Integrand size = 21, antiderivative size = 205 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^6}{32 d (a-a \cos (c+d x))^4}-\frac {7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac {15 a^4}{32 d (a-a \cos (c+d x))^2}-\frac {51 a^3}{32 d (a-a \cos (c+d x))}+\frac {a^4}{64 d (a+a \cos (c+d x))^2}+\frac {9 a^3}{64 d (a+a \cos (c+d x))}+\frac {303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {47 a^2 \log (1+\cos (c+d x))}{128 d}+\frac {a^2 \sec (c+d x)}{d} \]
-1/32*a^6/d/(a-a*cos(d*x+c))^4-7/48*a^5/d/(a-a*cos(d*x+c))^3-15/32*a^4/d/( a-a*cos(d*x+c))^2-51/32*a^3/d/(a-a*cos(d*x+c))+1/64*a^4/d/(a+a*cos(d*x+c)) ^2+9/64*a^3/d/(a+a*cos(d*x+c))+303/128*a^2*ln(1-cos(d*x+c))/d-2*a^2*ln(cos (d*x+c))/d-47/128*a^2*ln(1+cos(d*x+c))/d+a^2*sec(d*x+c)/d
Time = 2.62 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.80 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (1224 \csc ^2\left (\frac {1}{2} (c+d x)\right )+180 \csc ^4\left (\frac {1}{2} (c+d x)\right )+28 \csc ^6\left (\frac {1}{2} (c+d x)\right )+3 \csc ^8\left (\frac {1}{2} (c+d x)\right )-6 \left (18 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+4 \left (-47 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-128 \log (\cos (c+d x))+303 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+64 \sec (c+d x)\right )\right )\right )}{6144 d} \]
-1/6144*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(1224*Csc[(c + d*x)/2 ]^2 + 180*Csc[(c + d*x)/2]^4 + 28*Csc[(c + d*x)/2]^6 + 3*Csc[(c + d*x)/2]^ 8 - 6*(18*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4 + 4*(-47*Log[Cos[(c + d* x)/2]] - 128*Log[Cos[c + d*x]] + 303*Log[Sin[(c + d*x)/2]] + 64*Sec[c + d* x]))))/d
Time = 0.52 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4360, 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.
| \(\displaystyle \int \csc ^9(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^9}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^9(c+d x) \sec ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}{\sin \left (c+d x-\frac {\pi }{2}\right )^2 \cos \left (c+d x-\frac {\pi }{2}\right )^9}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^9 \int \frac {\sec ^2(c+d x)}{(a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^3}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^{11} \int \frac {\sec ^2(c+d x)}{a^2 (a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^3}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^{11} \int \left (\frac {\sec ^2(c+d x)}{a^{10}}+\frac {2 \sec (c+d x)}{a^{10}}+\frac {303}{128 a^9 (a-a \cos (c+d x))}+\frac {47}{128 a^9 (\cos (c+d x) a+a)}+\frac {51}{32 a^8 (a-a \cos (c+d x))^2}+\frac {9}{64 a^8 (\cos (c+d x) a+a)^2}+\frac {15}{16 a^7 (a-a \cos (c+d x))^3}+\frac {1}{32 a^7 (\cos (c+d x) a+a)^3}+\frac {7}{16 a^6 (a-a \cos (c+d x))^4}+\frac {1}{8 a^5 (a-a \cos (c+d x))^5}\right )d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{11} \left (\frac {\sec (c+d x)}{a^9}-\frac {2 \log (-a \cos (c+d x))}{a^9}+\frac {303 \log (a-a \cos (c+d x))}{128 a^9}-\frac {47 \log (a \cos (c+d x)+a)}{128 a^9}-\frac {51}{32 a^8 (a-a \cos (c+d x))}+\frac {9}{64 a^8 (a \cos (c+d x)+a)}-\frac {15}{32 a^7 (a-a \cos (c+d x))^2}+\frac {1}{64 a^7 (a \cos (c+d x)+a)^2}-\frac {7}{48 a^6 (a-a \cos (c+d x))^3}-\frac {1}{32 a^5 (a-a \cos (c+d x))^4}\right )}{d}\) |
(a^11*(-1/32*1/(a^5*(a - a*Cos[c + d*x])^4) - 7/(48*a^6*(a - a*Cos[c + d*x ])^3) - 15/(32*a^7*(a - a*Cos[c + d*x])^2) - 51/(32*a^8*(a - a*Cos[c + d*x ])) + 1/(64*a^7*(a + a*Cos[c + d*x])^2) + 9/(64*a^8*(a + a*Cos[c + d*x])) - (2*Log[-(a*Cos[c + d*x])])/a^9 + (303*Log[a - a*Cos[c + d*x]])/(128*a^9) - (47*Log[a + a*Cos[c + d*x]])/(128*a^9) + Sec[c + d*x]/a^9))/d
3.1.28.3.1 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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 1.27 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.02
| method | result | size |
| norman | \(\frac {\frac {a^{2}}{512 d}+\frac {37 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{1536 d}+\frac {121 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{768 d}+\frac {233 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{256 d}+\frac {19 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{256 d}+\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{256 d}-\frac {203 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {303 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(210\) |
| parallelrisch | \(\frac {a^{2} \left (3 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+37 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+114 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+242 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7272 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3072 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3072 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1398 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7272 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3072 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3072 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-4872\right )}{1536 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1536 d}\) | \(218\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )}-\frac {3}{16 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {21}{64 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {105}{128 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {315}{128 \cos \left (d x +c \right )}+\frac {315 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\left (-\frac {\csc \left (d x +c \right )^{7}}{8}-\frac {7 \csc \left (d x +c \right )^{5}}{48}-\frac {35 \csc \left (d x +c \right )^{3}}{192}-\frac {35 \csc \left (d x +c \right )}{128}\right ) \cot \left (d x +c \right )+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )}{d}\) | \(233\) |
| default | \(\frac {a^{2} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )}-\frac {3}{16 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {21}{64 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {105}{128 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {315}{128 \cos \left (d x +c \right )}+\frac {315 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\left (-\frac {\csc \left (d x +c \right )^{7}}{8}-\frac {7 \csc \left (d x +c \right )^{5}}{48}-\frac {35 \csc \left (d x +c \right )^{3}}{192}-\frac {35 \csc \left (d x +c \right )}{128}\right ) \cot \left (d x +c \right )+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )}{d}\) | \(233\) |
| risch | \(\frac {a^{2} \left (525 \,{\mathrm e}^{13 i \left (d x +c \right )}-1716 \,{\mathrm e}^{12 i \left (d x +c \right )}+214 \,{\mathrm e}^{11 i \left (d x +c \right )}+4652 \,{\mathrm e}^{10 i \left (d x +c \right )}-4173 \,{\mathrm e}^{9 i \left (d x +c \right )}-2552 \,{\mathrm e}^{8 i \left (d x +c \right )}+4564 \,{\mathrm e}^{7 i \left (d x +c \right )}-2552 \,{\mathrm e}^{6 i \left (d x +c \right )}-4173 \,{\mathrm e}^{5 i \left (d x +c \right )}+4652 \,{\mathrm e}^{4 i \left (d x +c \right )}+214 \,{\mathrm e}^{3 i \left (d x +c \right )}-1716 \,{\mathrm e}^{2 i \left (d x +c \right )}+525 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{96 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {303 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d}-\frac {47 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(253\) |
(1/512/d*a^2+37/1536*a^2/d*tan(1/2*d*x+1/2*c)^2+121/768*a^2/d*tan(1/2*d*x+ 1/2*c)^4+233/256*a^2/d*tan(1/2*d*x+1/2*c)^6+19/256*a^2/d*tan(1/2*d*x+1/2*c )^12+1/256*a^2/d*tan(1/2*d*x+1/2*c)^14-203/64/d*a^2*tan(1/2*d*x+1/2*c)^8)/ tan(1/2*d*x+1/2*c)^8/(-1+tan(1/2*d*x+1/2*c)^2)+303/64/d*a^2*ln(tan(1/2*d*x +1/2*c))-2*a^2/d*ln(tan(1/2*d*x+1/2*c)-1)-2*a^2/d*ln(tan(1/2*d*x+1/2*c)+1)
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (193) = 386\).
Time = 0.29 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.25 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {1050 \, a^{2} \cos \left (d x + c\right )^{6} - 1716 \, a^{2} \cos \left (d x + c\right )^{5} - 1468 \, a^{2} \cos \left (d x + c\right )^{4} + 3308 \, a^{2} \cos \left (d x + c\right )^{3} - 38 \, a^{2} \cos \left (d x + c\right )^{2} - 1568 \, a^{2} \cos \left (d x + c\right ) + 384 \, a^{2} - 768 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right )\right ) - 141 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 909 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{384 \, {\left (d \cos \left (d x + c\right )^{7} - 2 \, d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{5} + 4 \, d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}} \]
1/384*(1050*a^2*cos(d*x + c)^6 - 1716*a^2*cos(d*x + c)^5 - 1468*a^2*cos(d* x + c)^4 + 3308*a^2*cos(d*x + c)^3 - 38*a^2*cos(d*x + c)^2 - 1568*a^2*cos( d*x + c) + 384*a^2 - 768*(a^2*cos(d*x + c)^7 - 2*a^2*cos(d*x + c)^6 - a^2* cos(d*x + c)^5 + 4*a^2*cos(d*x + c)^4 - a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(-cos(d*x + c)) - 141*(a^2*cos(d*x + c)^7 - 2*a^2*cos(d*x + c)^6 - a^2*cos(d*x + c)^5 + 4*a^2*cos(d*x + c)^4 - a^2*co s(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 909*(a^2*cos(d*x + c)^7 - 2*a^2*cos(d*x + c)^6 - a^2*cos(d*x + c)^5 + 4*a^2*cos(d*x + c)^4 - a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^7 - 2*d*c os(d*x + c)^6 - d*cos(d*x + c)^5 + 4*d*cos(d*x + c)^4 - d*cos(d*x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos(d*x + c))
Timed out. \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.96 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {141 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) - 909 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) + 768 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (525 \, a^{2} \cos \left (d x + c\right )^{6} - 858 \, a^{2} \cos \left (d x + c\right )^{5} - 734 \, a^{2} \cos \left (d x + c\right )^{4} + 1654 \, a^{2} \cos \left (d x + c\right )^{3} - 19 \, a^{2} \cos \left (d x + c\right )^{2} - 784 \, a^{2} \cos \left (d x + c\right ) + 192 \, a^{2}\right )}}{\cos \left (d x + c\right )^{7} - 2 \, \cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )}}{384 \, d} \]
-1/384*(141*a^2*log(cos(d*x + c) + 1) - 909*a^2*log(cos(d*x + c) - 1) + 76 8*a^2*log(cos(d*x + c)) - 2*(525*a^2*cos(d*x + c)^6 - 858*a^2*cos(d*x + c) ^5 - 734*a^2*cos(d*x + c)^4 + 1654*a^2*cos(d*x + c)^3 - 19*a^2*cos(d*x + c )^2 - 784*a^2*cos(d*x + c) + 192*a^2)/(cos(d*x + c)^7 - 2*cos(d*x + c)^6 - cos(d*x + c)^5 + 4*cos(d*x + c)^4 - cos(d*x + c)^3 - 2*cos(d*x + c)^2 + c os(d*x + c)))/d
Time = 0.38 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.42 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {3636 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 3072 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {120 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (3 \, a^{2} - \frac {40 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {282 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1680 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {7575 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}} + \frac {3072 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{1536 \, d} \]
1/1536*(3636*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 3072* a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - 120*a^2*(cos(d* x + c) - 1)/(cos(d*x + c) + 1) + 6*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - (3*a^2 - 40*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 282*a^2*( cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1680*a^2*(cos(d*x + c) - 1)^3/( cos(d*x + c) + 1)^3 + 7575*a^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)* (cos(d*x + c) + 1)^4/(cos(d*x + c) - 1)^4 + 3072*(2*a^2 + a^2*(cos(d*x + c ) - 1)/(cos(d*x + c) + 1))/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/d
Time = 14.14 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.99 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {-\frac {175\,a^2\,{\cos \left (c+d\,x\right )}^6}{64}+\frac {143\,a^2\,{\cos \left (c+d\,x\right )}^5}{32}+\frac {367\,a^2\,{\cos \left (c+d\,x\right )}^4}{96}-\frac {827\,a^2\,{\cos \left (c+d\,x\right )}^3}{96}+\frac {19\,a^2\,{\cos \left (c+d\,x\right )}^2}{192}+\frac {49\,a^2\,\cos \left (c+d\,x\right )}{12}-a^2}{d\,\left (-{\cos \left (c+d\,x\right )}^7+2\,{\cos \left (c+d\,x\right )}^6+{\cos \left (c+d\,x\right )}^5-4\,{\cos \left (c+d\,x\right )}^4+{\cos \left (c+d\,x\right )}^3+2\,{\cos \left (c+d\,x\right )}^2-\cos \left (c+d\,x\right )\right )}+\frac {303\,a^2\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{128\,d}-\frac {47\,a^2\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{128\,d}-\frac {2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
((49*a^2*cos(c + d*x))/12 - a^2 + (19*a^2*cos(c + d*x)^2)/192 - (827*a^2*c os(c + d*x)^3)/96 + (367*a^2*cos(c + d*x)^4)/96 + (143*a^2*cos(c + d*x)^5) /32 - (175*a^2*cos(c + d*x)^6)/64)/(d*(2*cos(c + d*x)^2 - cos(c + d*x) + c os(c + d*x)^3 - 4*cos(c + d*x)^4 + cos(c + d*x)^5 + 2*cos(c + d*x)^6 - cos (c + d*x)^7)) + (303*a^2*log(cos(c + d*x) - 1))/(128*d) - (47*a^2*log(cos( c + d*x) + 1))/(128*d) - (2*a^2*log(cos(c + d*x)))/d