Integrand size = 29, antiderivative size = 228 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {13 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \] Output:
13/256*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-2/9*a^2*cot(d*x+c) ^9/d+13/256*a^2*cot(d*x+c)*csc(d*x+c)/d-9/128*a^2*cot(d*x+c)*csc(d*x+c)^3/ d+5/48*a^2*cot(d*x+c)^3*csc(d*x+c)^3/d-1/8*a^2*cot(d*x+c)^5*csc(d*x+c)^3/d -1/32*a^2*cot(d*x+c)*csc(d*x+c)^5/d+1/16*a^2*cot(d*x+c)^3*csc(d*x+c)^5/d-1 /10*a^2*cot(d*x+c)^5*csc(d*x+c)^5/d
Time = 7.81 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.55 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^{10}(c+d x) \left (2732940 \cos (c+d x)+1151640 \cos (3 (c+d x))+388248 \cos (5 (c+d x))-135870 \cos (7 (c+d x))-8190 \cos (9 (c+d x))-515970 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+859950 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-491400 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+184275 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-40950 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4095 \cos (10 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+515970 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-859950 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+491400 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-184275 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+40950 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4095 \cos (10 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1075200 \sin (2 (c+d x))+1044480 \sin (4 (c+d x))+414720 \sin (6 (c+d x))+51200 \sin (8 (c+d x))-5120 \sin (10 (c+d x))\right )}{41287680 d} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]
Output:
-1/41287680*(a^2*Csc[c + d*x]^10*(2732940*Cos[c + d*x] + 1151640*Cos[3*(c + d*x)] + 388248*Cos[5*(c + d*x)] - 135870*Cos[7*(c + d*x)] - 8190*Cos[9*( c + d*x)] - 515970*Log[Cos[(c + d*x)/2]] + 859950*Cos[2*(c + d*x)]*Log[Cos [(c + d*x)/2]] - 491400*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 184275*Co s[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 40950*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 4095*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 515970*Log[Sin[ (c + d*x)/2]] - 859950*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 491400*Cos [4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 184275*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 40950*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 4095*Cos[10*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 1075200*Sin[2*(c + d*x)] + 1044480*Sin[4*( c + d*x)] + 414720*Sin[6*(c + d*x)] + 51200*Sin[8*(c + d*x)] - 5120*Sin[10 *(c + d*x)]))/d
Time = 0.66 (sec) , antiderivative size = 228, 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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^2}{\sin (c+d x)^{11}}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \cot ^6(c+d x) \csc ^5(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^4(c+d x)+a^2 \cot ^6(c+d x) \csc ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {13 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]
Output:
(13*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2 *a^2*Cot[c + d*x]^9)/(9*d) + (13*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*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 = 0.58 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(-\frac {a^{2} \left (4095 \,{\mathrm e}^{19 i \left (d x +c \right )}+67935 \,{\mathrm e}^{17 i \left (d x +c \right )}-194124 \,{\mathrm e}^{15 i \left (d x +c \right )}+215040 i {\mathrm e}^{14 i \left (d x +c \right )}-575820 \,{\mathrm e}^{13 i \left (d x +c \right )}-829440 i {\mathrm e}^{6 i \left (d x +c \right )}-1366470 \,{\mathrm e}^{11 i \left (d x +c \right )}+1075200 i {\mathrm e}^{12 i \left (d x +c \right )}-1366470 \,{\mathrm e}^{9 i \left (d x +c \right )}+322560 i {\mathrm e}^{16 i \left (d x +c \right )}-575820 \,{\mathrm e}^{7 i \left (d x +c \right )}-194124 \,{\mathrm e}^{5 i \left (d x +c \right )}-51200 i {\mathrm e}^{2 i \left (d x +c \right )}+67935 \,{\mathrm e}^{3 i \left (d x +c \right )}-645120 i {\mathrm e}^{10 i \left (d x +c \right )}+4095 \,{\mathrm e}^{i \left (d x +c \right )}-92160 i {\mathrm e}^{4 i \left (d x +c \right )}+5120 i\right )}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}-\frac {13 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}+\frac {13 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\) | \(260\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \cos \left (d x +c \right )^{7}}{63 \sin \left (d x +c \right )^{7}}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\) | \(312\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \cos \left (d x +c \right )^{7}}{63 \sin \left (d x +c \right )^{7}}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\) | \(312\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/40320*a^2*(4095*exp(19*I*(d*x+c))+67935*exp(17*I*(d*x+c))-194124*exp(15 *I*(d*x+c))+215040*I*exp(14*I*(d*x+c))-575820*exp(13*I*(d*x+c))-829440*I*e xp(6*I*(d*x+c))-1366470*exp(11*I*(d*x+c))+1075200*I*exp(12*I*(d*x+c))-1366 470*exp(9*I*(d*x+c))+322560*I*exp(16*I*(d*x+c))-575820*exp(7*I*(d*x+c))-19 4124*exp(5*I*(d*x+c))-51200*I*exp(2*I*(d*x+c))+67935*exp(3*I*(d*x+c))-6451 20*I*exp(10*I*(d*x+c))+4095*exp(I*(d*x+c))-92160*I*exp(4*I*(d*x+c))+5120*I )/d/(exp(2*I*(d*x+c))-1)^10-13/256*a^2/d*ln(exp(I*(d*x+c))-1)+13/256*a^2/d *ln(exp(I*(d*x+c))+1)
Time = 0.11 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {8190 \, a^{2} \cos \left (d x + c\right )^{9} + 15540 \, a^{2} \cos \left (d x + c\right )^{7} - 69888 \, a^{2} \cos \left (d x + c\right )^{5} + 38220 \, a^{2} \cos \left (d x + c\right )^{3} - 8190 \, a^{2} \cos \left (d x + c\right ) - 4095 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4095 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5120 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{9} - 9 \, a^{2} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/161280*(8190*a^2*cos(d*x + c)^9 + 15540*a^2*cos(d*x + c)^7 - 69888*a^2* cos(d*x + c)^5 + 38220*a^2*cos(d*x + c)^3 - 8190*a^2*cos(d*x + c) - 4095*( a^2*cos(d*x + c)^10 - 5*a^2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^6 - 10*a^ 2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2) + 4095*(a^2*cos(d*x + c)^10 - 5*a^2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^ 6 - 10*a^2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + c) + 1/2) + 5120*(2*a^2*cos(d*x + c)^9 - 9*a^2*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*c os(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**5*(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.20 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {63 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {5120 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/161280*(63*a^2*(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*cos(d*x + c)^2 - 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 210*a^2*(2*(15*cos( d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos (d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 5120*(9*tan(d*x + c)^2 + 7)*a^2/tan(d*x + c)^9)/d
Time = 0.22 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.42 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {126 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3990 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 13440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 65520 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {191906 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 13440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3990 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/1290240*(126*a^2*tan(1/2*d*x + 1/2*c)^10 + 560*a^2*tan(1/2*d*x + 1/2*c)^ 9 + 315*a^2*tan(1/2*d*x + 1/2*c)^8 - 2160*a^2*tan(1/2*d*x + 1/2*c)^7 - 399 0*a^2*tan(1/2*d*x + 1/2*c)^6 + 7560*a^2*tan(1/2*d*x + 1/2*c)^4 + 13440*a^2 *tan(1/2*d*x + 1/2*c)^3 + 11340*a^2*tan(1/2*d*x + 1/2*c)^2 - 65520*a^2*log (abs(tan(1/2*d*x + 1/2*c))) - 30240*a^2*tan(1/2*d*x + 1/2*c) + (191906*a^2 *tan(1/2*d*x + 1/2*c)^10 + 30240*a^2*tan(1/2*d*x + 1/2*c)^9 - 11340*a^2*ta n(1/2*d*x + 1/2*c)^8 - 13440*a^2*tan(1/2*d*x + 1/2*c)^7 - 7560*a^2*tan(1/2 *d*x + 1/2*c)^6 + 3990*a^2*tan(1/2*d*x + 1/2*c)^4 + 2160*a^2*tan(1/2*d*x + 1/2*c)^3 - 315*a^2*tan(1/2*d*x + 1/2*c)^2 - 560*a^2*tan(1/2*d*x + 1/2*c) - 126*a^2)/tan(1/2*d*x + 1/2*c)^10)/d
Time = 34.55 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.57 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {19\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {9\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {19\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {13\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}+\frac {3\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^5,x)
Output:
(19*a^2*cot(c/2 + (d*x)/2)^6)/(6144*d) - (a^2*cot(c/2 + (d*x)/2)^3)/(96*d) - (3*a^2*cot(c/2 + (d*x)/2)^4)/(512*d) - (9*a^2*cot(c/2 + (d*x)/2)^2)/(10 24*d) + (3*a^2*cot(c/2 + (d*x)/2)^7)/(1792*d) - (a^2*cot(c/2 + (d*x)/2)^8) /(4096*d) - (a^2*cot(c/2 + (d*x)/2)^9)/(2304*d) - (a^2*cot(c/2 + (d*x)/2)^ 10)/(10240*d) + (9*a^2*tan(c/2 + (d*x)/2)^2)/(1024*d) + (a^2*tan(c/2 + (d* x)/2)^3)/(96*d) + (3*a^2*tan(c/2 + (d*x)/2)^4)/(512*d) - (19*a^2*tan(c/2 + (d*x)/2)^6)/(6144*d) - (3*a^2*tan(c/2 + (d*x)/2)^7)/(1792*d) + (a^2*tan(c /2 + (d*x)/2)^8)/(4096*d) + (a^2*tan(c/2 + (d*x)/2)^9)/(2304*d) + (a^2*tan (c/2 + (d*x)/2)^10)/(10240*d) - (13*a^2*log(tan(c/2 + (d*x)/2)))/(256*d) + (3*a^2*cot(c/2 + (d*x)/2))/(128*d) - (3*a^2*tan(c/2 + (d*x)/2))/(128*d)
Time = 0.15 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.82 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (5120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+4095 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+2560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-24150 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-38400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+12936 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+48640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+11088 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-17920 \cos \left (d x +c \right ) \sin \left (d x +c \right )-8064 \cos \left (d x +c \right )-4095 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{10}\right )}{80640 \sin \left (d x +c \right )^{10} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*(5120*cos(c + d*x)*sin(c + d*x)**9 + 4095*cos(c + d*x)*sin(c + d*x)* *8 + 2560*cos(c + d*x)*sin(c + d*x)**7 - 24150*cos(c + d*x)*sin(c + d*x)** 6 - 38400*cos(c + d*x)*sin(c + d*x)**5 + 12936*cos(c + d*x)*sin(c + d*x)** 4 + 48640*cos(c + d*x)*sin(c + d*x)**3 + 11088*cos(c + d*x)*sin(c + d*x)** 2 - 17920*cos(c + d*x)*sin(c + d*x) - 8064*cos(c + d*x) - 4095*log(tan((c + d*x)/2))*sin(c + d*x)**10))/(80640*sin(c + d*x)**10*d)