Integrand size = 21, antiderivative size = 120 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {14 \text {arctanh}(\cos (c+d x))}{a^4 d}-\frac {9 \cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {2 \cot (c+d x) \csc (c+d x)}{a^4 d}+\frac {4 \cot (c+d x)}{3 a^4 d (1+\csc (c+d x))^2}-\frac {44 \cot (c+d x)}{3 a^4 d (1+\csc (c+d x))} \] Output:
14*arctanh(cos(d*x+c))/a^4/d-9*cot(d*x+c)/a^4/d-1/3*cot(d*x+c)^3/a^4/d+2*c ot(d*x+c)*csc(d*x+c)/a^4/d+4/3*cot(d*x+c)/a^4/d/(1+csc(d*x+c))^2-44/3*cot( d*x+c)/a^4/d/(1+csc(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(589\) vs. \(2(120)=240\).
Time = 6.85 (sec) , antiderivative size = 589, normalized size of antiderivative = 4.91 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {8 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}{3 d (a+a \sin (c+d x))^4}-\frac {4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}{3 d (a+a \sin (c+d x))^4}+\frac {80 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^7}{3 d (a+a \sin (c+d x))^4}-\frac {13 \cot \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{3 d (a+a \sin (c+d x))^4}+\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{2 d (a+a \sin (c+d x))^4}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{24 d (a+a \sin (c+d x))^4}+\frac {14 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{d (a+a \sin (c+d x))^4}-\frac {14 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{d (a+a \sin (c+d x))^4}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{2 d (a+a \sin (c+d x))^4}+\frac {13 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8 \tan \left (\frac {1}{2} (c+d x)\right )}{3 d (a+a \sin (c+d x))^4}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8 \tan \left (\frac {1}{2} (c+d x)\right )}{24 d (a+a \sin (c+d x))^4} \] Input:
Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^4,x]
Output:
(8*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)/(3*d*(a + a*S in[c + d*x])^4) - (4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(3*d*(a + a* Sin[c + d*x])^4) + (80*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/ 2])^7)/(3*d*(a + a*Sin[c + d*x])^4) - (13*Cot[(c + d*x)/2]*(Cos[(c + d*x)/ 2] + Sin[(c + d*x)/2])^8)/(3*d*(a + a*Sin[c + d*x])^4) + (Csc[(c + d*x)/2] ^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2*d*(a + a*Sin[c + d*x])^4) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2] )^8)/(24*d*(a + a*Sin[c + d*x])^4) + (14*Log[Cos[(c + d*x)/2]]*(Cos[(c + d *x)/2] + Sin[(c + d*x)/2])^8)/(d*(a + a*Sin[c + d*x])^4) - (14*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(d*(a + a*Sin[c + d*x] )^4) - (Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2*d*( a + a*Sin[c + d*x])^4) + (13*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[( c + d*x)/2])/(3*d*(a + a*Sin[c + d*x])^4) + (Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(24*d*(a + a*Sin[c + d*x]) ^4)
Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3188, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x)}{(a \sin (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^4 (a \sin (c+d x)+a)^4}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (\csc ^4(c+d x)-4 \csc ^3(c+d x)+8 \csc ^2(c+d x)-12 \csc (c+d x)-\frac {20}{\csc (c+d x)+1}+\frac {4}{(\csc (c+d x)+1)^2}+16\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {14 \text {arctanh}(\cos (c+d x))}{d}-\frac {\cot ^3(c+d x)}{3 d}-\frac {9 \cot (c+d x)}{d}+\frac {2 \cot (c+d x) \csc (c+d x)}{d}-\frac {44 \cot (c+d x)}{3 d (\csc (c+d x)+1)}+\frac {4 \cot (c+d x)}{3 d (\csc (c+d x)+1)^2}}{a^4}\) |
Input:
Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^4,x]
Output:
((14*ArcTanh[Cos[c + d*x]])/d - (9*Cot[c + d*x])/d - Cot[c + d*x]^3/(3*d) + (2*Cot[c + d*x]*Csc[c + d*x])/d + (4*Cot[c + d*x])/(3*d*(1 + Csc[c + d*x ])^2) - (44*Cot[c + d*x])/(3*d*(1 + Csc[c + d*x])))/a^4
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[a^p Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Time = 22.68 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {35}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-112 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {128}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {64}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {256}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(143\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {35}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-112 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {128}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {64}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {256}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(143\) |
| risch | \(-\frac {4 \left (-119 \,{\mathrm e}^{6 i \left (d x +c \right )}+63 i {\mathrm e}^{7 i \left (d x +c \right )}+204 \,{\mathrm e}^{4 i \left (d x +c \right )}-192 i {\mathrm e}^{5 i \left (d x +c \right )}+21 \,{\mathrm e}^{8 i \left (d x +c \right )}-135 \,{\mathrm e}^{2 i \left (d x +c \right )}+211 i {\mathrm e}^{3 i \left (d x +c \right )}+33-78 i {\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} a^{4} d}+\frac {14 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{4} d}-\frac {14 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{4} d}\) | \(171\) |
Input:
int(cot(d*x+c)^4/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/8/d/a^4*(1/3*tan(1/2*d*x+1/2*c)^3-4*tan(1/2*d*x+1/2*c)^2+35*tan(1/2*d*x+ 1/2*c)-1/3/tan(1/2*d*x+1/2*c)^3+4/tan(1/2*d*x+1/2*c)^2-35/tan(1/2*d*x+1/2* c)-112*ln(tan(1/2*d*x+1/2*c))-128/3/(tan(1/2*d*x+1/2*c)+1)^3+64/(tan(1/2*d *x+1/2*c)+1)^2-256/(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (114) = 228\).
Time = 0.12 (sec) , antiderivative size = 445, normalized size of antiderivative = 3.71 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {66 \, \cos \left (d x + c\right )^{5} - 24 \, \cos \left (d x + c\right )^{4} - 147 \, \cos \left (d x + c\right )^{3} + 29 \, \cos \left (d x + c\right )^{2} - 21 \, {\left (\cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 2\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 21 \, {\left (\cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 2\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (66 \, \cos \left (d x + c\right )^{4} + 90 \, \cos \left (d x + c\right )^{3} - 57 \, \cos \left (d x + c\right )^{2} - 86 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) + 82 \, \cos \left (d x + c\right ) - 4}{3 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 2 \, a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{3} - 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right ) + 2 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{4} - a^{4} d \cos \left (d x + c\right )^{3} - 3 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right ) + 2 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)^4/(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
-1/3*(66*cos(d*x + c)^5 - 24*cos(d*x + c)^4 - 147*cos(d*x + c)^3 + 29*cos( d*x + c)^2 - 21*(cos(d*x + c)^5 + 2*cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 4* cos(d*x + c)^2 + (cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 + cos (d*x + c) + 2)*sin(d*x + c) + cos(d*x + c) + 2)*log(1/2*cos(d*x + c) + 1/2 ) + 21*(cos(d*x + c)^5 + 2*cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 4*cos(d*x + c)^2 + (cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 + cos(d*x + c) + 2)*sin(d*x + c) + cos(d*x + c) + 2)*log(-1/2*cos(d*x + c) + 1/2) - (66* cos(d*x + c)^4 + 90*cos(d*x + c)^3 - 57*cos(d*x + c)^2 - 86*cos(d*x + c) - 4)*sin(d*x + c) + 82*cos(d*x + c) - 4)/(a^4*d*cos(d*x + c)^5 + 2*a^4*d*co s(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^3 - 4*a^4*d*cos(d*x + c)^2 + a^4*d*cos (d*x + c) + 2*a^4*d + (a^4*d*cos(d*x + c)^4 - a^4*d*cos(d*x + c)^3 - 3*a^4 *d*cos(d*x + c)^2 + a^4*d*cos(d*x + c) + 2*a^4*d)*sin(d*x + c))
\[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\int \frac {\cot ^{4}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:
integrate(cot(d*x+c)**4/(a+a*sin(d*x+c))**4,x)
Output:
Integral(cot(c + d*x)**4/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c + d*x) + 1), x)/a**4
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (114) = 228\).
Time = 0.04 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.38 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {72 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {984 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1647 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {873 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1}{\frac {a^{4} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{4}} - \frac {336 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^4/(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
1/24*((9*sin(d*x + c)/(cos(d*x + c) + 1) - 72*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 984*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1647*sin(d*x + c)^4/(c os(d*x + c) + 1)^4 - 873*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1)/(a^4*sin (d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*a^4*sin(d*x + c)^4/(cos(d*x + c) + 1) ^4 + 3*a^4*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + a^4*sin(d*x + c)^6/(cos(d *x + c) + 1)^6) + (105*sin(d*x + c)/(cos(d*x + c) + 1) - 12*sin(d*x + c)^2 /(cos(d*x + c) + 1)^2 + sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^4 - 336*log (sin(d*x + c)/(cos(d*x + c) + 1))/a^4)/d
Time = 0.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.49 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {\frac {336 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {308 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 51 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 723 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 676 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3} a^{4}} - \frac {a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^4/(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
-1/24*(336*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 - (308*tan(1/2*d*x + 1/2*c)^ 6 + 51*tan(1/2*d*x + 1/2*c)^5 - 723*tan(1/2*d*x + 1/2*c)^4 - 676*tan(1/2*d *x + 1/2*c)^3 - 72*tan(1/2*d*x + 1/2*c)^2 + 9*tan(1/2*d*x + 1/2*c) - 1)/(( tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c))^3*a^4) - (a^8*tan(1/2*d*x + 1/2*c)^3 - 12*a^8*tan(1/2*d*x + 1/2*c)^2 + 105*a^8*tan(1/2*d*x + 1/2*c))/ a^12)/d
Time = 18.47 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.42 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,a^4\,d}-\frac {14\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}+\frac {35\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^4\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {291\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}+\frac {549\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{8}+41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {1}{24}\right )}{a^4\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3} \] Input:
int(cot(c + d*x)^4/(a + a*sin(c + d*x))^4,x)
Output:
tan(c/2 + (d*x)/2)^3/(24*a^4*d) - tan(c/2 + (d*x)/2)^2/(2*a^4*d) - (14*log (tan(c/2 + (d*x)/2)))/(a^4*d) + (35*tan(c/2 + (d*x)/2))/(8*a^4*d) - (cot(c /2 + (d*x)/2)^3*(3*tan(c/2 + (d*x)/2)^2 - (3*tan(c/2 + (d*x)/2))/8 + 41*ta n(c/2 + (d*x)/2)^3 + (549*tan(c/2 + (d*x)/2)^4)/8 + (291*tan(c/2 + (d*x)/2 )^5)/8 + 1/24))/(a^4*d*(tan(c/2 + (d*x)/2) + 1)^3)
Time = 0.20 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.10 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {-336 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-1008 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-1008 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-336 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+470 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-972 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-794 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}{24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \] Input:
int(cot(d*x+c)^4/(a+a*sin(d*x+c))^4,x)
Output:
( - 336*log(tan((c + d*x)/2))*tan((c + d*x)/2)**6 - 1008*log(tan((c + d*x) /2))*tan((c + d*x)/2)**5 - 1008*log(tan((c + d*x)/2))*tan((c + d*x)/2)**4 - 336*log(tan((c + d*x)/2))*tan((c + d*x)/2)**3 + tan((c + d*x)/2)**9 - 9* tan((c + d*x)/2)**8 + 72*tan((c + d*x)/2)**7 + 470*tan((c + d*x)/2)**6 - 9 72*tan((c + d*x)/2)**4 - 794*tan((c + d*x)/2)**3 - 72*tan((c + d*x)/2)**2 + 9*tan((c + d*x)/2) - 1)/(24*tan((c + d*x)/2)**3*a**4*d*(tan((c + d*x)/2) **3 + 3*tan((c + d*x)/2)**2 + 3*tan((c + d*x)/2) + 1))