + d *x]])/(a^5*d) + (b*Cot[c + d*x])/(a^2*d) - (b*(3*a^2 - b^2)*Cot[c + d*x])/ (a^4*d) + (b*Cot[c + d*x]^3)/(3*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8* a*d) + ((3*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) - (Cot[c + d*x] *Csc[c + d*x]^3)/(4*a*d)
3.14.27.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(d*sin[ e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x] /; Fr eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && ( LtQ[m, -1] || (EqQ[m, -1] && GtQ[p, 0]))
Time = 0.62 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.64
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{4}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}+\frac {\left (32 a^{6}-96 a^{4} b^{2}+96 a^{2} b^{4}-32 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{16 a^{5} b \sqrt {a^{2}-b^{2}}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+4 b^{2}}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-40 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (9 a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(320\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{4}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}+\frac {\left (32 a^{6}-96 a^{4} b^{2}+96 a^{2} b^{4}-32 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{16 a^{5} b \sqrt {a^{2}-b^{2}}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+4 b^{2}}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-40 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (9 a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(320\) |
| risch | \(-\frac {x}{b}+\frac {i \left (-3 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+27 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-72 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+24 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+168 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-72 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+27 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-152 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+72 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+56 a^{2} b -24 b^{3}\right )}{12 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 a^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{a^{5} d}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d b a}-\frac {2 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{5}}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d b a}+\frac {2 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{5}}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 a^{3} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{a^{5} d}\) | \(704\) |
1/d*(1/16/a^4*(1/4*tan(1/2*d*x+1/2*c)^4*a^3-2/3*tan(1/2*d*x+1/2*c)^3*a^2*b -4*tan(1/2*d*x+1/2*c)^2*a^3+2*tan(1/2*d*x+1/2*c)^2*a*b^2+18*tan(1/2*d*x+1/ 2*c)*a^2*b-8*tan(1/2*d*x+1/2*c)*b^3)-2/b*arctan(tan(1/2*d*x+1/2*c))+1/16*( 32*a^6-96*a^4*b^2+96*a^2*b^4-32*b^6)/a^5/b/(a^2-b^2)^(1/2)*arctan(1/2*(2*a *tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-1/64/a/tan(1/2*d*x+1/2*c)^4-1/32 *(-8*a^2+4*b^2)/a^3/tan(1/2*d*x+1/2*c)^2+1/16/a^5*(30*a^4-40*a^2*b^2+16*b^ 4)*ln(tan(1/2*d*x+1/2*c))+1/24/a^2*b/tan(1/2*d*x+1/2*c)^3-1/8*b*(9*a^2-4*b ^2)/a^4/tan(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (185) = 370\).
Time = 0.78 (sec) , antiderivative size = 1034, normalized size of antiderivative = 5.30 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
[-1/48*(48*a^5*d*x*cos(d*x + c)^4 - 96*a^5*d*x*cos(d*x + c)^2 + 48*a^5*d*x + 6*(9*a^4*b - 4*a^2*b^3)*cos(d*x + c)^3 - 24*((a^4 - 2*a^2*b^2 + b^4)*co s(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt( -a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 6*(7 *a^4*b - 4*a^2*b^3)*cos(d*x + c) + 3*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15* a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8* b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 3*(15*a^4*b - 20*a^2*b^ 3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 16*((7 *a^3*b^2 - 3*a*b^4)*cos(d*x + c)^3 - 3*(2*a^3*b^2 - a*b^4)*cos(d*x + c))*s in(d*x + c))/(a^5*b*d*cos(d*x + c)^4 - 2*a^5*b*d*cos(d*x + c)^2 + a^5*b*d) , -1/48*(48*a^5*d*x*cos(d*x + c)^4 - 96*a^5*d*x*cos(d*x + c)^2 + 48*a^5*d* x + 6*(9*a^4*b - 4*a^2*b^3)*cos(d*x + c)^3 + 48*((a^4 - 2*a^2*b^2 + b^4)*c os(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos( d*x + c))) - 6*(7*a^4*b - 4*a^2*b^3)*cos(d*x + c) + 3*(15*a^4*b - 20*a^2*b ^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 3*(...
Timed out. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (185) = 370\).
Time = 0.44 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.03 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {192 \, {\left (d x + c\right )}}{b} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 216 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {24 \, {\left (15 \, a^{4} - 20 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {384 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5} b} + \frac {750 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1000 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 216 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
-1/192*(192*(d*x + c)/b - (3*a^3*tan(1/2*d*x + 1/2*c)^4 - 8*a^2*b*tan(1/2* d*x + 1/2*c)^3 - 48*a^3*tan(1/2*d*x + 1/2*c)^2 + 24*a*b^2*tan(1/2*d*x + 1/ 2*c)^2 + 216*a^2*b*tan(1/2*d*x + 1/2*c) - 96*b^3*tan(1/2*d*x + 1/2*c))/a^4 - 24*(15*a^4 - 20*a^2*b^2 + 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 - 3 84*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*s gn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^5*b) + (750*a^4*tan(1/2*d*x + 1/2*c)^4 - 1000*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 400*b^4*tan(1/2*d*x + 1/2*c)^4 + 216*a^3*b*tan(1/2*d*x + 1/2*c )^3 - 96*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 48*a^4*tan(1/2*d*x + 1/2*c)^2 + 24 *a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 8*a^3*b*tan(1/2*d*x + 1/2*c) + 3*a^4)/(a ^5*tan(1/2*d*x + 1/2*c)^4))/d
Time = 12.99 (sec) , antiderivative size = 4334, normalized size of antiderivative = 22.23 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
tan(c/2 + (d*x)/2)^4/(64*a*d) - (2*atan(-(((((((4*(64*a^9*b^9 - 208*a^11*b ^7 + 240*a^13*b^5 - 93*a^15*b^3))/a^12 + (((4*(32*a^14*b^5 - 24*a^16*b^3)) /a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^6 - 136*a^15*b^4 + 16*a^17*b^2)) /a^12)*1i)/b + (4*tan(c/2 + (d*x)/2)*(128*a^8*b^10 - 456*a^10*b^8 + 604*a^ 12*b^6 - 335*a^14*b^4 + 62*a^16*b^2))/a^12)*1i)/b - (4*(24*a^16*b - 32*a^4 *b^13 + 184*a^6*b^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58*a^14 *b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(16*a^17 + 8*a^5*b^12 - 40*a^7*b^10 + 100*a^9*b^8 - 148*a^11*b^6 + 252*a^13*b^4 - 180*a^15*b^2))/a^12)*1i)/b - ( 4*(53*a^15*b + 8*a^5*b^11 - 48*a^7*b^9 + 56*a^9*b^7 + 48*a^11*b^5 - 120*a^ 13*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(62*a^16 + 8*b^16 - 68*a^2*b^14 + 25 5*a^4*b^12 - 550*a^6*b^10 + 873*a^8*b^8 - 1096*a^10*b^6 + 929*a^12*b^4 - 4 10*a^14*b^2))/a^12)/b + ((((4*(24*a^16*b - 32*a^4*b^13 + 184*a^6*b^11 - 44 0*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58*a^14*b^3))/a^12 + (((4*(64*a^ 9*b^9 - 208*a^11*b^7 + 240*a^13*b^5 - 93*a^15*b^3))/a^12 - (((4*(32*a^14*b ^5 - 24*a^16*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^6 - 136*a^15*b ^4 + 16*a^17*b^2))/a^12)*1i)/b + (4*tan(c/2 + (d*x)/2)*(128*a^8*b^10 - 456 *a^10*b^8 + 604*a^12*b^6 - 335*a^14*b^4 + 62*a^16*b^2))/a^12)*1i)/b - (4*t an(c/2 + (d*x)/2)*(16*a^17 + 8*a^5*b^12 - 40*a^7*b^10 + 100*a^9*b^8 - 148* a^11*b^6 + 252*a^13*b^4 - 180*a^15*b^2))/a^12)*1i)/b - (4*(53*a^15*b + 8*a ^5*b^11 - 48*a^7*b^9 + 56*a^9*b^7 + 48*a^11*b^5 - 120*a^13*b^3))/a^12 +...