Integrand size = 29, antiderivative size = 172 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-3 a^3 x-\frac {15 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d} \] Output:
-3*a^3*x-15/16*a^3*arctanh(cos(d*x+c))/d+a^3*cos(d*x+c)/d-3*a^3*cot(d*x+c) /d+a^3*cot(d*x+c)^3/d-3/5*a^3*cot(d*x+c)^5/d-1/7*a^3*cot(d*x+c)^7/d-15/16* a^3*cot(d*x+c)*csc(d*x+c)/d+11/8*a^3*cot(d*x+c)*csc(d*x+c)^3/d-1/2*a^3*cot (d*x+c)*csc(d*x+c)^5/d
Time = 2.69 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.70 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-13440 c-13440 d x+4480 \cos (c+d x)-9984 \cot \left (\frac {1}{2} (c+d x)\right )-1050 \csc ^2\left (\frac {1}{2} (c+d x)\right )+350 \csc ^4\left (\frac {1}{2} (c+d x)\right )-35 \csc ^6\left (\frac {1}{2} (c+d x)\right )-4200 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4200 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1050 \sec ^2\left (\frac {1}{2} (c+d x)\right )-350 \sec ^4\left (\frac {1}{2} (c+d x)\right )+35 \sec ^6\left (\frac {1}{2} (c+d x)\right )-7664 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+479 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-17 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-\frac {5}{2} \csc ^8\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+9984 \tan \left (\frac {1}{2} (c+d x)\right )+34 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{4480 d} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(-13440*c - 13440*d*x + 4480*Cos[c + d*x] - 9984*Cot[(c + d*x)/2] - 1 050*Csc[(c + d*x)/2]^2 + 350*Csc[(c + d*x)/2]^4 - 35*Csc[(c + d*x)/2]^6 - 4200*Log[Cos[(c + d*x)/2]] + 4200*Log[Sin[(c + d*x)/2]] + 1050*Sec[(c + d* x)/2]^2 - 350*Sec[(c + d*x)/2]^4 + 35*Sec[(c + d*x)/2]^6 - 7664*Csc[c + d* x]^3*Sin[(c + d*x)/2]^4 + 479*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 17*Csc[(c + d*x)/2]^6*Sin[c + d*x] - (5*Csc[(c + d*x)/2]^8*Sin[c + d*x])/2 + 9984*Ta n[(c + d*x)/2] + 34*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 5*Sec[(c + d*x)/ 2]^6*Tan[(c + d*x)/2]))/(4480*d)
Time = 0.53 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3351, 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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^3}{\sin (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\csc ^8(c+d x) a^9+3 \csc ^7(c+d x) a^9-8 \csc ^5(c+d x) a^9-6 \csc ^4(c+d x) a^9+6 \csc ^3(c+d x) a^9+8 \csc ^2(c+d x) a^9-\sin (c+d x) a^9-3 a^9\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {15 a^9 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^9 \cos (c+d x)}{d}-\frac {a^9 \cot ^7(c+d x)}{7 d}-\frac {3 a^9 \cot ^5(c+d x)}{5 d}+\frac {a^9 \cot ^3(c+d x)}{d}-\frac {3 a^9 \cot (c+d x)}{d}-\frac {a^9 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {11 a^9 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {15 a^9 \cot (c+d x) \csc (c+d x)}{16 d}-3 a^9 x}{a^6}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
(-3*a^9*x - (15*a^9*ArcTanh[Cos[c + d*x]])/(16*d) + (a^9*Cos[c + d*x])/d - (3*a^9*Cot[c + d*x])/d + (a^9*Cot[c + d*x]^3)/d - (3*a^9*Cot[c + d*x]^5)/ (5*d) - (a^9*Cot[c + d*x]^7)/(7*d) - (15*a^9*Cot[c + d*x]*Csc[c + d*x])/(1 6*d) + (11*a^9*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^9*Cot[c + d*x]*Csc[ c + d*x]^5)/(2*d))/a^6
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(-3 a^{3} x +\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a^{3} \left (-4480 i {\mathrm e}^{12 i \left (d x +c \right )}+525 \,{\mathrm e}^{13 i \left (d x +c \right )}+20160 i {\mathrm e}^{10 i \left (d x +c \right )}+980 \,{\mathrm e}^{11 i \left (d x +c \right )}-38080 i {\mathrm e}^{8 i \left (d x +c \right )}+945 \,{\mathrm e}^{9 i \left (d x +c \right )}+49280 i {\mathrm e}^{6 i \left (d x +c \right )}-32256 i {\mathrm e}^{4 i \left (d x +c \right )}-945 \,{\mathrm e}^{5 i \left (d x +c \right )}+12992 i {\mathrm e}^{2 i \left (d x +c \right )}-980 \,{\mathrm e}^{3 i \left (d x +c \right )}-2496 i-525 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{280 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\) | \(244\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{3} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(261\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{3} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(261\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-3*a^3*x+1/2*a^3/d*exp(I*(d*x+c))+1/2*a^3/d*exp(-I*(d*x+c))+1/280*a^3*(-44 80*I*exp(12*I*(d*x+c))+525*exp(13*I*(d*x+c))+20160*I*exp(10*I*(d*x+c))+980 *exp(11*I*(d*x+c))-38080*I*exp(8*I*(d*x+c))+945*exp(9*I*(d*x+c))+49280*I*e xp(6*I*(d*x+c))-32256*I*exp(4*I*(d*x+c))-945*exp(5*I*(d*x+c))+12992*I*exp( 2*I*(d*x+c))-980*exp(3*I*(d*x+c))-2496*I-525*exp(I*(d*x+c)))/d/(exp(2*I*(d *x+c))-1)^7+15/16*a^3/d*ln(exp(I*(d*x+c))-1)-15/16*a^3/d*ln(exp(I*(d*x+c)) +1)
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (160) = 320\).
Time = 0.11 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.95 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {4992 \, a^{3} \cos \left (d x + c\right )^{7} - 12992 \, a^{3} \cos \left (d x + c\right )^{5} + 11200 \, a^{3} \cos \left (d x + c\right )^{3} - 3360 \, a^{3} \cos \left (d x + c\right ) + 525 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 525 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (48 \, a^{3} d x \cos \left (d x + c\right )^{6} - 16 \, a^{3} \cos \left (d x + c\right )^{7} - 144 \, a^{3} d x \cos \left (d x + c\right )^{4} + 33 \, a^{3} \cos \left (d x + c\right )^{5} + 144 \, a^{3} d x \cos \left (d x + c\right )^{2} - 40 \, a^{3} \cos \left (d x + c\right )^{3} - 48 \, a^{3} d x + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/1120*(4992*a^3*cos(d*x + c)^7 - 12992*a^3*cos(d*x + c)^5 + 11200*a^3*co s(d*x + c)^3 - 3360*a^3*cos(d*x + c) + 525*(a^3*cos(d*x + c)^6 - 3*a^3*cos (d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2)*sin( d*x + c) - 525*(a^3*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 70*(48*a^3*d*x*c os(d*x + c)^6 - 16*a^3*cos(d*x + c)^7 - 144*a^3*d*x*cos(d*x + c)^4 + 33*a^ 3*cos(d*x + c)^5 + 144*a^3*d*x*cos(d*x + c)^2 - 40*a^3*cos(d*x + c)^3 - 48 *a^3*d*x + 15*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^6 - 3*d*cos (d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**2*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.35 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {160 \, a^{3} \cot \left (d x + c\right )^{7} + 224 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} - 35 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 70 \, a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{1120 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/1120*(160*a^3*cot(d*x + c)^7 + 224*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^3 - 35*a^3*(2*(33*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 70*a^3*(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 = 291, normalized size of antiderivative = 1.69 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 49 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 245 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 875 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13440 \, {\left (d x + c\right )} a^{3} + 4200 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 9065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {8960 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {10890 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 875 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 245 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 49 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{4480 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/4480*(5*a^3*tan(1/2*d*x + 1/2*c)^7 + 35*a^3*tan(1/2*d*x + 1/2*c)^6 + 49* a^3*tan(1/2*d*x + 1/2*c)^5 - 245*a^3*tan(1/2*d*x + 1/2*c)^4 - 875*a^3*tan( 1/2*d*x + 1/2*c)^3 + 455*a^3*tan(1/2*d*x + 1/2*c)^2 - 13440*(d*x + c)*a^3 + 4200*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 9065*a^3*tan(1/2*d*x + 1/2*c) + 8960*a^3/(tan(1/2*d*x + 1/2*c)^2 + 1) - (10890*a^3*tan(1/2*d*x + 1/2*c)^ 7 + 9065*a^3*tan(1/2*d*x + 1/2*c)^6 + 455*a^3*tan(1/2*d*x + 1/2*c)^5 - 875 *a^3*tan(1/2*d*x + 1/2*c)^4 - 245*a^3*tan(1/2*d*x + 1/2*c)^3 + 49*a^3*tan( 1/2*d*x + 1/2*c)^2 + 35*a^3*tan(1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/ 2*c)^7)/d
Time = 33.53 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.26 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {15\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,d}+\frac {6\,a^3\,\mathrm {atan}\left (\frac {36\,a^6}{\frac {45\,a^6}{4}+36\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {45\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {45\,a^6}{4}+36\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}-\frac {259\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-243\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+234\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {118\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{7}}{d\,\left (128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}+\frac {259\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^2,x)
Output:
(13*a^3*tan(c/2 + (d*x)/2)^2)/(128*d) - (25*a^3*tan(c/2 + (d*x)/2)^3)/(128 *d) - (7*a^3*tan(c/2 + (d*x)/2)^4)/(128*d) + (7*a^3*tan(c/2 + (d*x)/2)^5)/ (640*d) + (a^3*tan(c/2 + (d*x)/2)^6)/(128*d) + (a^3*tan(c/2 + (d*x)/2)^7)/ (896*d) + (15*a^3*log(tan(c/2 + (d*x)/2)))/(16*d) + (6*a^3*atan((36*a^6)/( (45*a^6)/4 + 36*a^6*tan(c/2 + (d*x)/2)) - (45*a^6*tan(c/2 + (d*x)/2))/(4*( (45*a^6)/4 + 36*a^6*tan(c/2 + (d*x)/2)))))/d - ((54*a^3*tan(c/2 + (d*x)/2) ^2)/35 - 6*a^3*tan(c/2 + (d*x)/2)^3 - (118*a^3*tan(c/2 + (d*x)/2)^4)/5 + 6 *a^3*tan(c/2 + (d*x)/2)^5 + 234*a^3*tan(c/2 + (d*x)/2)^6 - 243*a^3*tan(c/2 + (d*x)/2)^7 + 259*a^3*tan(c/2 + (d*x)/2)^8 + a^3/7 + a^3*tan(c/2 + (d*x) /2))/(d*(128*tan(c/2 + (d*x)/2)^7 + 128*tan(c/2 + (d*x)/2)^9)) + (259*a^3* tan(c/2 + (d*x)/2))/(128*d)
Time = 0.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.03 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (4480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-19968 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-4200 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+7936 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+6160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-2240 \cos \left (d x +c \right ) \sin \left (d x +c \right )-640 \cos \left (d x +c \right )+4200 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{7}-13440 \sin \left (d x +c \right )^{7} d x -4025 \sin \left (d x +c \right )^{7}\right )}{4480 \sin \left (d x +c \right )^{7} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(4480*cos(c + d*x)*sin(c + d*x)**7 - 19968*cos(c + d*x)*sin(c + d*x) **6 - 4200*cos(c + d*x)*sin(c + d*x)**5 + 7936*cos(c + d*x)*sin(c + d*x)** 4 + 6160*cos(c + d*x)*sin(c + d*x)**3 - 768*cos(c + d*x)*sin(c + d*x)**2 - 2240*cos(c + d*x)*sin(c + d*x) - 640*cos(c + d*x) + 4200*log(tan((c + d*x )/2))*sin(c + d*x)**7 - 13440*sin(c + d*x)**7*d*x - 4025*sin(c + d*x)**7)) /(4480*sin(c + d*x)**7*d)