Integrand size = 29, antiderivative size = 216 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {21 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d} \] Output:
-21/256*a^3*arctanh(cos(d*x+c))/d-4/5*a^3*cot(d*x+c)^5/d-a^3*cot(d*x+c)^7/ d-1/3*a^3*cot(d*x+c)^9/d-21/256*a^3*cot(d*x+c)*csc(d*x+c)/d-7/128*a^3*cot( d*x+c)*csc(d*x+c)^3/d+29/160*a^3*cot(d*x+c)*csc(d*x+c)^5/d-3/8*a^3*cot(d*x +c)^3*csc(d*x+c)^5/d+3/80*a^3*cot(d*x+c)*csc(d*x+c)^7/d-1/10*a^3*cot(d*x+c )^3*csc(d*x+c)^7/d
Time = 7.72 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.69 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (-4096 \cot \left (\frac {1}{2} (c+d x)\right )-1260 \csc ^2\left (\frac {1}{2} (c+d x)\right )-5040 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+5040 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1260 \sec ^2\left (\frac {1}{2} (c+d x)\right )-180 \sec ^4\left (\frac {1}{2} (c+d x)\right )-390 \sec ^6\left (\frac {1}{2} (c+d x)\right )+75 \sec ^8\left (\frac {1}{2} (c+d x)\right )+6 \sec ^{10}\left (\frac {1}{2} (c+d x)\right )+64 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-4 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (-45+\sin (c+d x))+5 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (-15+4 \sin (c+d x))-2 \csc ^{10}\left (\frac {1}{2} (c+d x)\right ) (3+10 \sin (c+d x))+6 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (65+42 \sin (c+d x))+4096 \tan \left (\frac {1}{2} (c+d x)\right )-504 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )-40 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+40 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{61440 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(1 + Sin[c + d*x])^3*(-4096*Cot[(c + d*x)/2] - 1260*Csc[(c + d*x)/2]^ 2 - 5040*Log[Cos[(c + d*x)/2]] + 5040*Log[Sin[(c + d*x)/2]] + 1260*Sec[(c + d*x)/2]^2 - 180*Sec[(c + d*x)/2]^4 - 390*Sec[(c + d*x)/2]^6 + 75*Sec[(c + d*x)/2]^8 + 6*Sec[(c + d*x)/2]^10 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*(-45 + Sin[c + d*x]) + 5*Csc[(c + d*x)/2]^8*(-15 + 4*Sin[c + d*x]) - 2*Csc[(c + d*x)/2]^10*(3 + 10*Sin[c + d*x]) + 6*Csc[(c + d*x)/2]^6*(65 + 42*Sin[c + d*x]) + 4096*Tan[(c + d*x)/2] - 504*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] - 40*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2] + 40*S ec[(c + d*x)/2]^8*Tan[(c + d*x)/2]))/(61440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.67 (sec) , antiderivative size = 216, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^3}{\sin (c+d x)^{11}}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^4(c+d x) \csc ^7(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^6(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^5(c+d x)+a^3 \cot ^4(c+d x) \csc ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {21 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]
Output:
(-21*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - ( a^3*Cot[c + d*x]^7)/d - (a^3*Cot[c + d*x]^9)/(3*d) - (21*a^3*Cot[c + d*x]* Csc[c + d*x])/(256*d) - (7*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (29* a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d *x]^5)/(8*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(80*d) - (a^3*Cot[c + d *x]^3*Csc[c + d*x]^7)/(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 = 1.14 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(\frac {a^{3} \left (315 \,{\mathrm e}^{19 i \left (d x +c \right )}-3045 \,{\mathrm e}^{17 i \left (d x +c \right )}-23676 \,{\mathrm e}^{15 i \left (d x +c \right )}+122880 i {\mathrm e}^{8 i \left (d x +c \right )}+27780 \,{\mathrm e}^{13 i \left (d x +c \right )}-76800 i {\mathrm e}^{14 i \left (d x +c \right )}+96930 \,{\mathrm e}^{11 i \left (d x +c \right )}+15360 i {\mathrm e}^{6 i \left (d x +c \right )}+96930 \,{\mathrm e}^{9 i \left (d x +c \right )}-15360 i {\mathrm e}^{12 i \left (d x +c \right )}+27780 \,{\mathrm e}^{7 i \left (d x +c \right )}+7680 i {\mathrm e}^{16 i \left (d x +c \right )}-23676 \,{\mathrm e}^{5 i \left (d x +c \right )}-5120 i {\mathrm e}^{2 i \left (d x +c \right )}-3045 \,{\mathrm e}^{3 i \left (d x +c \right )}-64512 i {\mathrm e}^{10 i \left (d x +c \right )}+315 \,{\mathrm e}^{i \left (d x +c \right )}+15360 i {\mathrm e}^{4 i \left (d x +c \right )}+512 i\right )}{1920 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}+\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}-\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\) | \(272\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \cos \left (d x +c \right )^{5}}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \cos \left (d x +c \right )^{5}}{315 \sin \left (d x +c \right )^{5}}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{256 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\) | \(352\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \cos \left (d x +c \right )^{5}}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \cos \left (d x +c \right )^{5}}{315 \sin \left (d x +c \right )^{5}}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{256 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\) | \(352\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/1920*a^3*(315*exp(19*I*(d*x+c))-3045*exp(17*I*(d*x+c))-23676*exp(15*I*(d *x+c))+122880*I*exp(8*I*(d*x+c))+27780*exp(13*I*(d*x+c))-76800*I*exp(14*I* (d*x+c))+96930*exp(11*I*(d*x+c))+15360*I*exp(6*I*(d*x+c))+96930*exp(9*I*(d *x+c))-15360*I*exp(12*I*(d*x+c))+27780*exp(7*I*(d*x+c))+7680*I*exp(16*I*(d *x+c))-23676*exp(5*I*(d*x+c))-5120*I*exp(2*I*(d*x+c))-3045*exp(3*I*(d*x+c) )-64512*I*exp(10*I*(d*x+c))+315*exp(I*(d*x+c))+15360*I*exp(4*I*(d*x+c))+51 2*I)/d/(exp(2*I*(d*x+c))-1)^10+21/256*a^3/d*ln(exp(I*(d*x+c))-1)-21/256*a^ 3/d*ln(exp(I*(d*x+c))+1)
Time = 0.10 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.57 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {630 \, a^{3} \cos \left (d x + c\right )^{9} - 2940 \, a^{3} \cos \left (d x + c\right )^{7} + 768 \, a^{3} \cos \left (d x + c\right )^{5} + 2940 \, a^{3} \cos \left (d x + c\right )^{3} - 630 \, a^{3} \cos \left (d x + c\right ) - 315 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 512 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{9} - 9 \, a^{3} \cos \left (d x + c\right )^{7} + 12 \, a^{3} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{7680 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/7680*(630*a^3*cos(d*x + c)^9 - 2940*a^3*cos(d*x + c)^7 + 768*a^3*cos(d*x + c)^5 + 2940*a^3*cos(d*x + c)^3 - 630*a^3*cos(d*x + c) - 315*(a^3*cos(d* x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3*cos(d*x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 315*(a^ 3*cos(d*x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3* cos(d*x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2) + 512*(2*a^3*cos(d*x + c)^9 - 9*a^3*cos(d*x + c)^7 + 12*a^3*cos(d*x + c)^5 )*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*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**7*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.43 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 630 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1536 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}} - \frac {512 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{53760 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/53760*(21*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)) + 630*a^3*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^3 + 3*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) - 3* log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1536*(7*tan(d*x + c)^2 + 5)*a^3/tan(d*x + c)^7 - 512*(63*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35) *a^3/tan(d*x + c)^9)/d
Time = 0.24 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.65 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5040 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 3600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {14762 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 3600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{61440 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/61440*(6*a^3*tan(1/2*d*x + 1/2*c)^10 + 40*a^3*tan(1/2*d*x + 1/2*c)^9 + 1 05*a^3*tan(1/2*d*x + 1/2*c)^8 + 120*a^3*tan(1/2*d*x + 1/2*c)^7 - 30*a^3*ta n(1/2*d*x + 1/2*c)^6 - 384*a^3*tan(1/2*d*x + 1/2*c)^5 - 840*a^3*tan(1/2*d* x + 1/2*c)^4 - 960*a^3*tan(1/2*d*x + 1/2*c)^3 + 60*a^3*tan(1/2*d*x + 1/2*c )^2 + 5040*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 3600*a^3*tan(1/2*d*x + 1/2 *c) - (14762*a^3*tan(1/2*d*x + 1/2*c)^10 + 3600*a^3*tan(1/2*d*x + 1/2*c)^9 + 60*a^3*tan(1/2*d*x + 1/2*c)^8 - 960*a^3*tan(1/2*d*x + 1/2*c)^7 - 840*a^ 3*tan(1/2*d*x + 1/2*c)^6 - 384*a^3*tan(1/2*d*x + 1/2*c)^5 - 30*a^3*tan(1/2 *d*x + 1/2*c)^4 + 120*a^3*tan(1/2*d*x + 1/2*c)^3 + 105*a^3*tan(1/2*d*x + 1 /2*c)^2 + 40*a^3*tan(1/2*d*x + 1/2*c) + 6*a^3)/tan(1/2*d*x + 1/2*c)^10)/d
Time = 18.55 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.83 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512\,d}-\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {21\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}-\frac {15\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}+\frac {15\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x)^7,x)
Output:
(a^3*cot(c/2 + (d*x)/2)^3)/(64*d) - (a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (7*a^3*cot(c/2 + (d*x)/2)^4)/(512*d) + (a^3*cot(c/2 + (d*x)/2)^5)/(160*d) + (a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) - (a^3*cot(c/2 + (d*x)/2)^7)/(512*d) - (7*a^3*cot(c/2 + (d*x)/2)^8)/(4096*d) - (a^3*cot(c/2 + (d*x)/2)^9)/(153 6*d) - (a^3*cot(c/2 + (d*x)/2)^10)/(10240*d) + (a^3*tan(c/2 + (d*x)/2)^2)/ (1024*d) - (a^3*tan(c/2 + (d*x)/2)^3)/(64*d) - (7*a^3*tan(c/2 + (d*x)/2)^4 )/(512*d) - (a^3*tan(c/2 + (d*x)/2)^5)/(160*d) - (a^3*tan(c/2 + (d*x)/2)^6 )/(2048*d) + (a^3*tan(c/2 + (d*x)/2)^7)/(512*d) + (7*a^3*tan(c/2 + (d*x)/2 )^8)/(4096*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(1536*d) + (a^3*tan(c/2 + (d*x) /2)^10)/(10240*d) + (21*a^3*log(tan(c/2 + (d*x)/2)))/(256*d) - (15*a^3*cot (c/2 + (d*x)/2))/(256*d) + (15*a^3*tan(c/2 + (d*x)/2))/(256*d)
Time = 0.16 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-512 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}-315 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-210 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+2136 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+1280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-912 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-1280 \cos \left (d x +c \right ) \sin \left (d x +c \right )-384 \cos \left (d x +c \right )+315 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{10}\right )}{3840 \sin \left (d x +c \right )^{10} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 512*cos(c + d*x)*sin(c + d*x)**9 - 315*cos(c + d*x)*sin(c + d*x) **8 - 256*cos(c + d*x)*sin(c + d*x)**7 - 210*cos(c + d*x)*sin(c + d*x)**6 + 768*cos(c + d*x)*sin(c + d*x)**5 + 2136*cos(c + d*x)*sin(c + d*x)**4 + 1 280*cos(c + d*x)*sin(c + d*x)**3 - 912*cos(c + d*x)*sin(c + d*x)**2 - 1280 *cos(c + d*x)*sin(c + d*x) - 384*cos(c + d*x) + 315*log(tan((c + d*x)/2))* sin(c + d*x)**10))/(3840*sin(c + d*x)**10*d)