Integrand size = 21, antiderivative size = 232 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {17 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {16 a^3 \cot (c+d x)}{d}-\frac {34 a^3 \cot ^3(c+d x)}{3 d}-\frac {36 a^3 \cot ^5(c+d x)}{5 d}-\frac {19 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {17 a^3 \csc (c+d x)}{2 d}-\frac {17 a^3 \csc ^3(c+d x)}{6 d}-\frac {17 a^3 \csc ^5(c+d x)}{10 d}-\frac {17 a^3 \csc ^7(c+d x)}{14 d}-\frac {17 a^3 \csc ^9(c+d x)}{18 d}+\frac {a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \tan (c+d x)}{d} \]
17/2*a^3*arctanh(sin(d*x+c))/d-16*a^3*cot(d*x+c)/d-34/3*a^3*cot(d*x+c)^3/d -36/5*a^3*cot(d*x+c)^5/d-19/7*a^3*cot(d*x+c)^7/d-4/9*a^3*cot(d*x+c)^9/d-17 /2*a^3*csc(d*x+c)/d-17/6*a^3*csc(d*x+c)^3/d-17/10*a^3*csc(d*x+c)^5/d-17/14 *a^3*csc(d*x+c)^7/d-17/18*a^3*csc(d*x+c)^9/d+1/2*a^3*csc(d*x+c)^9*sec(d*x+ c)^2/d+3*a^3*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(1000\) vs. \(2(232)=464\).
Time = 12.60 (sec) , antiderivative size = 1000, normalized size of antiderivative = 4.31 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {9833 \cos ^3(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{80640 d}-\frac {979 \cos ^3(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{53760 d}-\frac {5 \cos ^3(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{2016 d}-\frac {\cos ^3(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{4608 d}-\frac {17 \cos ^3(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{16 d}+\frac {17 \cos ^3(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{16 d}+\frac {197147 \cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{161280 d}+\frac {9833 \cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{80640 d}+\frac {979 \cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{53760 d}+\frac {5 \cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{2016 d}+\frac {\cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{4608 d}-\frac {35 \cos ^3(c+d x) \sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{1536 d}-\frac {\cos ^3(c+d x) \sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{1536 d}+\frac {\cos (c+d x) \sec (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin (d x)}{16 d}+\frac {\cos ^2(c+d x) \sec (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 (\sin (c)+6 \sin (d x))}{16 d}-\frac {\cos ^3(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \tan \left (\frac {c}{2}\right )}{1536 d} \]
(-9833*Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^2*Sec[c/2 + (d*x)/2]^6*( a + a*Sec[c + d*x])^3)/(80640*d) - (979*Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(53760*d) - (5*Cos [c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^6*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[ c + d*x])^3)/(2016*d) - (Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^8*Sec[ c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(4608*d) - (17*Cos[c + d*x]^3*Log [Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[ c + d*x])^3)/(16*d) + (17*Cos[c + d*x]^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(16*d) + (197147* Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec [c + d*x])^3*Sin[(d*x)/2])/(161280*d) + (9833*Cos[c + d*x]^3*Csc[c/2]*Csc[ c/2 + (d*x)/2]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2]) /(80640*d) + (979*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^5*Sec[c/2 + ( d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(53760*d) + (5*Cos[c + d*x] ^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^7*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x]) ^3*Sin[(d*x)/2])/(2016*d) + (Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^9* Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(4608*d) - (35*C os[c + d*x]^3*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(a + a*Sec[c + d*x])^3*Sin[(d* x)/2])/(1536*d) - (Cos[c + d*x]^3*Sec[c/2]*Sec[c/2 + (d*x)/2]^9*(a + a*Sec [c + d*x])^3*Sin[(d*x)/2])/(1536*d) + (Cos[c + d*x]*Sec[c]*Sec[c/2 + (d...
Time = 0.99 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4360, 25, 25, 3042, 25, 3352, 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.
| \(\displaystyle \int \csc ^{10}(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^{10}}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^{10}(c+d x) \sec ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \csc ^{10}(c+d x) \sec ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^{10}(c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )^{10}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^{10} \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle -\int \left (-a^3 \sec ^{10}\left (\frac {1}{2} (2 c-\pi )+d x\right )-a^3 \sec ^3(c+d x) \sec ^{10}\left (\frac {1}{2} (2 c-\pi )+d x\right )-3 a^3 \sec ^2(c+d x) \sec ^{10}\left (\frac {1}{2} (2 c-\pi )+d x\right )-3 a^3 \sec (c+d x) \sec ^{10}\left (\frac {1}{2} (2 c-\pi )+d x\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {17 a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {3 a^3 \tan (c+d x)}{d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {19 a^3 \cot ^7(c+d x)}{7 d}-\frac {36 a^3 \cot ^5(c+d x)}{5 d}-\frac {34 a^3 \cot ^3(c+d x)}{3 d}-\frac {16 a^3 \cot (c+d x)}{d}-\frac {17 a^3 \csc ^9(c+d x)}{18 d}-\frac {17 a^3 \csc ^7(c+d x)}{14 d}-\frac {17 a^3 \csc ^5(c+d x)}{10 d}-\frac {17 a^3 \csc ^3(c+d x)}{6 d}-\frac {17 a^3 \csc (c+d x)}{2 d}+\frac {a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}\) |
(17*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (16*a^3*Cot[c + d*x])/d - (34*a^3*C ot[c + d*x]^3)/(3*d) - (36*a^3*Cot[c + d*x]^5)/(5*d) - (19*a^3*Cot[c + d*x ]^7)/(7*d) - (4*a^3*Cot[c + d*x]^9)/(9*d) - (17*a^3*Csc[c + d*x])/(2*d) - (17*a^3*Csc[c + d*x]^3)/(6*d) - (17*a^3*Csc[c + d*x]^5)/(10*d) - (17*a^3*C sc[c + d*x]^7)/(14*d) - (17*a^3*Csc[c + d*x]^9)/(18*d) + (a^3*Csc[c + d*x] ^9*Sec[c + d*x]^2)/(2*d) + (3*a^3*Tan[c + d*x])/d
3.1.56.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 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]
Result contains complex when optimal does not.
Time = 1.75 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(-\frac {i a^{3} \left (5355 \,{\mathrm e}^{15 i \left (d x +c \right )}-32130 \,{\mathrm e}^{14 i \left (d x +c \right )}+73185 \,{\mathrm e}^{13 i \left (d x +c \right )}-64260 \,{\mathrm e}^{12 i \left (d x +c \right )}-34629 \,{\mathrm e}^{11 i \left (d x +c \right )}+157794 \,{\mathrm e}^{10 i \left (d x +c \right )}-207111 \,{\mathrm e}^{9 i \left (d x +c \right )}+125256 \,{\mathrm e}^{8 i \left (d x +c \right )}+62713 \,{\mathrm e}^{7 i \left (d x +c \right )}-175518 \,{\mathrm e}^{6 i \left (d x +c \right )}+171707 \,{\mathrm e}^{5 i \left (d x +c \right )}-80132 \,{\mathrm e}^{4 i \left (d x +c \right )}-37919 \,{\mathrm e}^{3 i \left (d x +c \right )}+78974 \,{\mathrm e}^{2 i \left (d x +c \right )}-42261 \,{\mathrm e}^{i \left (d x +c \right )}+7936\right )}{315 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\) | \(259\) |
| parallelrisch | \(-\frac {1673 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\frac {31314}{11711}+\frac {765 \left (\frac {\sin \left (8 d x +8 c \right )}{14}-\frac {3 \sin \left (7 d x +7 c \right )}{7}-\sin \left (5 d x +5 c \right )+\sin \left (6 d x +6 c \right )-\frac {\sin \left (4 d x +4 c \right )}{7}+\frac {13 \sin \left (3 d x +3 c \right )}{7}+\frac {17 \sin \left (d x +c \right )}{7}-3 \sin \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{478}+\frac {765 \left (\frac {3 \sin \left (7 d x +7 c \right )}{7}-\frac {\sin \left (8 d x +8 c \right )}{14}+\frac {\sin \left (4 d x +4 c \right )}{7}-\frac {13 \sin \left (3 d x +3 c \right )}{7}-\sin \left (6 d x +6 c \right )+\sin \left (5 d x +5 c \right )-\frac {17 \sin \left (d x +c \right )}{7}+3 \sin \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{478}-\frac {18453 \cos \left (7 d x +7 c \right )}{23422}+\cos \left (6 d x +6 c \right )-\frac {36098 \cos \left (4 d x +4 c \right )}{11711}-\frac {72199 \cos \left (d x +c \right )}{23422}-\frac {633 \cos \left (2 d x +2 c \right )}{1673}+\frac {2519 \cos \left (5 d x +5 c \right )}{3346}+\frac {68539 \cos \left (3 d x +3 c \right )}{23422}+\frac {1984 \cos \left (8 d x +8 c \right )}{11711}\right ) a^{3} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{92160 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(325\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )^{2}}-\frac {11}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )^{2}}-\frac {11}{35 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {11}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {11}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {11}{2 \sin \left (d x +c \right )}+\frac {11 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )}-\frac {10}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {16}{63 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {32}{63 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {128}{63 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {256 \cot \left (d x +c \right )}{63}\right )+3 a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {128}{315}-\frac {\csc \left (d x +c \right )^{8}}{9}-\frac {8 \csc \left (d x +c \right )^{6}}{63}-\frac {16 \csc \left (d x +c \right )^{4}}{105}-\frac {64 \csc \left (d x +c \right )^{2}}{315}\right ) \cot \left (d x +c \right )}{d}\) | \(353\) |
| default | \(\frac {a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )^{2}}-\frac {11}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )^{2}}-\frac {11}{35 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {11}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {11}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {11}{2 \sin \left (d x +c \right )}+\frac {11 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )}-\frac {10}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {16}{63 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {32}{63 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {128}{63 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {256 \cot \left (d x +c \right )}{63}\right )+3 a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {128}{315}-\frac {\csc \left (d x +c \right )^{8}}{9}-\frac {8 \csc \left (d x +c \right )^{6}}{63}-\frac {16 \csc \left (d x +c \right )^{4}}{105}-\frac {64 \csc \left (d x +c \right )^{2}}{315}\right ) \cot \left (d x +c \right )}{d}\) | \(353\) |
-1/315*I*a^3*(5355*exp(15*I*(d*x+c))-32130*exp(14*I*(d*x+c))+73185*exp(13* I*(d*x+c))-64260*exp(12*I*(d*x+c))-34629*exp(11*I*(d*x+c))+157794*exp(10*I *(d*x+c))-207111*exp(9*I*(d*x+c))+125256*exp(8*I*(d*x+c))+62713*exp(7*I*(d *x+c))-175518*exp(6*I*(d*x+c))+171707*exp(5*I*(d*x+c))-80132*exp(4*I*(d*x+ c))-37919*exp(3*I*(d*x+c))+78974*exp(2*I*(d*x+c))-42261*exp(I*(d*x+c))+793 6)/d/(exp(I*(d*x+c))-1)^9/(exp(I*(d*x+c))+1)^3/(exp(2*I*(d*x+c))+1)^2-17/2 *a^3/d*ln(exp(I*(d*x+c))-I)+17/2*a^3/d*ln(exp(I*(d*x+c))+I)
Time = 0.29 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.62 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {15872 \, a^{3} \cos \left (d x + c\right )^{8} - 36906 \, a^{3} \cos \left (d x + c\right )^{7} - 8322 \, a^{3} \cos \left (d x + c\right )^{6} + 73402 \, a^{3} \cos \left (d x + c\right )^{5} - 33342 \, a^{3} \cos \left (d x + c\right )^{4} - 34746 \, a^{3} \cos \left (d x + c\right )^{3} + 26702 \, a^{3} \cos \left (d x + c\right )^{2} - 1890 \, a^{3} \cos \left (d x + c\right ) - 630 \, a^{3} - 5355 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 5355 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{1260 \, {\left (d \cos \left (d x + c\right )^{7} - 3 \, d \cos \left (d x + c\right )^{6} + 2 \, d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]
-1/1260*(15872*a^3*cos(d*x + c)^8 - 36906*a^3*cos(d*x + c)^7 - 8322*a^3*co s(d*x + c)^6 + 73402*a^3*cos(d*x + c)^5 - 33342*a^3*cos(d*x + c)^4 - 34746 *a^3*cos(d*x + c)^3 + 26702*a^3*cos(d*x + c)^2 - 1890*a^3*cos(d*x + c) - 6 30*a^3 - 5355*(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(sin(d*x + c) + 1)*sin(d*x + c) + 5355*(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(-sin(d*x + c) + 1)*sin(d*x + c))/((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)*sin(d*x + c))
Timed out. \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.33 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^{3} {\left (\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{10} - 2310 \, \sin \left (d x + c\right )^{8} - 462 \, \sin \left (d x + c\right )^{6} - 198 \, \sin \left (d x + c\right )^{4} - 110 \, \sin \left (d x + c\right )^{2} - 70\right )}}{\sin \left (d x + c\right )^{11} - \sin \left (d x + c\right )^{9}} - 3465 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3465 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{6} + 63 \, \sin \left (d x + c\right )^{4} + 45 \, \sin \left (d x + c\right )^{2} + 35\right )}}{\sin \left (d x + c\right )^{9}} - 315 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 315 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 60 \, a^{3} {\left (\frac {315 \, \tan \left (d x + c\right )^{8} + 210 \, \tan \left (d x + c\right )^{6} + 126 \, \tan \left (d x + c\right )^{4} + 45 \, \tan \left (d x + c\right )^{2} + 7}{\tan \left (d x + c\right )^{9}} - 63 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (315 \, \tan \left (d x + c\right )^{8} + 420 \, \tan \left (d x + c\right )^{6} + 378 \, \tan \left (d x + c\right )^{4} + 180 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{1260 \, d} \]
-1/1260*(a^3*(2*(3465*sin(d*x + c)^10 - 2310*sin(d*x + c)^8 - 462*sin(d*x + c)^6 - 198*sin(d*x + c)^4 - 110*sin(d*x + c)^2 - 70)/(sin(d*x + c)^11 - sin(d*x + c)^9) - 3465*log(sin(d*x + c) + 1) + 3465*log(sin(d*x + c) - 1)) + 6*a^3*(2*(315*sin(d*x + c)^8 + 105*sin(d*x + c)^6 + 63*sin(d*x + c)^4 + 45*sin(d*x + c)^2 + 35)/sin(d*x + c)^9 - 315*log(sin(d*x + c) + 1) + 315* log(sin(d*x + c) - 1)) + 60*a^3*((315*tan(d*x + c)^8 + 210*tan(d*x + c)^6 + 126*tan(d*x + c)^4 + 45*tan(d*x + c)^2 + 7)/tan(d*x + c)^9 - 63*tan(d*x + c)) + 4*(315*tan(d*x + c)^8 + 420*tan(d*x + c)^6 + 378*tan(d*x + c)^4 + 180*tan(d*x + c)^2 + 35)*a^3/tan(d*x + c)^9)/d
Time = 0.46 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.87 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 171360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 171360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3780 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {20160 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {220185 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 26880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4347 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 540 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{20160 \, d} \]
-1/20160*(105*a^3*tan(1/2*d*x + 1/2*c)^3 - 171360*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) + 171360*a^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 3780*a^3* tan(1/2*d*x + 1/2*c) + 20160*(5*a^3*tan(1/2*d*x + 1/2*c)^3 - 7*a^3*tan(1/2 *d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2 + (220185*a^3*tan(1/2*d*x + 1/2*c)^8 + 26880*a^3*tan(1/2*d*x + 1/2*c)^6 + 4347*a^3*tan(1/2*d*x + 1/2*c )^4 + 540*a^3*tan(1/2*d*x + 1/2*c)^2 + 35*a^3)/tan(1/2*d*x + 1/2*c)^9)/d
Time = 14.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.88 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {17\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {1019\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {5282\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+\frac {8132\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{15}+\frac {6242\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{105}+\frac {3302\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{315}+\frac {94\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}+\frac {a^3}{9}}{d\,\left (64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\right )} \]
(17*a^3*atanh(tan(c/2 + (d*x)/2)))/d - (a^3*tan(c/2 + (d*x)/2)^3)/(192*d) - (3*a^3*tan(c/2 + (d*x)/2))/(16*d) - ((94*a^3*tan(c/2 + (d*x)/2)^2)/63 + (3302*a^3*tan(c/2 + (d*x)/2)^4)/315 + (6242*a^3*tan(c/2 + (d*x)/2)^6)/105 + (8132*a^3*tan(c/2 + (d*x)/2)^8)/15 - (5282*a^3*tan(c/2 + (d*x)/2)^10)/3 + 1019*a^3*tan(c/2 + (d*x)/2)^12 + a^3/9)/(d*(64*tan(c/2 + (d*x)/2)^9 - 12 8*tan(c/2 + (d*x)/2)^11 + 64*tan(c/2 + (d*x)/2)^13))