Integrand size = 29, antiderivative size = 97 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 x}{2 a}+\frac {3 \text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cos (c+d x)}{a d}+\frac {\cot (c+d x)}{a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d} \] Output:
3/2*x/a+3/2*arctanh(cos(d*x+c))/a/d-cos(d*x+c)/a/d+cot(d*x+c)/a/d-1/2*cot( d*x+c)*csc(d*x+c)/a/d+1/2*cos(d*x+c)*sin(d*x+c)/a/d
Time = 1.71 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-12 c-12 d x+12 \cos (c+d x)-4 \cos (3 (c+d x))-12 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+12 \cos (2 (c+d x)) \left (c+d x+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+12 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-10 \sin (2 (c+d x))+\sin (4 (c+d x))\right )}{64 a d (1+\sin (c+d x))} \] Input:
Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
-1/64*((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(-12*c - 12*d*x + 12*Cos[c + d*x] - 4*Cos[3*(c + d*x)] - 12*Log[Cos[(c + d*x)/2]] + 12*Cos[2*(c + d*x )]*(c + d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + 12*Log[Sin[ (c + d*x)/2]] - 10*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))
Time = 0.48 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {3042, 3318, 3042, 25, 3071, 252, 262, 216, 3072, 252, 262, 219}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^3 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos (c+d x) \cot ^3(c+d x)dx}{a}-\frac {\int \cos ^2(c+d x) \cot ^2(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\sin \left (c+d x+\frac {\pi }{2}\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^3dx}{a}-\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2dx}{a}-\frac {\int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx}{a}\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle \frac {\int \frac {\cot ^4(c+d x)}{\left (\cot ^2(c+d x)+1\right )^2}d\cot (c+d x)}{a d}-\frac {\int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx}{a}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {3}{2} \int \frac {\cot ^2(c+d x)}{\cot ^2(c+d x)+1}d\cot (c+d x)-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}}{a d}-\frac {\int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx}{a}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {3}{2} \left (\cot (c+d x)-\int \frac {1}{\cot ^2(c+d x)+1}d\cot (c+d x)\right )-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}}{a d}-\frac {\int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx}{a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}}{a d}-\frac {\int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx}{a}\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle \frac {\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}}{a d}-\frac {\int \frac {\cos ^4(c+d x)}{\left (1-\cos ^2(c+d x)\right )^2}d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}}{a d}-\frac {\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} \int \frac {\cos ^2(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}}{a d}-\frac {\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} \left (\int \frac {1}{1-\cos ^2(c+d x)}d\cos (c+d x)-\cos (c+d x)\right )}{a d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}}{a d}-\frac {\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))}{a d}\) |
Input:
Int[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
-(((-3*(ArcTanh[Cos[c + d*x]] - Cos[c + d*x]))/2 + Cos[c + d*x]^3/(2*(1 - Cos[c + d*x]^2)))/(a*d)) + ((3*(-ArcTan[Cot[c + d*x]] + Cot[c + d*x]))/2 - Cot[c + d*x]^3/(2*(1 + Cot[c + d*x]^2)))/(a*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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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[((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[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Time = 1.24 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.44
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+12 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}\) | \(140\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+12 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}\) | \(140\) |
| risch | \(\frac {3 x}{2 a}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d a}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d a}+\frac {{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+2 i {\mathrm e}^{2 i \left (d x +c \right )}-2 i}{a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}\) | \(171\) |
Input:
int(cos(d*x+c)^3*cot(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/4/d/a*(1/2*tan(1/2*d*x+1/2*c)^2-2*tan(1/2*d*x+1/2*c)+16*(-1/4*tan(1/2*d* x+1/2*c)^3-1/2*tan(1/2*d*x+1/2*c)^2+1/4*tan(1/2*d*x+1/2*c)-1/2)/(1+tan(1/2 *d*x+1/2*c)^2)^2+12*arctan(tan(1/2*d*x+1/2*c))-1/2/tan(1/2*d*x+1/2*c)^2+2/ tan(1/2*d*x+1/2*c)-6*ln(tan(1/2*d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {6 \, d x \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right )^{3} - 6 \, d x + 3 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right )}{4 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/4*(6*d*x*cos(d*x + c)^2 - 4*cos(d*x + c)^3 - 6*d*x + 3*(cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) - 3*(cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2) + 2*(cos(d*x + c)^3 - 3*cos(d*x + c))*sin(d*x + c) + 6*cos(d*x + c))/(a*d*cos(d*x + c)^2 - a*d)
Timed out. \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*cot(d*x+c)**3/(a+a*sin(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (89) = 178\).
Time = 0.13 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.69 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a} - \frac {\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {18 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {17 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1}{\frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{8 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/8*((4*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/a - (4*sin(d*x + c)/(cos(d*x + c) + 1) - 18*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 16*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 17*sin(d*x + c)^4/( cos(d*x + c) + 1)^4 - 4*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1)/(a*sin(d* x + c)^2/(cos(d*x + c) + 1)^2 + 2*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) - 24*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 12*log(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.72 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {12 \, {\left (d x + c\right )}}{a} - \frac {12 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} + \frac {6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2} a}}{8 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/8*(12*(d*x + c)/a - 12*log(abs(tan(1/2*d*x + 1/2*c)))/a + (a*tan(1/2*d*x + 1/2*c)^2 - 4*a*tan(1/2*d*x + 1/2*c))/a^2 + (6*tan(1/2*d*x + 1/2*c)^6 - 4*tan(1/2*d*x + 1/2*c)^5 - 5*tan(1/2*d*x + 1/2*c)^4 + 16*tan(1/2*d*x + 1/2 *c)^3 - 12*tan(1/2*d*x + 1/2*c)^2 + 4*tan(1/2*d*x + 1/2*c) - 1)/((tan(1/2* d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))^2*a))/d
Time = 33.54 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.30 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {3\,\mathrm {atan}\left (\frac {9}{9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+9}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+9}\right )}{a\,d}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{2}}{d\,\left (4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d} \] Input:
int((cos(c + d*x)^3*cot(c + d*x)^3)/(a + a*sin(c + d*x)),x)
Output:
tan(c/2 + (d*x)/2)^2/(8*a*d) - (3*atan(9/(9*tan(c/2 + (d*x)/2) + 9) - (9*t an(c/2 + (d*x)/2))/(9*tan(c/2 + (d*x)/2) + 9)))/(a*d) - (3*log(tan(c/2 + ( d*x)/2)))/(2*a*d) - (9*tan(c/2 + (d*x)/2)^2 - 2*tan(c/2 + (d*x)/2) - 8*tan (c/2 + (d*x)/2)^3 + (17*tan(c/2 + (d*x)/2)^4)/2 + 2*tan(c/2 + (d*x)/2)^5 + 1/2)/(d*(4*a*tan(c/2 + (d*x)/2)^2 + 8*a*tan(c/2 + (d*x)/2)^4 + 4*a*tan(c/ 2 + (d*x)/2)^6)) - tan(c/2 + (d*x)/2)/(2*a*d)
Time = 116.88 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-4 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+4 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2 \cos \left (d x +c \right )-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+6 \sin \left (d x +c \right )^{2} d x +5 \sin \left (d x +c \right )^{2}}{4 \sin \left (d x +c \right )^{2} a d} \] Input:
int(cos(d*x+c)^3*cot(d*x+c)^3/(a+a*sin(d*x+c)),x)
Output:
(2*cos(c + d*x)*sin(c + d*x)**3 - 4*cos(c + d*x)*sin(c + d*x)**2 + 4*cos(c + d*x)*sin(c + d*x) - 2*cos(c + d*x) - 6*log(tan((c + d*x)/2))*sin(c + d* x)**2 + 6*sin(c + d*x)**2*d*x + 5*sin(c + d*x)**2)/(4*sin(c + d*x)**2*a*d)