Integrand size = 27, antiderivative size = 124 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {15 a x}{8}+\frac {5 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a \cos (c+d x)}{d}-\frac {a \cos ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}-\frac {9 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \cos (c+d x) \sin ^3(c+d x)}{4 d} \] Output:
-15/8*a*x+5/2*a*arctanh(cos(d*x+c))/d-2*a*cos(d*x+c)/d-1/3*a*cos(d*x+c)^3/ d-a*cot(d*x+c)/d-1/2*a*cot(d*x+c)*csc(d*x+c)/d-9/8*a*cos(d*x+c)*sin(d*x+c) /d+1/4*a*cos(d*x+c)*sin(d*x+c)^3/d
Time = 3.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \left (216 \cos (c+d x)+8 \cos (3 (c+d x))+3 \left (60 c+60 d x+32 \cot (c+d x)+4 \csc ^2\left (\frac {1}{2} (c+d x)\right )-80 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+80 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 \sec ^2\left (\frac {1}{2} (c+d x)\right )+16 \sin (2 (c+d x))+\sin (4 (c+d x))\right )\right )}{96 d} \] Input:
Integrate[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
-1/96*(a*(216*Cos[c + d*x] + 8*Cos[3*(c + d*x)] + 3*(60*c + 60*d*x + 32*Co t[c + d*x] + 4*Csc[(c + d*x)/2]^2 - 80*Log[Cos[(c + d*x)/2]] + 80*Log[Sin[ (c + d*x)/2]] - 4*Sec[(c + d*x)/2]^2 + 16*Sin[2*(c + d*x)] + Sin[4*(c + d* x)])))/d
Time = 0.50 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.18, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3317, 3042, 25, 3071, 252, 252, 262, 216, 3072, 252, 254, 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.
| \(\displaystyle \int \cos ^3(c+d x) \cot ^3(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)}{\sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cos ^4(c+d x) \cot ^2(c+d x)dx+a \int \cos ^3(c+d x) \cot ^3(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4 \tan \left (c+d x+\frac {\pi }{2}\right )^2dx+a \int -\sin \left (c+d x+\frac {\pi }{2}\right )^3 \tan \left (c+d x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4 \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle -\frac {a \int \frac {\cot ^6(c+d x)}{\left (\cot ^2(c+d x)+1\right )^3}d\cot (c+d x)}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -\frac {a \left (\frac {5}{4} \int \frac {\cot ^4(c+d x)}{\left (\cot ^2(c+d x)+1\right )^2}d\cot (c+d x)-\frac {\cot ^5(c+d x)}{4 \left (\cot ^2(c+d x)+1\right )^2}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -\frac {a \left (\frac {5}{4} \left (\frac {3}{2} \int \frac {\cot ^2(c+d x)}{\cot ^2(c+d x)+1}d\cot (c+d x)-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )-\frac {\cot ^5(c+d x)}{4 \left (\cot ^2(c+d x)+1\right )^2}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {a \left (\frac {5}{4} \left (\frac {3}{2} \left (\cot (c+d x)-\int \frac {1}{\cot ^2(c+d x)+1}d\cot (c+d x)\right )-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )-\frac {\cot ^5(c+d x)}{4 \left (\cot ^2(c+d x)+1\right )^2}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx-\frac {a \left (\frac {5}{4} \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )-\frac {\cot ^5(c+d x)}{4 \left (\cot ^2(c+d x)+1\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle -\frac {a \int \frac {\cos ^6(c+d x)}{\left (1-\cos ^2(c+d x)\right )^2}d\cos (c+d x)}{d}-\frac {a \left (\frac {5}{4} \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )-\frac {\cot ^5(c+d x)}{4 \left (\cot ^2(c+d x)+1\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -\frac {a \left (\frac {\cos ^5(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {5}{2} \int \frac {\cos ^4(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)\right )}{d}-\frac {a \left (\frac {5}{4} \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )-\frac {\cot ^5(c+d x)}{4 \left (\cot ^2(c+d x)+1\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle -\frac {a \left (\frac {\cos ^5(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {5}{2} \int \left (-\cos ^2(c+d x)+\frac {1}{1-\cos ^2(c+d x)}-1\right )d\cos (c+d x)\right )}{d}-\frac {a \left (\frac {5}{4} \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )-\frac {\cot ^5(c+d x)}{4 \left (\cot ^2(c+d x)+1\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (\frac {5}{4} \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )-\frac {\cot ^5(c+d x)}{4 \left (\cot ^2(c+d x)+1\right )^2}\right )}{d}-\frac {a \left (\frac {\cos ^5(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {5}{2} \left (\text {arctanh}(\cos (c+d x))-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)\right )\right )}{d}\) |
Input:
Int[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
-((a*(Cos[c + d*x]^5/(2*(1 - Cos[c + d*x]^2)) - (5*(ArcTanh[Cos[c + d*x]] - Cos[c + d*x] - Cos[c + d*x]^3/3))/2))/d) - (a*(-1/4*Cot[c + d*x]^5/(1 + Cot[c + d*x]^2)^2 + (5*((3*(-ArcTan[Cot[c + d*x]] + Cot[c + d*x]))/2 - Cot [c + d*x]^3/(2*(1 + Cot[c + d*x]^2))))/4))/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[I nt[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 3.60 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{7}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+a \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(136\) |
| default | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{7}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+a \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(136\) |
| risch | \(-\frac {15 a x}{8}-\frac {a \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 d}-\frac {9 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {9 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}-\frac {a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {a \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-2 i {\mathrm e}^{2 i \left (d x +c \right )}+2 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {a \sin \left (4 d x +4 c \right )}{32 d}\) | \(200\) |
Input:
int(cos(d*x+c)^3*cot(d*x+c)^3*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/sin(d*x+c)*cos(d*x+c)^7-(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos (d*x+c))*sin(d*x+c)-15/8*d*x-15/8*c)+a*(-1/2/sin(d*x+c)^2*cos(d*x+c)^7-1/2 *cos(d*x+c)^5-5/6*cos(d*x+c)^3-5/2*cos(d*x+c)-5/2*ln(csc(d*x+c)-cot(d*x+c) )))
Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.31 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {8 \, a \cos \left (d x + c\right )^{5} + 45 \, a d x \cos \left (d x + c\right )^{2} + 40 \, a \cos \left (d x + c\right )^{3} - 45 \, a d x - 60 \, a \cos \left (d x + c\right ) - 30 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 30 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (2 \, a \cos \left (d x + c\right )^{5} + 5 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/24*(8*a*cos(d*x + c)^5 + 45*a*d*x*cos(d*x + c)^2 + 40*a*cos(d*x + c)^3 - 45*a*d*x - 60*a*cos(d*x + c) - 30*(a*cos(d*x + c)^2 - a)*log(1/2*cos(d*x + c) + 1/2) + 30*(a*cos(d*x + c)^2 - a)*log(-1/2*cos(d*x + c) + 1/2) + 3* (2*a*cos(d*x + c)^5 + 5*a*cos(d*x + c)^3 - 15*a*cos(d*x + c))*sin(d*x + c) )/(d*cos(d*x + c)^2 - d)
\[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \cos ^{3}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)**3*cot(d*x+c)**3*(a+a*sin(d*x+c)),x)
Output:
a*(Integral(cos(c + d*x)**3*cot(c + d*x)**3, x) + Integral(sin(c + d*x)*co s(c + d*x)**3*cot(c + d*x)**3, x))
Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.06 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {2 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a + 3 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a}{24 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/24*(2*(4*cos(d*x + c)^3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos( d*x + c) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a + 3*(15* d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)^5 + 2*tan(d*x + c)^3 + tan(d*x + c)))*a)/d
Time = 0.15 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.73 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, {\left (d x + c\right )} a - 60 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3 \, {\left (30 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, {\left (27 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 152 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 56 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/24*(3*a*tan(1/2*d*x + 1/2*c)^2 - 45*(d*x + c)*a - 60*a*log(abs(tan(1/2*d *x + 1/2*c))) + 12*a*tan(1/2*d*x + 1/2*c) + 3*(30*a*tan(1/2*d*x + 1/2*c)^2 - 4*a*tan(1/2*d*x + 1/2*c) - a)/tan(1/2*d*x + 1/2*c)^2 + 2*(27*a*tan(1/2* d*x + 1/2*c)^7 - 72*a*tan(1/2*d*x + 1/2*c)^6 + 3*a*tan(1/2*d*x + 1/2*c)^5 - 168*a*tan(1/2*d*x + 1/2*c)^4 - 3*a*tan(1/2*d*x + 1/2*c)^3 - 152*a*tan(1/ 2*d*x + 1/2*c)^2 - 27*a*tan(1/2*d*x + 1/2*c) - 56*a)/(tan(1/2*d*x + 1/2*c) ^2 + 1)^4)/d
Time = 33.60 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.59 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {49\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+58\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {161\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {62\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {15\,a\,\mathrm {atan}\left (\frac {225\,a^2}{16\,\left (\frac {75\,a^2}{4}-\frac {225\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {75\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {75\,a^2}{4}-\frac {225\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d} \] Input:
int(cos(c + d*x)^3*cot(c + d*x)^3*(a + a*sin(c + d*x)),x)
Output:
(a*tan(c/2 + (d*x)/2))/(2*d) + (a*tan(c/2 + (d*x)/2)^2)/(8*d) - (5*a*log(t an(c/2 + (d*x)/2)))/(2*d) - (a/2 + 2*a*tan(c/2 + (d*x)/2) + (62*a*tan(c/2 + (d*x)/2)^2)/3 + 17*a*tan(c/2 + (d*x)/2)^3 + (161*a*tan(c/2 + (d*x)/2)^4) /3 + 13*a*tan(c/2 + (d*x)/2)^5 + 58*a*tan(c/2 + (d*x)/2)^6 + 7*a*tan(c/2 + (d*x)/2)^7 + (49*a*tan(c/2 + (d*x)/2)^8)/2 - 7*a*tan(c/2 + (d*x)/2)^9)/(d *(4*tan(c/2 + (d*x)/2)^2 + 16*tan(c/2 + (d*x)/2)^4 + 24*tan(c/2 + (d*x)/2) ^6 + 16*tan(c/2 + (d*x)/2)^8 + 4*tan(c/2 + (d*x)/2)^10)) - (15*a*atan((225 *a^2)/(16*((75*a^2)/4 - (225*a^2*tan(c/2 + (d*x)/2))/16)) + (75*a^2*tan(c/ 2 + (d*x)/2))/(4*((75*a^2)/4 - (225*a^2*tan(c/2 + (d*x)/2))/16))))/(4*d)
Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.15 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-27 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-56 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-24 \cos \left (d x +c \right ) \sin \left (d x +c \right )-12 \cos \left (d x +c \right )-60 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}-45 \sin \left (d x +c \right )^{2} d x +68 \sin \left (d x +c \right )^{2}\right )}{24 \sin \left (d x +c \right )^{2} d} \] Input:
int(cos(d*x+c)^3*cot(d*x+c)^3*(a+a*sin(d*x+c)),x)
Output:
(a*(6*cos(c + d*x)*sin(c + d*x)**5 + 8*cos(c + d*x)*sin(c + d*x)**4 - 27*c os(c + d*x)*sin(c + d*x)**3 - 56*cos(c + d*x)*sin(c + d*x)**2 - 24*cos(c + d*x)*sin(c + d*x) - 12*cos(c + d*x) - 60*log(tan((c + d*x)/2))*sin(c + d* x)**2 - 45*sin(c + d*x)**2*d*x + 68*sin(c + d*x)**2))/(24*sin(c + d*x)**2* d)