Integrand size = 21, antiderivative size = 165 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {b \left (6 a^2-b^2\right ) \csc (c+d x)}{d}+\frac {a \left (2 a^2-3 b^2\right ) \csc ^2(c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {a^3 \csc ^4(c+d x)}{4 d}+\frac {a \left (a^2-6 b^2\right ) \log (\sin (c+d x))}{d}+\frac {b \left (3 a^2-2 b^2\right ) \sin (c+d x)}{d}+\frac {3 a b^2 \sin ^2(c+d x)}{2 d}+\frac {b^3 \sin ^3(c+d x)}{3 d} \] Output:
b*(6*a^2-b^2)*csc(d*x+c)/d+1/2*a*(2*a^2-3*b^2)*csc(d*x+c)^2/d-a^2*b*csc(d* x+c)^3/d-1/4*a^3*csc(d*x+c)^4/d+a*(a^2-6*b^2)*ln(sin(d*x+c))/d+b*(3*a^2-2* b^2)*sin(d*x+c)/d+3/2*a*b^2*sin(d*x+c)^2/d+1/3*b^3*sin(d*x+c)^3/d
Time = 0.10 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.18 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {6 a^2 b \csc (c+d x)}{d}-\frac {b^3 \csc (c+d x)}{d}+\frac {a^3 \csc ^2(c+d x)}{d}-\frac {3 a b^2 \csc ^2(c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {a^3 \csc ^4(c+d x)}{4 d}+\frac {a^3 \log (\sin (c+d x))}{d}-\frac {6 a b^2 \log (\sin (c+d x))}{d}+\frac {3 a^2 b \sin (c+d x)}{d}-\frac {2 b^3 \sin (c+d x)}{d}+\frac {3 a b^2 \sin ^2(c+d x)}{2 d}+\frac {b^3 \sin ^3(c+d x)}{3 d} \] Input:
Integrate[Cot[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]
Output:
(6*a^2*b*Csc[c + d*x])/d - (b^3*Csc[c + d*x])/d + (a^3*Csc[c + d*x]^2)/d - (3*a*b^2*Csc[c + d*x]^2)/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (a^3*Csc[c + d*x]^4)/(4*d) + (a^3*Log[Sin[c + d*x]])/d - (6*a*b^2*Log[Sin[c + d*x]])/d + (3*a^2*b*Sin[c + d*x])/d - (2*b^3*Sin[c + d*x])/d + (3*a*b^2*Sin[c + d*x ]^2)/(2*d) + (b^3*Sin[c + d*x]^3)/(3*d)
Time = 0.36 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3200, 522, 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.
| \(\displaystyle \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^3}{\tan (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3200 |
| \(\displaystyle \frac {\int \frac {\csc ^5(c+d x) (a+b \sin (c+d x))^3 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b^5}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {\int \left (\frac {a^3 \csc ^5(c+d x)}{b}+3 a^2 \csc ^4(c+d x)+\frac {\left (3 a b^4-2 a^3 b^2\right ) \csc ^3(c+d x)}{b^3}+\frac {\left (b^4-6 a^2 b^2\right ) \csc ^2(c+d x)}{b^2}+\frac {\left (a^3-6 a b^2\right ) \csc (c+d x)}{b}+b^2 \sin ^2(c+d x)+3 a^2 \left (1-\frac {2 b^2}{3 a^2}\right )+3 a b \sin (c+d x)\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{4} a^3 \csc ^4(c+d x)+b \left (3 a^2-2 b^2\right ) \sin (c+d x)+\frac {1}{2} a \left (2 a^2-3 b^2\right ) \csc ^2(c+d x)+b \left (6 a^2-b^2\right ) \csc (c+d x)+a \left (a^2-6 b^2\right ) \log (b \sin (c+d x))-a^2 b \csc ^3(c+d x)+\frac {3}{2} a b^2 \sin ^2(c+d x)+\frac {1}{3} b^3 \sin ^3(c+d x)}{d}\) |
Input:
Int[Cot[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]
Output:
(b*(6*a^2 - b^2)*Csc[c + d*x] + (a*(2*a^2 - 3*b^2)*Csc[c + d*x]^2)/2 - a^2 *b*Csc[c + d*x]^3 - (a^3*Csc[c + d*x]^4)/4 + a*(a^2 - 6*b^2)*Log[b*Sin[c + d*x]] + b*(3*a^2 - 2*b^2)*Sin[c + d*x] + (3*a*b^2*Sin[c + d*x]^2)/2 + (b^ 3*Sin[c + d*x]^3)/3)/d
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Time = 4.26 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{2} b \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )+b^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(212\) |
| default | \(\frac {a^{3} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{2} b \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )+b^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(212\) |
| risch | \(-\frac {3 i b \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 d}+\frac {3 i b \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 d}-\frac {7 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+6 i a \,b^{2} x -\frac {2 i \left (-2 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-6 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+2 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+14 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-3 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-2 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-14 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+3 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+6 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {i b^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {12 i a \,b^{2} c}{d}-\frac {3 a \,b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i b^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {2 i a^{3} c}{d}-i a^{3} x +\frac {7 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {6 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(452\) |
Input:
int(cot(d*x+c)^5*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-1/4*cot(d*x+c)^4+1/2*cot(d*x+c)^2+ln(sin(d*x+c)))+3*a^2*b*(-1/3 /sin(d*x+c)^3*cos(d*x+c)^6+1/sin(d*x+c)*cos(d*x+c)^6+(8/3+cos(d*x+c)^4+4/3 *cos(d*x+c)^2)*sin(d*x+c))+3*a*b^2*(-1/2/sin(d*x+c)^2*cos(d*x+c)^6-1/2*cos (d*x+c)^4-cos(d*x+c)^2-2*ln(sin(d*x+c)))+b^3*(-1/sin(d*x+c)*cos(d*x+c)^6-( 8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))
Time = 0.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.36 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {18 \, a b^{2} \cos \left (d x + c\right )^{6} - 45 \, a b^{2} \cos \left (d x + c\right )^{4} - 9 \, a^{3} + 9 \, a b^{2} + 6 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} - 12 \, {\left ({\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} - 6 \, a b^{2} - 2 \, {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (b^{3} \cos \left (d x + c\right )^{6} - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{4} - 24 \, a^{2} b + 8 \, b^{3} + 12 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-1/12*(18*a*b^2*cos(d*x + c)^6 - 45*a*b^2*cos(d*x + c)^4 - 9*a^3 + 9*a*b^2 + 6*(2*a^3 + 3*a*b^2)*cos(d*x + c)^2 - 12*((a^3 - 6*a*b^2)*cos(d*x + c)^4 + a^3 - 6*a*b^2 - 2*(a^3 - 6*a*b^2)*cos(d*x + c)^2)*log(1/2*sin(d*x + c)) + 4*(b^3*cos(d*x + c)^6 - 3*(3*a^2*b - b^3)*cos(d*x + c)^4 - 24*a^2*b + 8 *b^3 + 12*(3*a^2*b - b^3)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)
\[ \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \cot ^{5}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**5*(a+b*sin(d*x+c))**3,x)
Output:
Integral((a + b*sin(c + d*x))**3*cot(c + d*x)**5, x)
Time = 0.03 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {4 \, b^{3} \sin \left (d x + c\right )^{3} + 18 \, a b^{2} \sin \left (d x + c\right )^{2} + 12 \, {\left (a^{3} - 6 \, a b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) + 12 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \sin \left (d x + c\right ) - \frac {3 \, {\left (4 \, a^{2} b \sin \left (d x + c\right ) - 4 \, {\left (6 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{3} + a^{3} - 2 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{2}\right )}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="maxima")
Output:
1/12*(4*b^3*sin(d*x + c)^3 + 18*a*b^2*sin(d*x + c)^2 + 12*(a^3 - 6*a*b^2)* log(sin(d*x + c)) + 12*(3*a^2*b - 2*b^3)*sin(d*x + c) - 3*(4*a^2*b*sin(d*x + c) - 4*(6*a^2*b - b^3)*sin(d*x + c)^3 + a^3 - 2*(2*a^3 - 3*a*b^2)*sin(d *x + c)^2)/sin(d*x + c)^4)/d
Time = 0.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {4 \, b^{3} \sin \left (d x + c\right )^{3} + 18 \, a b^{2} \sin \left (d x + c\right )^{2} + 36 \, a^{2} b \sin \left (d x + c\right ) - 24 \, b^{3} \sin \left (d x + c\right ) + 12 \, {\left (a^{3} - 6 \, a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {3 \, {\left (4 \, a^{2} b \sin \left (d x + c\right ) - 4 \, {\left (6 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{3} + a^{3} - 2 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{2}\right )}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/12*(4*b^3*sin(d*x + c)^3 + 18*a*b^2*sin(d*x + c)^2 + 36*a^2*b*sin(d*x + c) - 24*b^3*sin(d*x + c) + 12*(a^3 - 6*a*b^2)*log(abs(sin(d*x + c))) - 3*( 4*a^2*b*sin(d*x + c) - 4*(6*a^2*b - b^3)*sin(d*x + c)^3 + a^3 - 2*(2*a^3 - 3*a*b^2)*sin(d*x + c)^2)/sin(d*x + c)^4)/d
Time = 17.80 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.57 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (6\,a\,b^2-a^3\right )}{d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a\,b^2}{8}-\frac {3\,a^3}{16}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (6\,a\,b^2-a^3\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (3\,a^3+90\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (18\,a\,b^2-\frac {33\,a^3}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (6\,a\,b^2-\frac {9\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {35\,a^3}{4}+78\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (36\,a^2\,b-8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (138\,a^2\,b-72\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (216\,a^2\,b-88\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (316\,a^2\,b-\frac {328\,b^3}{3}\right )-\frac {a^3}{4}-2\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {21\,a^2\,b}{8}-\frac {b^3}{2}\right )}{d}-\frac {a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d} \] Input:
int(cot(c + d*x)^5*(a + b*sin(c + d*x))^3,x)
Output:
(log(tan(c/2 + (d*x)/2)^2 + 1)*(6*a*b^2 - a^3))/d - (a^3*tan(c/2 + (d*x)/2 )^4)/(64*d) - (tan(c/2 + (d*x)/2)^2*((3*a*b^2)/8 - (3*a^3)/16))/d - (log(t an(c/2 + (d*x)/2))*(6*a*b^2 - a^3))/d + (tan(c/2 + (d*x)/2)^8*(90*a*b^2 + 3*a^3) - tan(c/2 + (d*x)/2)^4*(18*a*b^2 - (33*a^3)/4) - tan(c/2 + (d*x)/2) ^2*(6*a*b^2 - (9*a^3)/4) + tan(c/2 + (d*x)/2)^6*(78*a*b^2 + (35*a^3)/4) + tan(c/2 + (d*x)/2)^3*(36*a^2*b - 8*b^3) + tan(c/2 + (d*x)/2)^9*(138*a^2*b - 72*b^3) + tan(c/2 + (d*x)/2)^5*(216*a^2*b - 88*b^3) + tan(c/2 + (d*x)/2) ^7*(316*a^2*b - (328*b^3)/3) - a^3/4 - 2*a^2*b*tan(c/2 + (d*x)/2))/(d*(16* tan(c/2 + (d*x)/2)^4 + 48*tan(c/2 + (d*x)/2)^6 + 48*tan(c/2 + (d*x)/2)^8 + 16*tan(c/2 + (d*x)/2)^10)) + (tan(c/2 + (d*x)/2)*((21*a^2*b)/8 - b^3/2))/ d - (a^2*b*tan(c/2 + (d*x)/2)^3)/(8*d)
Time = 0.16 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.62 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-192 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4} a^{3}+1152 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4} a \,b^{2}+192 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a^{3}-1152 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a \,b^{2}+64 \sin \left (d x +c \right )^{7} b^{3}+288 \sin \left (d x +c \right )^{6} a \,b^{2}+576 \sin \left (d x +c \right )^{5} a^{2} b -384 \sin \left (d x +c \right )^{5} b^{3}-177 \sin \left (d x +c \right )^{4} a^{3}+360 \sin \left (d x +c \right )^{4} a \,b^{2}+1152 \sin \left (d x +c \right )^{3} a^{2} b -192 \sin \left (d x +c \right )^{3} b^{3}+192 \sin \left (d x +c \right )^{2} a^{3}-288 \sin \left (d x +c \right )^{2} a \,b^{2}-192 \sin \left (d x +c \right ) a^{2} b -48 a^{3}}{192 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^5*(a+b*sin(d*x+c))^3,x)
Output:
( - 192*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a**3 + 1152*log(tan(( c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a*b**2 + 192*log(tan((c + d*x)/2))*sin (c + d*x)**4*a**3 - 1152*log(tan((c + d*x)/2))*sin(c + d*x)**4*a*b**2 + 64 *sin(c + d*x)**7*b**3 + 288*sin(c + d*x)**6*a*b**2 + 576*sin(c + d*x)**5*a **2*b - 384*sin(c + d*x)**5*b**3 - 177*sin(c + d*x)**4*a**3 + 360*sin(c + d*x)**4*a*b**2 + 1152*sin(c + d*x)**3*a**2*b - 192*sin(c + d*x)**3*b**3 + 192*sin(c + d*x)**2*a**3 - 288*sin(c + d*x)**2*a*b**2 - 192*sin(c + d*x)*a **2*b - 48*a**3)/(192*sin(c + d*x)**4*d)