Integrand size = 29, antiderivative size = 270 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {41 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d} \] Output:
41/1024*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-7/9*a^3*cot(d*x+c )^9/d-3/11*a^3*cot(d*x+c)^11/d+41/1024*a^3*cot(d*x+c)*csc(d*x+c)/d+41/1536 *a^3*cot(d*x+c)*csc(d*x+c)^3/d-35/384*a^3*cot(d*x+c)*csc(d*x+c)^5/d+3/16*a ^3*cot(d*x+c)^3*csc(d*x+c)^5/d-3/10*a^3*cot(d*x+c)^5*csc(d*x+c)^5/d-1/64*a ^3*cot(d*x+c)*csc(d*x+c)^7/d+1/24*a^3*cot(d*x+c)^3*csc(d*x+c)^7/d-1/12*a^3 *cot(d*x+c)^5*csc(d*x+c)^7/d
Time = 11.78 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.73 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (72737280 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{11}(c+d x) (91311066+62609778 \cos (2 (c+d x))+22551144 \cos (4 (c+d x))-23426403 \cos (6 (c+d x))-1799490 \cos (8 (c+d x))+142065 \cos (10 (c+d x))+49776640 \sin (c+d x)+84039680 \sin (3 (c+d x))+38118400 \sin (5 (c+d x))+2206720 \sin (7 (c+d x))-1530880 \sin (9 (c+d x))+117760 \sin (11 (c+d x)))\right )}{1816657920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(1 + Sin[c + d*x])^3*(72737280*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^11*(91311066 + 62609778*Cos[2*(c + d *x)] + 22551144*Cos[4*(c + d*x)] - 23426403*Cos[6*(c + d*x)] - 1799490*Cos [8*(c + d*x)] + 142065*Cos[10*(c + d*x)] + 49776640*Sin[c + d*x] + 8403968 0*Sin[3*(c + d*x)] + 38118400*Sin[5*(c + d*x)] + 2206720*Sin[7*(c + d*x)] - 1530880*Sin[9*(c + d*x)] + 117760*Sin[11*(c + d*x)])))/(1816657920*d*(Co s[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.75 (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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^3}{\sin (c+d x)^{13}}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^6(c+d x) \csc ^7(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^5(c+d x)+a^3 \cot ^6(c+d x) \csc ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {41 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]
Output:
(41*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - ( 7*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3*Cot[c + d*x]^11)/(11*d) + (41*a^3*Cot [c + d*x]*Csc[c + d*x])/(1024*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(1 536*d) - (35*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(384*d) + (3*a^3*Cot[c + d*x ]^3*Csc[c + d*x]^5)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (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)/(24*d) - (a^3*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 = 1.19 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(-\frac {a^{3} \left (142065 \,{\mathrm e}^{23 i \left (d x +c \right )}-1657425 \,{\mathrm e}^{21 i \left (d x +c \right )}-25225893 \,{\mathrm e}^{19 i \left (d x +c \right )}+180449280 i {\mathrm e}^{8 i \left (d x +c \right )}-875259 \,{\mathrm e}^{17 i \left (d x +c \right )}-66232320 i {\mathrm e}^{18 i \left (d x +c \right )}+85160922 \,{\mathrm e}^{15 i \left (d x +c \right )}-227082240 i {\mathrm e}^{14 i \left (d x +c \right )}+245231910 \,{\mathrm e}^{13 i \left (d x +c \right )}+14417920 i {\mathrm e}^{6 i \left (d x +c \right )}+245231910 \,{\mathrm e}^{11 i \left (d x +c \right )}+108810240 i {\mathrm e}^{12 i \left (d x +c \right )}+85160922 \,{\mathrm e}^{9 i \left (d x +c \right )}+7096320 i {\mathrm e}^{20 i \left (d x +c \right )}-875259 \,{\mathrm e}^{7 i \left (d x +c \right )}-63866880 i {\mathrm e}^{16 i \left (d x +c \right )}-25225893 \,{\mathrm e}^{5 i \left (d x +c \right )}-2826240 i {\mathrm e}^{2 i \left (d x +c \right )}-1657425 \,{\mathrm e}^{3 i \left (d x +c \right )}+40550400 i {\mathrm e}^{10 i \left (d x +c \right )}+142065 \,{\mathrm e}^{i \left (d x +c \right )}+8448000 i {\mathrm e}^{4 i \left (d x +c \right )}+235520 i\right )}{1774080 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}+\frac {41 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{1024 d}-\frac {41 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{1024 d}\) | \(318\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \cos \left (d x +c \right )^{7}}{63 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \cos \left (d x +c \right )^{7}}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \cos \left (d x +c \right )^{7}}{693 \sin \left (d x +c \right )^{7}}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{7}}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{1024 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{1024}-\frac {5 \cos \left (d x +c \right )^{3}}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )}{d}\) | \(408\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \cos \left (d x +c \right )^{7}}{63 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \cos \left (d x +c \right )^{7}}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \cos \left (d x +c \right )^{7}}{693 \sin \left (d x +c \right )^{7}}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{7}}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{1024 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{1024}-\frac {5 \cos \left (d x +c \right )^{3}}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )}{d}\) | \(408\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/1774080*a^3*(142065*exp(23*I*(d*x+c))-1657425*exp(21*I*(d*x+c))-2522589 3*exp(19*I*(d*x+c))+180449280*I*exp(8*I*(d*x+c))-875259*exp(17*I*(d*x+c))- 66232320*I*exp(18*I*(d*x+c))+85160922*exp(15*I*(d*x+c))-227082240*I*exp(14 *I*(d*x+c))+245231910*exp(13*I*(d*x+c))+14417920*I*exp(6*I*(d*x+c))+245231 910*exp(11*I*(d*x+c))+108810240*I*exp(12*I*(d*x+c))+85160922*exp(9*I*(d*x+ c))+7096320*I*exp(20*I*(d*x+c))-875259*exp(7*I*(d*x+c))-63866880*I*exp(16* I*(d*x+c))-25225893*exp(5*I*(d*x+c))-2826240*I*exp(2*I*(d*x+c))-1657425*ex p(3*I*(d*x+c))+40550400*I*exp(10*I*(d*x+c))+142065*exp(I*(d*x+c))+8448000* I*exp(4*I*(d*x+c))+235520*I)/d/(exp(2*I*(d*x+c))-1)^12+41/1024*a^3/d*ln(ex p(I*(d*x+c))+1)-41/1024*a^3/d*ln(exp(I*(d*x+c))-1)
Time = 0.12 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.42 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {284130 \, a^{3} \cos \left (d x + c\right )^{11} - 1610070 \, a^{3} \cos \left (d x + c\right )^{9} - 507276 \, a^{3} \cos \left (d x + c\right )^{7} + 3750516 \, a^{3} \cos \left (d x + c\right )^{5} - 1610070 \, a^{3} \cos \left (d x + c\right )^{3} + 284130 \, a^{3} \cos \left (d x + c\right ) - 142065 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 142065 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 10240 \, {\left (46 \, a^{3} \cos \left (d x + c\right )^{11} - 253 \, a^{3} \cos \left (d x + c\right )^{9} + 396 \, a^{3} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{7096320 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/7096320*(284130*a^3*cos(d*x + c)^11 - 1610070*a^3*cos(d*x + c)^9 - 5072 76*a^3*cos(d*x + c)^7 + 3750516*a^3*cos(d*x + c)^5 - 1610070*a^3*cos(d*x + c)^3 + 284130*a^3*cos(d*x + c) - 142065*(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) + 1420 65*(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) + 10240*(46*a^3*cos(d*x + c)^11 - 253*a^3*c os(d*x + c)^9 + 396*a^3*cos(d*x + c)^7)*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*co s(d*x + c)^4 - 6*d*cos(d*x + c)^2 + d)
Timed out. \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**7*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.29 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {1155 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 8316 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {112640 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}} + \frac {30720 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}}}{7096320 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/7096320*(1155*a^3*(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)) + 8316*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*cos( d*x + c)^2 - 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 1 12640*(9*tan(d*x + c)^2 + 7)*a^3/tan(d*x + c)^9 + 30720*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^3/tan(d*x + c)^11)/d
Time = 0.32 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.56 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/56770560*(1155*a^3*tan(1/2*d*x + 1/2*c)^12 + 7560*a^3*tan(1/2*d*x + 1/2* c)^11 + 16632*a^3*tan(1/2*d*x + 1/2*c)^10 + 3080*a^3*tan(1/2*d*x + 1/2*c)^ 9 - 51975*a^3*tan(1/2*d*x + 1/2*c)^8 - 106920*a^3*tan(1/2*d*x + 1/2*c)^7 - 83160*a^3*tan(1/2*d*x + 1/2*c)^6 + 83160*a^3*tan(1/2*d*x + 1/2*c)^5 + 384 615*a^3*tan(1/2*d*x + 1/2*c)^4 + 572880*a^3*tan(1/2*d*x + 1/2*c)^3 + 16632 0*a^3*tan(1/2*d*x + 1/2*c)^2 - 2273040*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 1496880*a^3*tan(1/2*d*x + 1/2*c) + (7053722*a^3*tan(1/2*d*x + 1/2*c)^12 + 1496880*a^3*tan(1/2*d*x + 1/2*c)^11 - 166320*a^3*tan(1/2*d*x + 1/2*c)^10 - 572880*a^3*tan(1/2*d*x + 1/2*c)^9 - 384615*a^3*tan(1/2*d*x + 1/2*c)^8 - 83160*a^3*tan(1/2*d*x + 1/2*c)^7 + 83160*a^3*tan(1/2*d*x + 1/2*c)^6 + 106 920*a^3*tan(1/2*d*x + 1/2*c)^5 + 51975*a^3*tan(1/2*d*x + 1/2*c)^4 - 3080*a ^3*tan(1/2*d*x + 1/2*c)^3 - 16632*a^3*tan(1/2*d*x + 1/2*c)^2 - 7560*a^3*ta n(1/2*d*x + 1/2*c) - 1155*a^3)/tan(1/2*d*x + 1/2*c)^12)/d
Time = 35.51 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.74 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^7,x)
Output:
(3*a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) - (31*a^3*cot(c/2 + (d*x)/2)^3)/(307 2*d) - (111*a^3*cot(c/2 + (d*x)/2)^4)/(16384*d) - (3*a^3*cot(c/2 + (d*x)/2 )^5)/(2048*d) - (3*a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (27*a^3*cot(c/2 + (d*x)/2)^7)/(14336*d) + (15*a^3*cot(c/2 + (d*x)/2)^8)/(16384*d) - (a^3*cot (c/2 + (d*x)/2)^9)/(18432*d) - (3*a^3*cot(c/2 + (d*x)/2)^10)/(10240*d) - ( 3*a^3*cot(c/2 + (d*x)/2)^11)/(22528*d) - (a^3*cot(c/2 + (d*x)/2)^12)/(4915 2*d) + (3*a^3*tan(c/2 + (d*x)/2)^2)/(1024*d) + (31*a^3*tan(c/2 + (d*x)/2)^ 3)/(3072*d) + (111*a^3*tan(c/2 + (d*x)/2)^4)/(16384*d) + (3*a^3*tan(c/2 + (d*x)/2)^5)/(2048*d) - (3*a^3*tan(c/2 + (d*x)/2)^6)/(2048*d) - (27*a^3*tan (c/2 + (d*x)/2)^7)/(14336*d) - (15*a^3*tan(c/2 + (d*x)/2)^8)/(16384*d) + ( a^3*tan(c/2 + (d*x)/2)^9)/(18432*d) + (3*a^3*tan(c/2 + (d*x)/2)^10)/(10240 *d) + (3*a^3*tan(c/2 + (d*x)/2)^11)/(22528*d) + (a^3*tan(c/2 + (d*x)/2)^12 )/(49152*d) - (41*a^3*log(tan(c/2 + (d*x)/2)))/(1024*d) + (27*a^3*cot(c/2 + (d*x)/2))/(1024*d) - (27*a^3*tan(c/2 + (d*x)/2))/(1024*d)
Time = 0.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.81 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (235520 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11}+142065 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}+117760 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+94710 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-798720 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-2053128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-665600 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+2295216 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+2078720 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-325248 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-967680 \cos \left (d x +c \right ) \sin \left (d x +c \right )-295680 \cos \left (d x +c \right )-142065 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{12}\right )}{3548160 \sin \left (d x +c \right )^{12} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(235520*cos(c + d*x)*sin(c + d*x)**11 + 142065*cos(c + d*x)*sin(c + d*x)**10 + 117760*cos(c + d*x)*sin(c + d*x)**9 + 94710*cos(c + d*x)*sin(c + d*x)**8 - 798720*cos(c + d*x)*sin(c + d*x)**7 - 2053128*cos(c + d*x)*sin (c + d*x)**6 - 665600*cos(c + d*x)*sin(c + d*x)**5 + 2295216*cos(c + d*x)* sin(c + d*x)**4 + 2078720*cos(c + d*x)*sin(c + d*x)**3 - 325248*cos(c + d* x)*sin(c + d*x)**2 - 967680*cos(c + d*x)*sin(c + d*x) - 295680*cos(c + d*x ) - 142065*log(tan((c + d*x)/2))*sin(c + d*x)**12))/(3548160*sin(c + d*x)* *12*d)