Integrand size = 27, antiderivative size = 145 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {4 a^4 \csc (c+d x)}{d}-\frac {2 a^4 \csc ^2(c+d x)}{d}-\frac {a^4 \csc ^3(c+d x)}{3 d}-\frac {4 a^4 \log (\sin (c+d x))}{d}-\frac {10 a^4 \sin (c+d x)}{d}-\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^4(c+d x)}{d}+\frac {a^4 \sin ^5(c+d x)}{5 d} \] Output:
-4*a^4*csc(d*x+c)/d-2*a^4*csc(d*x+c)^2/d-1/3*a^4*csc(d*x+c)^3/d-4*a^4*ln(s in(d*x+c))/d-10*a^4*sin(d*x+c)/d-2*a^4*sin(d*x+c)^2/d+4/3*a^4*sin(d*x+c)^3 /d+a^4*sin(d*x+c)^4/d+1/5*a^4*sin(d*x+c)^5/d
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.66 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {a^4 \left (60 \csc (c+d x)+30 \csc ^2(c+d x)+5 \csc ^3(c+d x)+60 \log (\sin (c+d x))+150 \sin (c+d x)+30 \sin ^2(c+d x)-20 \sin ^3(c+d x)-15 \sin ^4(c+d x)-3 \sin ^5(c+d x)\right )}{15 d} \] Input:
Integrate[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]
Output:
-1/15*(a^4*(60*Csc[c + d*x] + 30*Csc[c + d*x]^2 + 5*Csc[c + d*x]^3 + 60*Lo g[Sin[c + d*x]] + 150*Sin[c + d*x] + 30*Sin[c + d*x]^2 - 20*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 - 3*Sin[c + d*x]^5))/d
Time = 0.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3315, 27, 99, 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.
\(\displaystyle \int \cos (c+d x) \cot ^4(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)^4}{\sin (c+d x)^4}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \csc ^4(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^6d(a \sin (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\csc ^4(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^6}{a^4}d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (\csc ^4(c+d x) a^4+\sin ^4(c+d x) a^4+4 \csc ^3(c+d x) a^4+4 \sin ^3(c+d x) a^4+4 \csc ^2(c+d x) a^4+4 \sin ^2(c+d x) a^4-4 \csc (c+d x) a^4-4 \sin (c+d x) a^4-10 a^4\right )d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{5} a^5 \sin ^5(c+d x)+a^5 \sin ^4(c+d x)+\frac {4}{3} a^5 \sin ^3(c+d x)-2 a^5 \sin ^2(c+d x)-10 a^5 \sin (c+d x)-\frac {1}{3} a^5 \csc ^3(c+d x)-2 a^5 \csc ^2(c+d x)-4 a^5 \csc (c+d x)-4 a^5 \log (a \sin (c+d x))}{a d}\) |
Input:
Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]
Output:
(-4*a^5*Csc[c + d*x] - 2*a^5*Csc[c + d*x]^2 - (a^5*Csc[c + d*x]^3)/3 - 4*a ^5*Log[a*Sin[c + d*x]] - 10*a^5*Sin[c + d*x] - 2*a^5*Sin[c + d*x]^2 + (4*a ^5*Sin[c + d*x]^3)/3 + a^5*Sin[c + d*x]^4 + (a^5*Sin[c + d*x]^5)/5)/(a*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 10.24 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.67
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )+a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(242\) |
default | \(\frac {\frac {a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )+a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(242\) |
risch | \(4 i a^{4} x -\frac {i a^{4} {\mathrm e}^{5 i \left (d x +c \right )}}{160 d}+\frac {a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{16 d}+\frac {19 i a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{96 d}+\frac {a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}+\frac {71 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {71 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}-\frac {19 i a^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{96 d}+\frac {a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{16 d}+\frac {i a^{4} {\mathrm e}^{-5 i \left (d x +c \right )}}{160 d}+\frac {8 i a^{4} c}{d}-\frac {8 i a^{4} \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-7 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}-3 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {4 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(295\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(1/5*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4*a^4*(1/4*cos (d*x+c)^4+1/2*cos(d*x+c)^2+ln(sin(d*x+c)))+6*a^4*(-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))+4*a^4*(-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)))+a^4*(-1/3/s in(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*c os(d*x+c)^2)*sin(d*x+c)))
Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {24 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \, a^{4} \cos \left (d x + c\right )^{6} - 576 \, a^{4} \cos \left (d x + c\right )^{4} + 2304 \, a^{4} \cos \left (d x + c\right )^{2} - 1536 \, a^{4} + 480 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 15 \, {\left (8 \, a^{4} \cos \left (d x + c\right )^{6} - 8 \, a^{4} \cos \left (d x + c\right )^{4} - 3 \, a^{4} \cos \left (d x + c\right )^{2} + 19 \, a^{4}\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="fricas" )
Output:
-1/120*(24*a^4*cos(d*x + c)^8 - 256*a^4*cos(d*x + c)^6 - 576*a^4*cos(d*x + c)^4 + 2304*a^4*cos(d*x + c)^2 - 1536*a^4 + 480*(a^4*cos(d*x + c)^2 - a^4 )*log(1/2*sin(d*x + c))*sin(d*x + c) - 15*(8*a^4*cos(d*x + c)^6 - 8*a^4*co s(d*x + c)^4 - 3*a^4*cos(d*x + c)^2 + 19*a^4)*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))
\[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=a^{4} \left (\int \cos {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int 4 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**4*(a+a*sin(d*x+c))**4,x)
Output:
a**4*(Integral(cos(c + d*x)*cot(c + d*x)**4, x) + Integral(4*sin(c + d*x)* cos(c + d*x)*cot(c + d*x)**4, x) + Integral(6*sin(c + d*x)**2*cos(c + d*x) *cot(c + d*x)**4, x) + Integral(4*sin(c + d*x)**3*cos(c + d*x)*cot(c + d*x )**4, x) + Integral(sin(c + d*x)**4*cos(c + d*x)*cot(c + d*x)**4, x))
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.82 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{4} \sin \left (d x + c\right )^{4} + 20 \, a^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) - 150 \, a^{4} \sin \left (d x + c\right ) - \frac {5 \, {\left (12 \, a^{4} \sin \left (d x + c\right )^{2} + 6 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{3}}}{15 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="maxima" )
Output:
1/15*(3*a^4*sin(d*x + c)^5 + 15*a^4*sin(d*x + c)^4 + 20*a^4*sin(d*x + c)^3 - 30*a^4*sin(d*x + c)^2 - 60*a^4*log(sin(d*x + c)) - 150*a^4*sin(d*x + c) - 5*(12*a^4*sin(d*x + c)^2 + 6*a^4*sin(d*x + c) + a^4)/sin(d*x + c)^3)/d
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{4} \sin \left (d x + c\right )^{4} + 20 \, a^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 150 \, a^{4} \sin \left (d x + c\right ) - \frac {5 \, {\left (12 \, a^{4} \sin \left (d x + c\right )^{2} + 6 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{3}}}{15 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
1/15*(3*a^4*sin(d*x + c)^5 + 15*a^4*sin(d*x + c)^4 + 20*a^4*sin(d*x + c)^3 - 30*a^4*sin(d*x + c)^2 - 60*a^4*log(abs(sin(d*x + c))) - 150*a^4*sin(d*x + c) - 5*(12*a^4*sin(d*x + c)^2 + 6*a^4*sin(d*x + c) + a^4)/sin(d*x + c)^ 3)/d
Time = 32.88 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.61 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {4\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {4\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {177\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+68\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+640\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+84\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {4549\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}+104\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+728\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+104\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {745\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+20\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {56\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {17\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d} \] Input:
int(cos(c + d*x)*cot(c + d*x)^4*(a + a*sin(c + d*x))^4,x)
Output:
(4*a^4*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (a^4*tan(c/2 + (d*x)/2)^3)/(24*d ) - (4*a^4*log(tan(c/2 + (d*x)/2)))/d - ((56*a^4*tan(c/2 + (d*x)/2)^2)/3 + 20*a^4*tan(c/2 + (d*x)/2)^3 + (745*a^4*tan(c/2 + (d*x)/2)^4)/3 + 104*a^4* tan(c/2 + (d*x)/2)^5 + 728*a^4*tan(c/2 + (d*x)/2)^6 + 104*a^4*tan(c/2 + (d *x)/2)^7 + (4549*a^4*tan(c/2 + (d*x)/2)^8)/5 + 84*a^4*tan(c/2 + (d*x)/2)^9 + 640*a^4*tan(c/2 + (d*x)/2)^10 + 68*a^4*tan(c/2 + (d*x)/2)^11 + 177*a^4* tan(c/2 + (d*x)/2)^12 + a^4/3 + 4*a^4*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3 + 40*tan(c/2 + (d*x)/2)^5 + 80*tan(c/2 + (d*x)/2)^7 + 80*tan(c/ 2 + (d*x)/2)^9 + 40*tan(c/2 + (d*x)/2)^11 + 8*tan(c/2 + (d*x)/2)^13)) - (1 7*a^4*tan(c/2 + (d*x)/2))/(8*d) - (a^4*tan(c/2 + (d*x)/2)^2)/(2*d)
Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \left (120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{3}-120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}+6 \sin \left (d x +c \right )^{8}+30 \sin \left (d x +c \right )^{7}+40 \sin \left (d x +c \right )^{6}-60 \sin \left (d x +c \right )^{5}-300 \sin \left (d x +c \right )^{4}+105 \sin \left (d x +c \right )^{3}-120 \sin \left (d x +c \right )^{2}-60 \sin \left (d x +c \right )-10\right )}{30 \sin \left (d x +c \right )^{3} d} \] Input:
int(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x)
Output:
(a**4*(120*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3 - 120*log(tan((c + d*x)/2))*sin(c + d*x)**3 + 6*sin(c + d*x)**8 + 30*sin(c + d*x)**7 + 40*si n(c + d*x)**6 - 60*sin(c + d*x)**5 - 300*sin(c + d*x)**4 + 105*sin(c + d*x )**3 - 120*sin(c + d*x)**2 - 60*sin(c + d*x) - 10))/(30*sin(c + d*x)**3*d)