Integrand size = 29, antiderivative size = 176 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {11 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d} \] Output:
-11/128*a^2*arctanh(cos(d*x+c))/d-2/5*a^2*cot(d*x+c)^5/d-2/7*a^2*cot(d*x+c )^7/d-11/128*a^2*cot(d*x+c)*csc(d*x+c)/d+7/64*a^2*cot(d*x+c)*csc(d*x+c)^3/ d-1/6*a^2*cot(d*x+c)^3*csc(d*x+c)^3/d+1/16*a^2*cot(d*x+c)*csc(d*x+c)^5/d-1 /8*a^2*cot(d*x+c)^3*csc(d*x+c)^5/d
Time = 6.53 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.65 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^8(c+d x) \left (158270 \cos (c+d x)+77210 \cos (3 (c+d x))-18130 \cos (5 (c+d x))-2310 \cos (7 (c+d x))+40425 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-64680 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+32340 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-9240 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1155 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-40425 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+64680 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-32340 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9240 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1155 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+86016 \sin (2 (c+d x))+64512 \sin (4 (c+d x))+12288 \sin (6 (c+d x))-1536 \sin (8 (c+d x))\right )}{1720320 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]
Output:
-1/1720320*(a^2*Csc[c + d*x]^8*(158270*Cos[c + d*x] + 77210*Cos[3*(c + d*x )] - 18130*Cos[5*(c + d*x)] - 2310*Cos[7*(c + d*x)] + 40425*Log[Cos[(c + d *x)/2]] - 64680*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 32340*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 9240*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 1155*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 40425*Log[Sin[(c + d*x)/2] ] + 64680*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 32340*Cos[4*(c + d*x)]* Log[Sin[(c + d*x)/2]] + 9240*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 1155 *Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 86016*Sin[2*(c + d*x)] + 64512*S in[4*(c + d*x)] + 12288*Sin[6*(c + d*x)] - 1536*Sin[8*(c + d*x)]))/d
Time = 0.55 (sec) , antiderivative size = 176, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^2}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \cot ^4(c+d x) \csc ^5(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^4(c+d x)+a^2 \cot ^4(c+d x) \csc ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {11 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]
Output:
(-11*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - ( 2*a^2*Cot[c + d*x]^7)/(7*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d* x]^3*Csc[c + d*x]^5)/(8*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 = 0.55 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(\frac {a^{2} \left (1155 \,{\mathrm e}^{15 i \left (d x +c \right )}+9065 \,{\mathrm e}^{13 i \left (d x +c \right )}-38605 \,{\mathrm e}^{11 i \left (d x +c \right )}+53760 i {\mathrm e}^{12 i \left (d x +c \right )}-79135 \,{\mathrm e}^{9 i \left (d x +c \right )}-79135 \,{\mathrm e}^{7 i \left (d x +c \right )}+53760 i {\mathrm e}^{8 i \left (d x +c \right )}-38605 \,{\mathrm e}^{5 i \left (d x +c \right )}-86016 i {\mathrm e}^{6 i \left (d x +c \right )}+9065 \,{\mathrm e}^{3 i \left (d x +c \right )}-10752 i {\mathrm e}^{4 i \left (d x +c \right )}+1155 \,{\mathrm e}^{i \left (d x +c \right )}-12288 i {\mathrm e}^{2 i \left (d x +c \right )}+1536 i\right )}{6720 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}-\frac {11 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}+\frac {11 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}\) | \(214\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(256\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(256\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/6720*a^2*(1155*exp(15*I*(d*x+c))+9065*exp(13*I*(d*x+c))-38605*exp(11*I*( d*x+c))+53760*I*exp(12*I*(d*x+c))-79135*exp(9*I*(d*x+c))-79135*exp(7*I*(d* x+c))+53760*I*exp(8*I*(d*x+c))-38605*exp(5*I*(d*x+c))-86016*I*exp(6*I*(d*x +c))+9065*exp(3*I*(d*x+c))-10752*I*exp(4*I*(d*x+c))+1155*exp(I*(d*x+c))-12 288*I*exp(2*I*(d*x+c))+1536*I)/d/(exp(2*I*(d*x+c))-1)^8-11/128*a^2/d*ln(ex p(I*(d*x+c))+1)+11/128*a^2/d*ln(exp(I*(d*x+c))-1)
Time = 0.12 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.54 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {2310 \, a^{2} \cos \left (d x + c\right )^{7} + 490 \, a^{2} \cos \left (d x + c\right )^{5} - 8470 \, a^{2} \cos \left (d x + c\right )^{3} + 2310 \, a^{2} \cos \left (d x + c\right ) - 1155 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1155 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1536 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{7} - 7 \, a^{2} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{26880 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
1/26880*(2310*a^2*cos(d*x + c)^7 + 490*a^2*cos(d*x + c)^5 - 8470*a^2*cos(d *x + c)^3 + 2310*a^2*cos(d*x + c) - 1155*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d *x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(1/2*cos (d*x + c) + 1/2) + 1155*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2 *cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2) + 1536*(2*a^2*cos(d*x + c)^7 - 7*a^2*cos(d*x + c)^5)*sin(d*x + c))/(d*cos (d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**5*(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {105 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1536 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}}}{26880 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/26880*(105*a^2*(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)) + 280*a^2*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/( cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*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^2/tan (d*x + c)^7)/d
Time = 0.25 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.66 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 480 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 18480 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 10080 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {50226 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 10080 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1680 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{215040 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/215040*(105*a^2*tan(1/2*d*x + 1/2*c)^8 + 480*a^2*tan(1/2*d*x + 1/2*c)^7 + 560*a^2*tan(1/2*d*x + 1/2*c)^6 - 672*a^2*tan(1/2*d*x + 1/2*c)^5 - 2520*a ^2*tan(1/2*d*x + 1/2*c)^4 - 3360*a^2*tan(1/2*d*x + 1/2*c)^3 - 1680*a^2*tan (1/2*d*x + 1/2*c)^2 + 18480*a^2*log(abs(tan(1/2*d*x + 1/2*c))) + 10080*a^2 *tan(1/2*d*x + 1/2*c) - (50226*a^2*tan(1/2*d*x + 1/2*c)^8 + 10080*a^2*tan( 1/2*d*x + 1/2*c)^7 - 1680*a^2*tan(1/2*d*x + 1/2*c)^6 - 3360*a^2*tan(1/2*d* x + 1/2*c)^5 - 2520*a^2*tan(1/2*d*x + 1/2*c)^4 - 672*a^2*tan(1/2*d*x + 1/2 *c)^3 + 560*a^2*tan(1/2*d*x + 1/2*c)^2 + 480*a^2*tan(1/2*d*x + 1/2*c) + 10 5*a^2)/tan(1/2*d*x + 1/2*c)^8)/d
Time = 18.19 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.81 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{448\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{448\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {11\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}-\frac {3\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,d}+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,d} \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^2)/sin(c + d*x)^5,x)
Output:
(a^2*cot(c/2 + (d*x)/2)^2)/(128*d) + (a^2*cot(c/2 + (d*x)/2)^3)/(64*d) + ( 3*a^2*cot(c/2 + (d*x)/2)^4)/(256*d) + (a^2*cot(c/2 + (d*x)/2)^5)/(320*d) - (a^2*cot(c/2 + (d*x)/2)^6)/(384*d) - (a^2*cot(c/2 + (d*x)/2)^7)/(448*d) - (a^2*cot(c/2 + (d*x)/2)^8)/(2048*d) - (a^2*tan(c/2 + (d*x)/2)^2)/(128*d) - (a^2*tan(c/2 + (d*x)/2)^3)/(64*d) - (3*a^2*tan(c/2 + (d*x)/2)^4)/(256*d) - (a^2*tan(c/2 + (d*x)/2)^5)/(320*d) + (a^2*tan(c/2 + (d*x)/2)^6)/(384*d) + (a^2*tan(c/2 + (d*x)/2)^7)/(448*d) + (a^2*tan(c/2 + (d*x)/2)^8)/(2048*d ) + (11*a^2*log(tan(c/2 + (d*x)/2)))/(128*d) - (3*a^2*cot(c/2 + (d*x)/2))/ (64*d) + (3*a^2*tan(c/2 + (d*x)/2))/(64*d)
Time = 0.17 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-1536 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-1155 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+3710 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+6144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-3840 \cos \left (d x +c \right ) \sin \left (d x +c \right )-1680 \cos \left (d x +c \right )+1155 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{8}\right )}{13440 \sin \left (d x +c \right )^{8} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*( - 1536*cos(c + d*x)*sin(c + d*x)**7 - 1155*cos(c + d*x)*sin(c + d* x)**6 - 768*cos(c + d*x)*sin(c + d*x)**5 + 3710*cos(c + d*x)*sin(c + d*x)* *4 + 6144*cos(c + d*x)*sin(c + d*x)**3 + 280*cos(c + d*x)*sin(c + d*x)**2 - 3840*cos(c + d*x)*sin(c + d*x) - 1680*cos(c + d*x) + 1155*log(tan((c + d *x)/2))*sin(c + d*x)**8))/(13440*sin(c + d*x)**8*d)