Integrand size = 21, antiderivative size = 192 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-a^2 x+\frac {5 b^2 x}{2}-\frac {15 a b \text {arctanh}(\cos (c+d x))}{4 d}+\frac {2 a b \cos (c+d x)}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {2 b^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {b^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {9 a b \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a b \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 d} \] Output:
-a^2*x+5/2*b^2*x-15/4*a*b*arctanh(cos(d*x+c))/d+2*a*b*cos(d*x+c)/d-a^2*cot (d*x+c)/d+2*b^2*cot(d*x+c)/d+1/3*a^2*cot(d*x+c)^3/d-1/3*b^2*cot(d*x+c)^3/d -1/5*a^2*cot(d*x+c)^5/d+9/4*a*b*cot(d*x+c)*csc(d*x+c)/d-1/2*a*b*cot(d*x+c) *csc(d*x+c)^3/d+1/2*b^2*cos(d*x+c)*sin(d*x+c)/d
Time = 1.14 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.83 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-480 a^2 c+1200 b^2 c-480 a^2 d x+1200 b^2 d x+960 a b \cos (c+d x)+\left (-368 a^2+560 b^2\right ) \cot \left (\frac {1}{2} (c+d x)\right )+270 a b \csc ^2\left (\frac {1}{2} (c+d x)\right )-15 a b \csc ^4\left (\frac {1}{2} (c+d x)\right )-1800 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1800 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-270 a b \sec ^2\left (\frac {1}{2} (c+d x)\right )+15 a b \sec ^4\left (\frac {1}{2} (c+d x)\right )-328 a^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+160 b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+96 a^2 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+\frac {41}{2} a^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-10 b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-\frac {3}{2} a^2 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+120 b^2 \sin (2 (c+d x))+368 a^2 \tan \left (\frac {1}{2} (c+d x)\right )-560 b^2 \tan \left (\frac {1}{2} (c+d x)\right )}{480 d} \] Input:
Integrate[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]
Output:
(-480*a^2*c + 1200*b^2*c - 480*a^2*d*x + 1200*b^2*d*x + 960*a*b*Cos[c + d* x] + (-368*a^2 + 560*b^2)*Cot[(c + d*x)/2] + 270*a*b*Csc[(c + d*x)/2]^2 - 15*a*b*Csc[(c + d*x)/2]^4 - 1800*a*b*Log[Cos[(c + d*x)/2]] + 1800*a*b*Log[ Sin[(c + d*x)/2]] - 270*a*b*Sec[(c + d*x)/2]^2 + 15*a*b*Sec[(c + d*x)/2]^4 - 328*a^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 160*b^2*Csc[c + d*x]^3*Sin[ (c + d*x)/2]^4 + 96*a^2*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + (41*a^2*Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 10*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - (3 *a^2*Csc[(c + d*x)/2]^6*Sin[c + d*x])/2 + 120*b^2*Sin[2*(c + d*x)] + 368*a ^2*Tan[(c + d*x)/2] - 560*b^2*Tan[(c + d*x)/2])/(480*d)
Time = 0.46 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3201, 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.
| \(\displaystyle \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^2}{\tan (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3201 |
| \(\displaystyle \int \left (a^2 \cot ^6(c+d x)+2 a b \cos (c+d x) \cot ^5(c+d x)+b^2 \cos ^2(c+d x) \cot ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}-a^2 x-\frac {15 a b \text {arctanh}(\cos (c+d x))}{4 d}+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac {5 b^2 x}{2}\) |
Input:
Int[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]
Output:
-(a^2*x) + (5*b^2*x)/2 - (15*a*b*ArcTanh[Cos[c + d*x]])/(4*d) + (15*a*b*Co s[c + d*x])/(4*d) - (a^2*Cot[c + d*x])/d + (5*b^2*Cot[c + d*x])/(2*d) + (5 *a*b*Cos[c + d*x]*Cot[c + d*x]^2)/(4*d) + (a^2*Cot[c + d*x]^3)/(3*d) - (5* b^2*Cot[c + d*x]^3)/(6*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (a *b*Cos[c + d*x]*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5)/(5*d)
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((g_.)*tan[(e_.) + (f_.)*( x_)])^(p_.), x_Symbol] :> Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Si n[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 2.68 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(216\) |
| default | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(216\) |
| risch | \(-a^{2} x +\frac {5 b^{2} x}{2}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{8 d}+\frac {a b \,{\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {a b \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{8 d}-\frac {180 i a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-180 i b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+135 a b \,{\mathrm e}^{9 i \left (d x +c \right )}-360 i a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+600 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-150 a b \,{\mathrm e}^{7 i \left (d x +c \right )}+560 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-800 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-280 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+520 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+150 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+92 i a^{2}-140 i b^{2}-135 a b \,{\mathrm e}^{i \left (d x +c \right )}}{30 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {15 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}+\frac {15 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d}\) | \(321\) |
Input:
int(cot(d*x+c)^6*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+2*a*b*(-1/4 /sin(d*x+c)^4*cos(d*x+c)^7+3/8/sin(d*x+c)^2*cos(d*x+c)^7+3/8*cos(d*x+c)^5+ 5/8*cos(d*x+c)^3+15/8*cos(d*x+c)+15/8*ln(csc(d*x+c)-cot(d*x+c)))+b^2*(-1/3 /sin(d*x+c)^3*cos(d*x+c)^7+4/3/sin(d*x+c)*cos(d*x+c)^7+4/3*(cos(d*x+c)^5+5 /4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/2*d*x+5/2*c))
Time = 0.10 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.59 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {60 \, b^{2} \cos \left (d x + c\right )^{7} + 92 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 140 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 225 \, {\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 225 \, {\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 60 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right ) + 30 \, {\left (2 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x \cos \left (d x + c\right )^{4} - 8 \, a b \cos \left (d x + c\right )^{5} - 4 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x \cos \left (d x + c\right )^{2} + 25 \, a b \cos \left (d x + c\right )^{3} + 2 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6*(a+b*sin(d*x+c))^2,x, algorithm="fricas")
Output:
-1/120*(60*b^2*cos(d*x + c)^7 + 92*(2*a^2 - 5*b^2)*cos(d*x + c)^5 - 140*(2 *a^2 - 5*b^2)*cos(d*x + c)^3 + 225*(a*b*cos(d*x + c)^4 - 2*a*b*cos(d*x + c )^2 + a*b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 225*(a*b*cos(d*x + c )^4 - 2*a*b*cos(d*x + c)^2 + a*b)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c ) + 60*(2*a^2 - 5*b^2)*cos(d*x + c) + 30*(2*(2*a^2 - 5*b^2)*d*x*cos(d*x + c)^4 - 8*a*b*cos(d*x + c)^5 - 4*(2*a^2 - 5*b^2)*d*x*cos(d*x + c)^2 + 25*a* b*cos(d*x + c)^3 + 2*(2*a^2 - 5*b^2)*d*x - 15*a*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))
\[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{2} \cot ^{6}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**6*(a+b*sin(d*x+c))**2,x)
Output:
Integral((a + b*sin(c + d*x))**2*cot(c + d*x)**6, x)
Time = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.95 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 20 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} b^{2} + 15 \, a b {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+b*sin(d*x+c))^2,x, algorithm="maxima")
Output:
-1/120*(8*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan( d*x + c)^5)*a^2 - 20*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c) ^2 - 2)/(tan(d*x + c)^5 + tan(d*x + c)^3))*b^2 + 15*a*b*(2*(9*cos(d*x + c) ^3 - 7*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d
Time = 0.26 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.76 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1800 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} {\left (d x + c\right )} - \frac {480 \, {\left (b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {4110 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/480*(3*a^2*tan(1/2*d*x + 1/2*c)^5 + 15*a*b*tan(1/2*d*x + 1/2*c)^4 - 35*a ^2*tan(1/2*d*x + 1/2*c)^3 + 20*b^2*tan(1/2*d*x + 1/2*c)^3 - 240*a*b*tan(1/ 2*d*x + 1/2*c)^2 + 1800*a*b*log(abs(tan(1/2*d*x + 1/2*c))) + 330*a^2*tan(1 /2*d*x + 1/2*c) - 540*b^2*tan(1/2*d*x + 1/2*c) - 240*(2*a^2 - 5*b^2)*(d*x + c) - 480*(b^2*tan(1/2*d*x + 1/2*c)^3 - 4*a*b*tan(1/2*d*x + 1/2*c)^2 - b^ 2*tan(1/2*d*x + 1/2*c) - 4*a*b)/(tan(1/2*d*x + 1/2*c)^2 + 1)^2 - (4110*a*b *tan(1/2*d*x + 1/2*c)^5 + 330*a^2*tan(1/2*d*x + 1/2*c)^4 - 540*b^2*tan(1/2 *d*x + 1/2*c)^4 - 240*a*b*tan(1/2*d*x + 1/2*c)^3 - 35*a^2*tan(1/2*d*x + 1/ 2*c)^2 + 20*b^2*tan(1/2*d*x + 1/2*c)^2 + 15*a*b*tan(1/2*d*x + 1/2*c) + 3*a ^2)/tan(1/2*d*x + 1/2*c)^5)/d
Time = 25.43 (sec) , antiderivative size = 888, normalized size of antiderivative = 4.62 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^6*(a + b*sin(c + d*x))^2,x)
Output:
((95*b^2*cos(c + d*x))/384 - (5*a^2*cos(c + d*x))/24 + (5*a^2*cos(3*c + 3* d*x))/48 - (23*a^2*cos(5*c + 5*d*x))/240 - (163*b^2*cos(3*c + 3*d*x))/384 + (71*b^2*cos(5*c + 5*d*x))/384 - (b^2*cos(7*c + 7*d*x))/128 + (5*a^2*atan ((10*b^2*cos(c/2 + (d*x)/2) - 4*a^2*cos(c/2 + (d*x)/2) + 15*a*b*sin(c/2 + (d*x)/2))/(4*a^2*sin(c/2 + (d*x)/2) - 10*b^2*sin(c/2 + (d*x)/2) + 15*a*b*c os(c/2 + (d*x)/2)))*sin(3*c + 3*d*x))/8 - (a^2*atan((10*b^2*cos(c/2 + (d*x )/2) - 4*a^2*cos(c/2 + (d*x)/2) + 15*a*b*sin(c/2 + (d*x)/2))/(4*a^2*sin(c/ 2 + (d*x)/2) - 10*b^2*sin(c/2 + (d*x)/2) + 15*a*b*cos(c/2 + (d*x)/2)))*sin (5*c + 5*d*x))/8 - (25*b^2*atan((10*b^2*cos(c/2 + (d*x)/2) - 4*a^2*cos(c/2 + (d*x)/2) + 15*a*b*sin(c/2 + (d*x)/2))/(4*a^2*sin(c/2 + (d*x)/2) - 10*b^ 2*sin(c/2 + (d*x)/2) + 15*a*b*cos(c/2 + (d*x)/2)))*sin(3*c + 3*d*x))/16 + (5*b^2*atan((10*b^2*cos(c/2 + (d*x)/2) - 4*a^2*cos(c/2 + (d*x)/2) + 15*a*b *sin(c/2 + (d*x)/2))/(4*a^2*sin(c/2 + (d*x)/2) - 10*b^2*sin(c/2 + (d*x)/2) + 15*a*b*cos(c/2 + (d*x)/2)))*sin(5*c + 5*d*x))/16 + (5*a*b*sin(c + d*x)) /4 - (5*a^2*atan((10*b^2*cos(c/2 + (d*x)/2) - 4*a^2*cos(c/2 + (d*x)/2) + 1 5*a*b*sin(c/2 + (d*x)/2))/(4*a^2*sin(c/2 + (d*x)/2) - 10*b^2*sin(c/2 + (d* x)/2) + 15*a*b*cos(c/2 + (d*x)/2)))*sin(c + d*x))/4 + (25*b^2*atan((10*b^2 *cos(c/2 + (d*x)/2) - 4*a^2*cos(c/2 + (d*x)/2) + 15*a*b*sin(c/2 + (d*x)/2) )/(4*a^2*sin(c/2 + (d*x)/2) - 10*b^2*sin(c/2 + (d*x)/2) + 15*a*b*cos(c/2 + (d*x)/2)))*sin(c + d*x))/8 + (5*a*b*sin(2*c + 2*d*x))/8 - (5*a*b*sin(3...
Time = 0.18 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.23 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}+960 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b -736 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}+1120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+1080 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b +352 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-240 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -96 \cos \left (d x +c \right ) a^{2}+1800 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} a b -480 \sin \left (d x +c \right )^{5} a^{2} d x -1425 \sin \left (d x +c \right )^{5} a b +1200 \sin \left (d x +c \right )^{5} b^{2} d x}{480 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^6*(a+b*sin(d*x+c))^2,x)
Output:
(240*cos(c + d*x)*sin(c + d*x)**6*b**2 + 960*cos(c + d*x)*sin(c + d*x)**5* a*b - 736*cos(c + d*x)*sin(c + d*x)**4*a**2 + 1120*cos(c + d*x)*sin(c + d* x)**4*b**2 + 1080*cos(c + d*x)*sin(c + d*x)**3*a*b + 352*cos(c + d*x)*sin( c + d*x)**2*a**2 - 160*cos(c + d*x)*sin(c + d*x)**2*b**2 - 240*cos(c + d*x )*sin(c + d*x)*a*b - 96*cos(c + d*x)*a**2 + 1800*log(tan((c + d*x)/2))*sin (c + d*x)**5*a*b - 480*sin(c + d*x)**5*a**2*d*x - 1425*sin(c + d*x)**5*a*b + 1200*sin(c + d*x)**5*b**2*d*x)/(480*sin(c + d*x)**5*d)