Integrand size = 29, antiderivative size = 270 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {17 a^2 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac {17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d} \] Output:
17/1024*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-4/9*a^2*cot(d*x+c )^9/d-2/11*a^2*cot(d*x+c)^11/d+17/1024*a^2*cot(d*x+c)*csc(d*x+c)/d+17/1536 *a^2*cot(d*x+c)*csc(d*x+c)^3/d-11/384*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-1/64*a ^2*cot(d*x+c)*csc(d*x+c)^7/d+1/24*a^2*cot(d*x+c)^3*csc(d*x+c)^7/d-1/12*a^2 *cot(d*x+c)^5*csc(d*x+c)^7/d
Time = 11.51 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.73 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (1+\sin (c+d x))^2 \left (30159360 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{11}(c+d x) (65553642+67499586 \cos (2 (c+d x))+25966248 \cos (4 (c+d x))-6944091 \cos (6 (c+d x))-746130 \cos (8 (c+d x))+58905 \cos (10 (c+d x))+29655040 \sin (c+d x)+51445760 \sin (3 (c+d x))+25600000 \sin (5 (c+d x))+3235840 \sin (7 (c+d x))-532480 \sin (9 (c+d x))+40960 \sin (11 (c+d x)))\right )}{1816657920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]
Output:
(a^2*(1 + Sin[c + d*x])^2*(30159360*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^11*(65553642 + 67499586*Cos[2*(c + d *x)] + 25966248*Cos[4*(c + d*x)] - 6944091*Cos[6*(c + d*x)] - 746130*Cos[8 *(c + d*x)] + 58905*Cos[10*(c + d*x)] + 29655040*Sin[c + d*x] + 51445760*S in[3*(c + d*x)] + 25600000*Sin[5*(c + d*x)] + 3235840*Sin[7*(c + d*x)] - 5 32480*Sin[9*(c + d*x)] + 40960*Sin[11*(c + d*x)])))/(1816657920*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)
Time = 0.74 (sec) , antiderivative size = 270, 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.
\(\displaystyle \int \cot ^6(c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^2}{\sin (c+d x)^{13}}dx\) |
\(\Big \downarrow \) 3352 |
\(\displaystyle \int \left (a^2 \cot ^6(c+d x) \csc ^7(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^6(c+d x)+a^2 \cot ^6(c+d x) \csc ^5(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {17 a^2 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {2 a^2 \cot ^{11}(c+d x)}{11 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac {17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]
Output:
(17*a^2*ArcTanh[Cos[c + d*x]])/(1024*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - ( 4*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Cot[c + d*x]^11)/(11*d) + (17*a^2*Cot [c + d*x]*Csc[c + d*x])/(1024*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(1 536*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(384*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) - (a ^2*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^ 7)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*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 = 1.02 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.13
method | result | size |
risch | \(-\frac {a^{2} \left (58905 \,{\mathrm e}^{23 i \left (d x +c \right )}-687225 \,{\mathrm e}^{21 i \left (d x +c \right )}-7690221 \,{\mathrm e}^{19 i \left (d x +c \right )}+19022157 \,{\mathrm e}^{17 i \left (d x +c \right )}+97320960 i {\mathrm e}^{8 i \left (d x +c \right )}+93465834 \,{\mathrm e}^{15 i \left (d x +c \right )}-37847040 i {\mathrm e}^{18 i \left (d x +c \right )}+198606870 \,{\mathrm e}^{13 i \left (d x +c \right )}-113541120 i {\mathrm e}^{14 i \left (d x +c \right )}+198606870 \,{\mathrm e}^{11 i \left (d x +c \right )}+19824640 i {\mathrm e}^{6 i \left (d x +c \right )}+93465834 \,{\mathrm e}^{9 i \left (d x +c \right )}+37847040 i {\mathrm e}^{12 i \left (d x +c \right )}+19022157 \,{\mathrm e}^{7 i \left (d x +c \right )}-56770560 i {\mathrm e}^{16 i \left (d x +c \right )}-7690221 \,{\mathrm e}^{5 i \left (d x +c \right )}-983040 i {\mathrm e}^{2 i \left (d x +c \right )}-687225 \,{\mathrm e}^{3 i \left (d x +c \right )}+48660480 i {\mathrm e}^{10 i \left (d x +c \right )}+58905 \,{\mathrm e}^{i \left (d x +c \right )}+5406720 i {\mathrm e}^{4 i \left (d x +c \right )}+81920 i\right )}{1774080 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}+\frac {17 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{1024 d}-\frac {17 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{1024 d}\) | \(306\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \cos \left (d x +c \right )^{7}}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \cos \left (d x +c \right )^{7}}{693 \sin \left (d x +c \right )^{7}}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{7}}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{1024 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{1024}-\frac {5 \cos \left (d x +c \right )^{3}}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )}{d}\) | \(366\) |
default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \cos \left (d x +c \right )^{7}}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \cos \left (d x +c \right )^{7}}{693 \sin \left (d x +c \right )^{7}}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{7}}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{1024 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{1024}-\frac {5 \cos \left (d x +c \right )^{3}}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )}{d}\) | \(366\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/1774080*a^2*(58905*exp(23*I*(d*x+c))-687225*exp(21*I*(d*x+c))-7690221*e xp(19*I*(d*x+c))+19022157*exp(17*I*(d*x+c))+97320960*I*exp(8*I*(d*x+c))+93 465834*exp(15*I*(d*x+c))-37847040*I*exp(18*I*(d*x+c))+198606870*exp(13*I*( d*x+c))-113541120*I*exp(14*I*(d*x+c))+198606870*exp(11*I*(d*x+c))+19824640 *I*exp(6*I*(d*x+c))+93465834*exp(9*I*(d*x+c))+37847040*I*exp(12*I*(d*x+c)) +19022157*exp(7*I*(d*x+c))-56770560*I*exp(16*I*(d*x+c))-7690221*exp(5*I*(d *x+c))-983040*I*exp(2*I*(d*x+c))-687225*exp(3*I*(d*x+c))+48660480*I*exp(10 *I*(d*x+c))+58905*exp(I*(d*x+c))+5406720*I*exp(4*I*(d*x+c))+81920*I)/d/(ex p(2*I*(d*x+c))-1)^12+17/1024*a^2/d*ln(exp(I*(d*x+c))+1)-17/1024*a^2/d*ln(e xp(I*(d*x+c))-1)
Time = 0.12 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.42 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {117810 \, a^{2} \cos \left (d x + c\right )^{11} - 667590 \, a^{2} \cos \left (d x + c\right )^{9} + 135828 \, a^{2} \cos \left (d x + c\right )^{7} + 1555092 \, a^{2} \cos \left (d x + c\right )^{5} - 667590 \, a^{2} \cos \left (d x + c\right )^{3} + 117810 \, a^{2} \cos \left (d x + c\right ) - 58905 \, {\left (a^{2} \cos \left (d x + c\right )^{12} - 6 \, a^{2} \cos \left (d x + c\right )^{10} + 15 \, a^{2} \cos \left (d x + c\right )^{8} - 20 \, a^{2} \cos \left (d x + c\right )^{6} + 15 \, a^{2} \cos \left (d x + c\right )^{4} - 6 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 58905 \, {\left (a^{2} \cos \left (d x + c\right )^{12} - 6 \, a^{2} \cos \left (d x + c\right )^{10} + 15 \, a^{2} \cos \left (d x + c\right )^{8} - 20 \, a^{2} \cos \left (d x + c\right )^{6} + 15 \, a^{2} \cos \left (d x + c\right )^{4} - 6 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 20480 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{11} - 44 \, a^{2} \cos \left (d x + c\right )^{9} + 99 \, a^{2} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{7096320 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/7096320*(117810*a^2*cos(d*x + c)^11 - 667590*a^2*cos(d*x + c)^9 + 13582 8*a^2*cos(d*x + c)^7 + 1555092*a^2*cos(d*x + c)^5 - 667590*a^2*cos(d*x + c )^3 + 117810*a^2*cos(d*x + c) - 58905*(a^2*cos(d*x + c)^12 - 6*a^2*cos(d*x + c)^10 + 15*a^2*cos(d*x + c)^8 - 20*a^2*cos(d*x + c)^6 + 15*a^2*cos(d*x + c)^4 - 6*a^2*cos(d*x + c)^2 + a^2)*log(1/2*cos(d*x + c) + 1/2) + 58905*( a^2*cos(d*x + c)^12 - 6*a^2*cos(d*x + c)^10 + 15*a^2*cos(d*x + c)^8 - 20*a ^2*cos(d*x + c)^6 + 15*a^2*cos(d*x + c)^4 - 6*a^2*cos(d*x + c)^2 + a^2)*lo g(-1/2*cos(d*x + c) + 1/2) + 20480*(8*a^2*cos(d*x + c)^11 - 44*a^2*cos(d*x + c)^9 + 99*a^2*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^12 - 6*d*co s(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)
Timed out. \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**7*(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.20 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {1155 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2772 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {20480 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{7096320 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/7096320*(1155*a^2*(2*(15*cos(d*x + c)^11 - 85*cos(d*x + c)^9 + 198*cos( d*x + c)^7 + 198*cos(d*x + c)^5 - 85*cos(d*x + c)^3 + 15*cos(d*x + c))/(co s(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) + 1 5*log(cos(d*x + c) - 1)) + 2772*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)) + 2 0480*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^2/tan(d*x + c)^11)/d
Time = 0.24 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.56 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/56770560*(1155*a^2*tan(1/2*d*x + 1/2*c)^12 + 5040*a^2*tan(1/2*d*x + 1/2* c)^11 + 5544*a^2*tan(1/2*d*x + 1/2*c)^10 - 6160*a^2*tan(1/2*d*x + 1/2*c)^9 - 24255*a^2*tan(1/2*d*x + 1/2*c)^8 - 39600*a^2*tan(1/2*d*x + 1/2*c)^7 - 2 7720*a^2*tan(1/2*d*x + 1/2*c)^6 + 55440*a^2*tan(1/2*d*x + 1/2*c)^5 + 16285 5*a^2*tan(1/2*d*x + 1/2*c)^4 + 184800*a^2*tan(1/2*d*x + 1/2*c)^3 + 55440*a ^2*tan(1/2*d*x + 1/2*c)^2 - 942480*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 55 4400*a^2*tan(1/2*d*x + 1/2*c) + (2924714*a^2*tan(1/2*d*x + 1/2*c)^12 + 554 400*a^2*tan(1/2*d*x + 1/2*c)^11 - 55440*a^2*tan(1/2*d*x + 1/2*c)^10 - 1848 00*a^2*tan(1/2*d*x + 1/2*c)^9 - 162855*a^2*tan(1/2*d*x + 1/2*c)^8 - 55440* a^2*tan(1/2*d*x + 1/2*c)^7 + 27720*a^2*tan(1/2*d*x + 1/2*c)^6 + 39600*a^2* tan(1/2*d*x + 1/2*c)^5 + 24255*a^2*tan(1/2*d*x + 1/2*c)^4 + 6160*a^2*tan(1 /2*d*x + 1/2*c)^3 - 5544*a^2*tan(1/2*d*x + 1/2*c)^2 - 5040*a^2*tan(1/2*d*x + 1/2*c) - 1155*a^2)/tan(1/2*d*x + 1/2*c)^12)/d
Time = 34.62 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.74 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {5\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1536\,d}-\frac {47\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1024\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {5\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{7168\,d}+\frac {7\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{9216\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{11264\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{49152\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1536\,d}+\frac {47\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1024\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{7168\,d}-\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{9216\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{11264\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{49152\,d}-\frac {17\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}+\frac {5\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512\,d}-\frac {5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512\,d} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^7,x)
Output:
(a^2*cot(c/2 + (d*x)/2)^6)/(2048*d) - (5*a^2*cot(c/2 + (d*x)/2)^3)/(1536*d ) - (47*a^2*cot(c/2 + (d*x)/2)^4)/(16384*d) - (a^2*cot(c/2 + (d*x)/2)^5)/( 1024*d) - (a^2*cot(c/2 + (d*x)/2)^2)/(1024*d) + (5*a^2*cot(c/2 + (d*x)/2)^ 7)/(7168*d) + (7*a^2*cot(c/2 + (d*x)/2)^8)/(16384*d) + (a^2*cot(c/2 + (d*x )/2)^9)/(9216*d) - (a^2*cot(c/2 + (d*x)/2)^10)/(10240*d) - (a^2*cot(c/2 + (d*x)/2)^11)/(11264*d) - (a^2*cot(c/2 + (d*x)/2)^12)/(49152*d) + (a^2*tan( c/2 + (d*x)/2)^2)/(1024*d) + (5*a^2*tan(c/2 + (d*x)/2)^3)/(1536*d) + (47*a ^2*tan(c/2 + (d*x)/2)^4)/(16384*d) + (a^2*tan(c/2 + (d*x)/2)^5)/(1024*d) - (a^2*tan(c/2 + (d*x)/2)^6)/(2048*d) - (5*a^2*tan(c/2 + (d*x)/2)^7)/(7168* d) - (7*a^2*tan(c/2 + (d*x)/2)^8)/(16384*d) - (a^2*tan(c/2 + (d*x)/2)^9)/( 9216*d) + (a^2*tan(c/2 + (d*x)/2)^10)/(10240*d) + (a^2*tan(c/2 + (d*x)/2)^ 11)/(11264*d) + (a^2*tan(c/2 + (d*x)/2)^12)/(49152*d) - (17*a^2*log(tan(c/ 2 + (d*x)/2)))/(1024*d) + (5*a^2*cot(c/2 + (d*x)/2))/(512*d) - (5*a^2*tan( c/2 + (d*x)/2))/(512*d)
Time = 0.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.81 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (81920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11}+58905 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}+40960 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+39270 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+30720 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-678216 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-1157120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+432432 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+1648640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+384384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-645120 \cos \left (d x +c \right ) \sin \left (d x +c \right )-295680 \cos \left (d x +c \right )-58905 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{12}\right )}{3548160 \sin \left (d x +c \right )^{12} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*(81920*cos(c + d*x)*sin(c + d*x)**11 + 58905*cos(c + d*x)*sin(c + d* x)**10 + 40960*cos(c + d*x)*sin(c + d*x)**9 + 39270*cos(c + d*x)*sin(c + d *x)**8 + 30720*cos(c + d*x)*sin(c + d*x)**7 - 678216*cos(c + d*x)*sin(c + d*x)**6 - 1157120*cos(c + d*x)*sin(c + d*x)**5 + 432432*cos(c + d*x)*sin(c + d*x)**4 + 1648640*cos(c + d*x)*sin(c + d*x)**3 + 384384*cos(c + d*x)*si n(c + d*x)**2 - 645120*cos(c + d*x)*sin(c + d*x) - 295680*cos(c + d*x) - 5 8905*log(tan((c + d*x)/2))*sin(c + d*x)**12))/(3548160*sin(c + d*x)**12*d)