Integrand size = 29, antiderivative size = 286 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d} \] Output:
27/1024*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-a^3*cot(d*x+c)^9/ d-6/11*a^3*cot(d*x+c)^11/d-1/13*a^3*cot(d*x+c)^13/d+27/1024*a^3*cot(d*x+c) *csc(d*x+c)/d+9/512*a^3*cot(d*x+c)*csc(d*x+c)^3/d-3/128*a^3*cot(d*x+c)*csc (d*x+c)^5/d+1/16*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d-1/10*a^3*cot(d*x+c)^5*csc (d*x+c)^5/d-3/64*a^3*cot(d*x+c)*csc(d*x+c)^7/d+1/8*a^3*cot(d*x+c)^3*csc(d* x+c)^7/d-1/4*a^3*cot(d*x+c)^5*csc(d*x+c)^7/d
Time = 13.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.99 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+a \sin (c+d x))^3}{1024 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {27 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+a \sin (c+d x))^3}{1024 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {\cot (c+d x) \csc ^{12}(c+d x) (a+a \sin (c+d x))^3 (-200294400-243712000 \cos (2 (c+d x))-11079680 \cos (4 (c+d x))+43294720 \cos (6 (c+d x))+9420800 \cos (8 (c+d x))-1433600 \cos (10 (c+d x))+102400 \cos (12 (c+d x))-194159966 \sin (c+d x)-182107926 \sin (3 (c+d x))-123736613 \sin (5 (c+d x))+4571567 \sin (7 (c+d x))+1846845 \sin (9 (c+d x))-135135 \sin (11 (c+d x)))}{5248122880 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3,x]
Output:
(27*Log[Cos[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(1024*d*(Cos[(c + d*x)/2 ] + Sin[(c + d*x)/2])^6) - (27*Log[Sin[(c + d*x)/2]]*(a + a*Sin[c + d*x])^ 3)/(1024*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Cot[c + d*x]*Csc[c + d*x]^12*(a + a*Sin[c + d*x])^3*(-200294400 - 243712000*Cos[2*(c + d*x)] - 11079680*Cos[4*(c + d*x)] + 43294720*Cos[6*(c + d*x)] + 9420800*Cos[8*(c + d*x)] - 1433600*Cos[10*(c + d*x)] + 102400*Cos[12*(c + d*x)] - 19415996 6*Sin[c + d*x] - 182107926*Sin[3*(c + d*x)] - 123736613*Sin[5*(c + d*x)] + 4571567*Sin[7*(c + d*x)] + 1846845*Sin[9*(c + d*x)] - 135135*Sin[11*(c + d*x)]))/(5248122880*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.77 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^8(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^3}{\sin (c+d x)^{14}}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^6(c+d x) \csc ^8(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^7(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+a^3 \cot ^6(c+d x) \csc ^5(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {27 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3,x]
Output:
(27*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - ( a^3*Cot[c + d*x]^9)/d - (6*a^3*Cot[c + d*x]^11)/(11*d) - (a^3*Cot[c + d*x] ^13)/(13*d) + (27*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (9*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(512*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(128*d ) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^3*Cot[ c + d*x]^3*Csc[c + d*x]^7)/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(4* d)
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]
Result contains complex when optimal does not.
Time = 1.65 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(-\frac {a^{3} \left (135135 \,{\mathrm e}^{25 i \left (d x +c \right )}-1711710 \,{\mathrm e}^{23 i \left (d x +c \right )}-105431040 i {\mathrm e}^{8 i \left (d x +c \right )}-6418412 \,{\mathrm e}^{21 i \left (d x +c \right )}+41000960 i {\mathrm e}^{18 i \left (d x +c \right )}+119165046 \,{\mathrm e}^{19 i \left (d x +c \right )}+820019200 i {\mathrm e}^{14 i \left (d x +c \right )}+305844539 \,{\mathrm e}^{17 i \left (d x +c \right )}-23429120 i {\mathrm e}^{6 i \left (d x +c \right )}+376267892 \,{\mathrm e}^{15 i \left (d x +c \right )}+468582400 i {\mathrm e}^{12 i \left (d x +c \right )}-82001920 i {\mathrm e}^{20 i \left (d x +c \right )}-376267892 \,{\mathrm e}^{11 i \left (d x +c \right )}+123002880 i {\mathrm e}^{16 i \left (d x +c \right )}-305844539 \,{\mathrm e}^{9 i \left (d x +c \right )}+2662400 i {\mathrm e}^{2 i \left (d x +c \right )}-119165046 \,{\mathrm e}^{7 i \left (d x +c \right )}+386580480 i {\mathrm e}^{10 i \left (d x +c \right )}+6418412 \,{\mathrm e}^{5 i \left (d x +c \right )}-15974400 i {\mathrm e}^{4 i \left (d x +c \right )}+1711710 \,{\mathrm e}^{3 i \left (d x +c \right )}-204800 i-135135 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2562560 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{13}}-\frac {27 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{1024 d}+\frac {27 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{1024 d}\) | \(318\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \cos \left (d x +c \right )^{7}}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \cos \left (d x +c \right )^{7}}{693 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{7}}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{1024 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{1024}-\frac {5 \cos \left (d x +c \right )^{3}}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{13 \sin \left (d x +c \right )^{13}}-\frac {6 \cos \left (d x +c \right )^{7}}{143 \sin \left (d x +c \right )^{11}}-\frac {8 \cos \left (d x +c \right )^{7}}{429 \sin \left (d x +c \right )^{9}}-\frac {16 \cos \left (d x +c \right )^{7}}{3003 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(444\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \cos \left (d x +c \right )^{7}}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \cos \left (d x +c \right )^{7}}{693 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{7}}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{1024 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{1024}-\frac {5 \cos \left (d x +c \right )^{3}}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{13 \sin \left (d x +c \right )^{13}}-\frac {6 \cos \left (d x +c \right )^{7}}{143 \sin \left (d x +c \right )^{11}}-\frac {8 \cos \left (d x +c \right )^{7}}{429 \sin \left (d x +c \right )^{9}}-\frac {16 \cos \left (d x +c \right )^{7}}{3003 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(444\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/2562560*a^3*(135135*exp(25*I*(d*x+c))-1711710*exp(23*I*(d*x+c))-1054310 40*I*exp(8*I*(d*x+c))-6418412*exp(21*I*(d*x+c))+41000960*I*exp(18*I*(d*x+c ))+119165046*exp(19*I*(d*x+c))+820019200*I*exp(14*I*(d*x+c))+305844539*exp (17*I*(d*x+c))-23429120*I*exp(6*I*(d*x+c))+376267892*exp(15*I*(d*x+c))+468 582400*I*exp(12*I*(d*x+c))-82001920*I*exp(20*I*(d*x+c))-376267892*exp(11*I *(d*x+c))+123002880*I*exp(16*I*(d*x+c))-305844539*exp(9*I*(d*x+c))+2662400 *I*exp(2*I*(d*x+c))-119165046*exp(7*I*(d*x+c))+386580480*I*exp(10*I*(d*x+c ))+6418412*exp(5*I*(d*x+c))-15974400*I*exp(4*I*(d*x+c))+1711710*exp(3*I*(d *x+c))-204800*I-135135*exp(I*(d*x+c)))/d/(exp(2*I*(d*x+c))-1)^13-27/1024*a ^3/d*ln(exp(I*(d*x+c))-1)+27/1024*a^3/d*ln(exp(I*(d*x+c))+1)
Time = 0.13 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.46 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {409600 \, a^{3} \cos \left (d x + c\right )^{13} - 2662400 \, a^{3} \cos \left (d x + c\right )^{11} + 7321600 \, a^{3} \cos \left (d x + c\right )^{9} - 5857280 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 135135 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2002 \, {\left (135 \, a^{3} \cos \left (d x + c\right )^{11} - 765 \, a^{3} \cos \left (d x + c\right )^{9} + 758 \, a^{3} \cos \left (d x + c\right )^{7} + 1782 \, a^{3} \cos \left (d x + c\right )^{5} - 765 \, a^{3} \cos \left (d x + c\right )^{3} + 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{10250240 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/10250240*(409600*a^3*cos(d*x + c)^13 - 2662400*a^3*cos(d*x + c)^11 + 732 1600*a^3*cos(d*x + c)^9 - 5857280*a^3*cos(d*x + c)^7 + 135135*(a^3*cos(d*x + c)^12 - 6*a^3*cos(d*x + c)^10 + 15*a^3*cos(d*x + c)^8 - 20*a^3*cos(d*x + c)^6 + 15*a^3*cos(d*x + c)^4 - 6*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d *x + c) + 1/2)*sin(d*x + c) - 135135*(a^3*cos(d*x + c)^12 - 6*a^3*cos(d*x + c)^10 + 15*a^3*cos(d*x + c)^8 - 20*a^3*cos(d*x + c)^6 + 15*a^3*cos(d*x + c)^4 - 6*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2002*(135*a^3*cos(d*x + c)^11 - 765*a^3*cos(d*x + c)^9 + 758*a^3*cos (d*x + c)^7 + 1782*a^3*cos(d*x + c)^5 - 765*a^3*cos(d*x + c)^3 + 135*a^3*c os(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^12 - 6*d*cos(d*x + c)^10 + 15* d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**8*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.29 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {15015 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12012 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {133120 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}} + \frac {10240 \, {\left (429 \, \tan \left (d x + c\right )^{6} + 1001 \, \tan \left (d x + c\right )^{4} + 819 \, \tan \left (d x + c\right )^{2} + 231\right )} a^{3}}{\tan \left (d x + c\right )^{13}}}{30750720 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/30750720*(15015*a^3*(2*(15*cos(d*x + c)^11 - 85*cos(d*x + c)^9 + 198*co s(d*x + c)^7 + 198*cos(d*x + c)^5 - 85*cos(d*x + c)^3 + 15*cos(d*x + c))/( cos(d*x + c)^12 - 6*cos(d*x + c)^10 + 15*cos(d*x + c)^8 - 20*cos(d*x + c)^ 6 + 15*cos(d*x + c)^4 - 6*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 12012*a^3*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*x + c))/(cos(d* x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*c os(d*x + c)^2 - 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 133120*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^3/tan(d*x + c)^11 + 10240*(429*tan(d*x + c)^6 + 1001*tan(d*x + c)^4 + 819*tan(d*x + c)^2 + 231)*a^3/tan(d*x + c)^13)/d
Time = 0.29 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.58 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/82001920*(770*a^3*tan(1/2*d*x + 1/2*c)^13 + 5005*a^3*tan(1/2*d*x + 1/2*c )^12 + 11830*a^3*tan(1/2*d*x + 1/2*c)^11 + 8008*a^3*tan(1/2*d*x + 1/2*c)^1 0 - 20020*a^3*tan(1/2*d*x + 1/2*c)^9 - 65065*a^3*tan(1/2*d*x + 1/2*c)^8 - 94380*a^3*tan(1/2*d*x + 1/2*c)^7 - 40040*a^3*tan(1/2*d*x + 1/2*c)^6 + 1501 50*a^3*tan(1/2*d*x + 1/2*c)^5 + 385385*a^3*tan(1/2*d*x + 1/2*c)^4 + 450450 *a^3*tan(1/2*d*x + 1/2*c)^3 + 80080*a^3*tan(1/2*d*x + 1/2*c)^2 - 2162160*a ^3*log(abs(tan(1/2*d*x + 1/2*c))) - 1401400*a^3*tan(1/2*d*x + 1/2*c) + (68 75958*a^3*tan(1/2*d*x + 1/2*c)^13 + 1401400*a^3*tan(1/2*d*x + 1/2*c)^12 - 80080*a^3*tan(1/2*d*x + 1/2*c)^11 - 450450*a^3*tan(1/2*d*x + 1/2*c)^10 - 3 85385*a^3*tan(1/2*d*x + 1/2*c)^9 - 150150*a^3*tan(1/2*d*x + 1/2*c)^8 + 400 40*a^3*tan(1/2*d*x + 1/2*c)^7 + 94380*a^3*tan(1/2*d*x + 1/2*c)^6 + 65065*a ^3*tan(1/2*d*x + 1/2*c)^5 + 20020*a^3*tan(1/2*d*x + 1/2*c)^4 - 8008*a^3*ta n(1/2*d*x + 1/2*c)^3 - 11830*a^3*tan(1/2*d*x + 1/2*c)^2 - 5005*a^3*tan(1/2 *d*x + 1/2*c) - 770*a^3)/tan(1/2*d*x + 1/2*c)^13)/d
Time = 35.19 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.78 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^8,x)
Output:
(a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) - (45*a^3*cot(c/2 + (d*x)/2)^3)/(8192* d) - (77*a^3*cot(c/2 + (d*x)/2)^4)/(16384*d) - (15*a^3*cot(c/2 + (d*x)/2)^ 5)/(8192*d) - (a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (33*a^3*cot(c/2 + (d*x )/2)^7)/(28672*d) + (13*a^3*cot(c/2 + (d*x)/2)^8)/(16384*d) + (a^3*cot(c/2 + (d*x)/2)^9)/(4096*d) - (a^3*cot(c/2 + (d*x)/2)^10)/(10240*d) - (13*a^3* cot(c/2 + (d*x)/2)^11)/(90112*d) - (a^3*cot(c/2 + (d*x)/2)^12)/(16384*d) - (a^3*cot(c/2 + (d*x)/2)^13)/(106496*d) + (a^3*tan(c/2 + (d*x)/2)^2)/(1024 *d) + (45*a^3*tan(c/2 + (d*x)/2)^3)/(8192*d) + (77*a^3*tan(c/2 + (d*x)/2)^ 4)/(16384*d) + (15*a^3*tan(c/2 + (d*x)/2)^5)/(8192*d) - (a^3*tan(c/2 + (d* x)/2)^6)/(2048*d) - (33*a^3*tan(c/2 + (d*x)/2)^7)/(28672*d) - (13*a^3*tan( c/2 + (d*x)/2)^8)/(16384*d) - (a^3*tan(c/2 + (d*x)/2)^9)/(4096*d) + (a^3*t an(c/2 + (d*x)/2)^10)/(10240*d) + (13*a^3*tan(c/2 + (d*x)/2)^11)/(90112*d) + (a^3*tan(c/2 + (d*x)/2)^12)/(16384*d) + (a^3*tan(c/2 + (d*x)/2)^13)/(10 6496*d) - (27*a^3*log(tan(c/2 + (d*x)/2)))/(1024*d) + (35*a^3*cot(c/2 + (d *x)/2))/(2048*d) - (35*a^3*tan(c/2 + (d*x)/2))/(2048*d)
Time = 0.18 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.82 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (204800 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{12}+135135 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11}+102400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}+90090 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+76800 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-952952 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-2498560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-816816 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+2938880 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+2690688 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-430080 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-1281280 \cos \left (d x +c \right ) \sin \left (d x +c \right )-394240 \cos \left (d x +c \right )-135135 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{13}\right )}{5125120 \sin \left (d x +c \right )^{13} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(204800*cos(c + d*x)*sin(c + d*x)**12 + 135135*cos(c + d*x)*sin(c + d*x)**11 + 102400*cos(c + d*x)*sin(c + d*x)**10 + 90090*cos(c + d*x)*sin(c + d*x)**9 + 76800*cos(c + d*x)*sin(c + d*x)**8 - 952952*cos(c + d*x)*sin( c + d*x)**7 - 2498560*cos(c + d*x)*sin(c + d*x)**6 - 816816*cos(c + d*x)*s in(c + d*x)**5 + 2938880*cos(c + d*x)*sin(c + d*x)**4 + 2690688*cos(c + d* x)*sin(c + d*x)**3 - 430080*cos(c + d*x)*sin(c + d*x)**2 - 1281280*cos(c + d*x)*sin(c + d*x) - 394240*cos(c + d*x) - 135135*log(tan((c + d*x)/2))*si n(c + d*x)**13))/(5125120*sin(c + d*x)**13*d)