Integrand size = 21, antiderivative size = 202 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^7}{16 d (a-a \cos (c+d x))^4}-\frac {a^6}{3 d (a-a \cos (c+d x))^3}-\frac {39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac {75 a^4}{16 d (a-a \cos (c+d x))}-\frac {a^4}{32 d (a+a \cos (c+d x))}+\frac {501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac {8 a^3 \log (\cos (c+d x))}{d}+\frac {11 a^3 \log (1+\cos (c+d x))}{64 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
-1/16*a^7/d/(a-a*cos(d*x+c))^4-1/3*a^6/d/(a-a*cos(d*x+c))^3-39/32*a^5/d/(a -a*cos(d*x+c))^2-75/16*a^4/d/(a-a*cos(d*x+c))-1/32*a^4/d/(a+a*cos(d*x+c))+ 501/64*a^3*ln(1-cos(d*x+c))/d-8*a^3*ln(cos(d*x+c))/d+11/64*a^3*ln(1+cos(d* x+c))/d+3*a^3*sec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d
Time = 0.89 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.79 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (1800 \csc ^2\left (\frac {1}{2} (c+d x)\right )+234 \csc ^4\left (\frac {1}{2} (c+d x)\right )+32 \csc ^6\left (\frac {1}{2} (c+d x)\right )+3 \csc ^8\left (\frac {1}{2} (c+d x)\right )-12 \left (22 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-512 \log (\cos (c+d x))+1002 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )+192 \sec (c+d x)+32 \sec ^2(c+d x)\right )\right )}{6144 d} \]
-1/6144*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(1800*Csc[(c + d*x)/2 ]^2 + 234*Csc[(c + d*x)/2]^4 + 32*Csc[(c + d*x)/2]^6 + 3*Csc[(c + d*x)/2]^ 8 - 12*(22*Log[Cos[(c + d*x)/2]] - 512*Log[Cos[c + d*x]] + 1002*Log[Sin[(c + d*x)/2]] - Sec[(c + d*x)/2]^2 + 192*Sec[c + d*x] + 32*Sec[c + d*x]^2))) /d
Time = 0.51 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.92, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 25, 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.
| \(\displaystyle \int \csc ^9(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^9}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^9(c+d x) \sec ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \csc ^9(c+d x) \sec ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^9(c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )^9}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^9 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^9 \int -\frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^9 \int \frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^{12} \int \frac {\sec ^3(c+d x)}{a^3 (a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^{12} \int \left (\frac {\sec ^3(c+d x)}{a^{10}}+\frac {3 \sec ^2(c+d x)}{a^{10}}+\frac {8 \sec (c+d x)}{a^{10}}+\frac {501}{64 a^9 (a-a \cos (c+d x))}-\frac {11}{64 a^9 (\cos (c+d x) a+a)}+\frac {75}{16 a^8 (a-a \cos (c+d x))^2}-\frac {1}{32 a^8 (\cos (c+d x) a+a)^2}+\frac {39}{16 a^7 (a-a \cos (c+d x))^3}+\frac {1}{a^6 (a-a \cos (c+d x))^4}+\frac {1}{4 a^5 (a-a \cos (c+d x))^5}\right )d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{12} \left (-\frac {\sec ^2(c+d x)}{2 a^9}-\frac {3 \sec (c+d x)}{a^9}+\frac {8 \log (a \cos (c+d x))}{a^9}-\frac {501 \log (a-a \cos (c+d x))}{64 a^9}-\frac {11 \log (a \cos (c+d x)+a)}{64 a^9}+\frac {75}{16 a^8 (a-a \cos (c+d x))}+\frac {1}{32 a^8 (a \cos (c+d x)+a)}+\frac {39}{32 a^7 (a-a \cos (c+d x))^2}+\frac {1}{3 a^6 (a-a \cos (c+d x))^3}+\frac {1}{16 a^5 (a-a \cos (c+d x))^4}\right )}{d}\) |
-((a^12*(1/(16*a^5*(a - a*Cos[c + d*x])^4) + 1/(3*a^6*(a - a*Cos[c + d*x]) ^3) + 39/(32*a^7*(a - a*Cos[c + d*x])^2) + 75/(16*a^8*(a - a*Cos[c + d*x]) ) + 1/(32*a^8*(a + a*Cos[c + d*x])) + (8*Log[a*Cos[c + d*x]])/a^9 - (501*L og[a - a*Cos[c + d*x]])/(64*a^9) - (11*Log[a + a*Cos[c + d*x]])/(64*a^9) - (3*Sec[c + d*x])/a^9 - Sec[c + d*x]^2/(2*a^9)))/d)
3.1.47.3.1 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 = 1.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.04
| method | result | size |
| norman | \(\frac {-\frac {a^{3}}{256 d}-\frac {19 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{384 d}-\frac {263 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{768 d}-\frac {431 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{192 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{64 d}-\frac {451 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64 d}+\frac {749 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {501 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(210\) |
| parallelrisch | \(-\frac {a^{3} \left (2048 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2048 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-4008 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {38 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {263 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3}+\frac {1724 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}+1804 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2996\right )}{256 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}\) | \(231\) |
| risch | \(\frac {a^{3} \left (735 \,{\mathrm e}^{13 i \left (d x +c \right )}-3642 \,{\mathrm e}^{12 i \left (d x +c \right )}+6662 \,{\mathrm e}^{11 i \left (d x +c \right )}-4650 \,{\mathrm e}^{10 i \left (d x +c \right )}-1983 \,{\mathrm e}^{9 i \left (d x +c \right )}+8868 \,{\mathrm e}^{8 i \left (d x +c \right )}-12748 \,{\mathrm e}^{7 i \left (d x +c \right )}+8868 \,{\mathrm e}^{6 i \left (d x +c \right )}-1983 \,{\mathrm e}^{5 i \left (d x +c \right )}-4650 \,{\mathrm e}^{4 i \left (d x +c \right )}+6662 \,{\mathrm e}^{3 i \left (d x +c \right )}-3642 \,{\mathrm e}^{2 i \left (d x +c \right )}+735 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{48 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {501 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{32 d}+\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{32 d}-\frac {8 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(253\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )^{2}}-\frac {5}{24 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {5}{12 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {5}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )^{2}}+5 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )}-\frac {3}{16 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {21}{64 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {105}{128 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {315}{128 \cos \left (d x +c \right )}+\frac {315 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{7}}{8}-\frac {7 \csc \left (d x +c \right )^{5}}{48}-\frac {35 \csc \left (d x +c \right )^{3}}{192}-\frac {35 \csc \left (d x +c \right )}{128}\right ) \cot \left (d x +c \right )+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )}{d}\) | \(330\) |
| default | \(\frac {a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )^{2}}-\frac {5}{24 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {5}{12 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {5}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )^{2}}+5 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )}-\frac {3}{16 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {21}{64 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {105}{128 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {315}{128 \cos \left (d x +c \right )}+\frac {315 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{7}}{8}-\frac {7 \csc \left (d x +c \right )^{5}}{48}-\frac {35 \csc \left (d x +c \right )^{3}}{192}-\frac {35 \csc \left (d x +c \right )}{128}\right ) \cot \left (d x +c \right )+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )}{d}\) | \(330\) |
(-1/256/d*a^3-19/384*a^3/d*tan(1/2*d*x+1/2*c)^2-263/768*a^3/d*tan(1/2*d*x+ 1/2*c)^4-431/192*a^3/d*tan(1/2*d*x+1/2*c)^6-1/64/d*a^3*tan(1/2*d*x+1/2*c)^ 14-451/64*a^3/d*tan(1/2*d*x+1/2*c)^10+749/64*a^3/d*tan(1/2*d*x+1/2*c)^8)/t an(1/2*d*x+1/2*c)^8/(-1+tan(1/2*d*x+1/2*c)^2)^2+501/32/d*a^3*ln(tan(1/2*d* x+1/2*c))-8/d*a^3*ln(tan(1/2*d*x+1/2*c)-1)-8/d*a^3*ln(tan(1/2*d*x+1/2*c)+1 )
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (190) = 380\).
Time = 0.29 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.07 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {1470 \, a^{3} \cos \left (d x + c\right )^{6} - 3642 \, a^{3} \cos \left (d x + c\right )^{5} + 1126 \, a^{3} \cos \left (d x + c\right )^{4} + 3390 \, a^{3} \cos \left (d x + c\right )^{3} - 2752 \, a^{3} \cos \left (d x + c\right )^{2} + 288 \, a^{3} \cos \left (d x + c\right ) + 96 \, a^{3} - 1536 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1503 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{192 \, {\left (d \cos \left (d x + c\right )^{7} - 3 \, d \cos \left (d x + c\right )^{6} + 2 \, d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}} \]
1/192*(1470*a^3*cos(d*x + c)^6 - 3642*a^3*cos(d*x + c)^5 + 1126*a^3*cos(d* x + c)^4 + 3390*a^3*cos(d*x + c)^3 - 2752*a^3*cos(d*x + c)^2 + 288*a^3*cos (d*x + c) + 96*a^3 - 1536*(a^3*cos(d*x + c)^7 - 3*a^3*cos(d*x + c)^6 + 2*a ^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^3 + a^3*cos( d*x + c)^2)*log(-cos(d*x + c)) + 33*(a^3*cos(d*x + c)^7 - 3*a^3*cos(d*x + c)^6 + 2*a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + 1503*(a^3*cos(d*x + c) ^7 - 3*a^3*cos(d*x + c)^6 + 2*a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/( d*cos(d*x + c)^7 - 3*d*cos(d*x + c)^6 + 2*d*cos(d*x + c)^5 + 2*d*cos(d*x + c)^4 - 3*d*cos(d*x + c)^3 + d*cos(d*x + c)^2)
Timed out. \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {33 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 1503 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 1536 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (735 \, a^{3} \cos \left (d x + c\right )^{6} - 1821 \, a^{3} \cos \left (d x + c\right )^{5} + 563 \, a^{3} \cos \left (d x + c\right )^{4} + 1695 \, a^{3} \cos \left (d x + c\right )^{3} - 1376 \, a^{3} \cos \left (d x + c\right )^{2} + 144 \, a^{3} \cos \left (d x + c\right ) + 48 \, a^{3}\right )}}{\cos \left (d x + c\right )^{7} - 3 \, \cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}}{192 \, d} \]
1/192*(33*a^3*log(cos(d*x + c) + 1) + 1503*a^3*log(cos(d*x + c) - 1) - 153 6*a^3*log(cos(d*x + c)) + 2*(735*a^3*cos(d*x + c)^6 - 1821*a^3*cos(d*x + c )^5 + 563*a^3*cos(d*x + c)^4 + 1695*a^3*cos(d*x + c)^3 - 1376*a^3*cos(d*x + c)^2 + 144*a^3*cos(d*x + c) + 48*a^3)/(cos(d*x + c)^7 - 3*cos(d*x + c)^6 + 2*cos(d*x + c)^5 + 2*cos(d*x + c)^4 - 3*cos(d*x + c)^3 + cos(d*x + c)^2 ))/d
Time = 0.45 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.45 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {6012 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 6144 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {12 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {{\left (3 \, a^{3} - \frac {44 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {348 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2376 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {12525 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}} + \frac {1536 \, {\left (9 \, a^{3} + \frac {14 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{768 \, d} \]
1/768*(6012*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 6144*a ^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + 12*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - (3*a^3 - 44*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 348*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 2376*a^3* (cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 12525*a^3*(cos(d*x + c) - 1)^4 /(cos(d*x + c) + 1)^4)*(cos(d*x + c) + 1)^4/(cos(d*x + c) - 1)^4 + 1536*(9 *a^3 + 14*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 6*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2 )/d
Time = 13.43 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.97 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {501\,a^3\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{64\,d}+\frac {11\,a^3\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{64\,d}+\frac {\frac {245\,a^3\,{\cos \left (c+d\,x\right )}^6}{32}-\frac {607\,a^3\,{\cos \left (c+d\,x\right )}^5}{32}+\frac {563\,a^3\,{\cos \left (c+d\,x\right )}^4}{96}+\frac {565\,a^3\,{\cos \left (c+d\,x\right )}^3}{32}-\frac {43\,a^3\,{\cos \left (c+d\,x\right )}^2}{3}+\frac {3\,a^3\,\cos \left (c+d\,x\right )}{2}+\frac {a^3}{2}}{d\,\left ({\cos \left (c+d\,x\right )}^7-3\,{\cos \left (c+d\,x\right )}^6+2\,{\cos \left (c+d\,x\right )}^5+2\,{\cos \left (c+d\,x\right )}^4-3\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {8\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
(501*a^3*log(cos(c + d*x) - 1))/(64*d) + (11*a^3*log(cos(c + d*x) + 1))/(6 4*d) + ((3*a^3*cos(c + d*x))/2 + a^3/2 - (43*a^3*cos(c + d*x)^2)/3 + (565* a^3*cos(c + d*x)^3)/32 + (563*a^3*cos(c + d*x)^4)/96 - (607*a^3*cos(c + d* x)^5)/32 + (245*a^3*cos(c + d*x)^6)/32)/(d*(cos(c + d*x)^2 - 3*cos(c + d*x )^3 + 2*cos(c + d*x)^4 + 2*cos(c + d*x)^5 - 3*cos(c + d*x)^6 + cos(c + d*x )^7)) - (8*a^3*log(cos(c + d*x)))/d