Integrand size = 25, antiderivative size = 117 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {5 a^4 x}{2}-\frac {a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^4 \cos (c+d x)}{d}-\frac {7 a^4 \cos ^3(c+d x)}{3 d}+\frac {a^4 \cos ^5(c+d x)}{5 d}+\frac {5 a^4 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{d} \] Output:
5/2*a^4*x-a^4*arctanh(cos(d*x+c))/d+a^4*cos(d*x+c)/d-7/3*a^4*cos(d*x+c)^3/ d+1/5*a^4*cos(d*x+c)^5/d+5/2*a^4*cos(d*x+c)*sin(d*x+c)/d-a^4*cos(d*x+c)^3* sin(d*x+c)/d
Time = 6.51 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.81 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \left (-150 \cos (c+d x)-125 \cos (3 (c+d x))+3 \cos (5 (c+d x))+30 \left (20 c+20 d x-8 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 \sin (2 (c+d x))-\sin (4 (c+d x))\right )\right )}{240 d} \] Input:
Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^4,x]
Output:
(a^4*(-150*Cos[c + d*x] - 125*Cos[3*(c + d*x)] + 3*Cos[5*(c + d*x)] + 30*( 20*c + 20*d*x - 8*Log[Cos[(c + d*x)/2]] + 8*Log[Sin[(c + d*x)/2]] + 8*Sin[ 2*(c + d*x)] - Sin[4*(c + d*x)])))/(240*d)
Time = 0.44 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3042, 3352, 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.
| \(\displaystyle \int \cos (c+d x) \cot (c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2 (a \sin (c+d x)+a)^4}{\sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (4 a^4 \cos ^2(c+d x)+a^4 \sin ^3(c+d x) \cos ^2(c+d x)+4 a^4 \sin ^2(c+d x) \cos ^2(c+d x)+6 a^4 \sin (c+d x) \cos ^2(c+d x)+a^4 \cos (c+d x) \cot (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^4 \cos ^5(c+d x)}{5 d}-\frac {7 a^4 \cos ^3(c+d x)}{3 d}+\frac {a^4 \cos (c+d x)}{d}-\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac {5 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {5 a^4 x}{2}\) |
Input:
Int[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^4,x]
Output:
(5*a^4*x)/2 - (a^4*ArcTanh[Cos[c + d*x]])/d + (a^4*Cos[c + d*x])/d - (7*a^ 4*Cos[c + d*x]^3)/(3*d) + (a^4*Cos[c + d*x]^5)/(5*d) + (5*a^4*Cos[c + d*x] *Sin[c + d*x])/(2*d) - (a^4*Cos[c + d*x]^3*Sin[c + d*x])/d
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Result contains complex when optimal does not.
Time = 3.93 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(\frac {5 a^{4} x}{2}-\frac {5 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {5 a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{4} \cos \left (5 d x +5 c \right )}{80 d}-\frac {a^{4} \sin \left (4 d x +4 c \right )}{8 d}-\frac {25 a^{4} \cos \left (3 d x +3 c \right )}{48 d}+\frac {a^{4} \sin \left (2 d x +2 c \right )}{d}\) | \(148\) |
| derivativedivides | \(\frac {a^{4} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+4 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-2 a^{4} \cos \left (d x +c \right )^{3}+4 a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(149\) |
| default | \(\frac {a^{4} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+4 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-2 a^{4} \cos \left (d x +c \right )^{3}+4 a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(149\) |
Input:
int(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
5/2*a^4*x-5/16*a^4/d*exp(I*(d*x+c))-5/16*a^4/d*exp(-I*(d*x+c))+a^4/d*ln(ex p(I*(d*x+c))-1)-a^4/d*ln(exp(I*(d*x+c))+1)+1/80*a^4/d*cos(5*d*x+5*c)-1/8*a ^4/d*sin(4*d*x+4*c)-25/48*a^4/d*cos(3*d*x+3*c)+a^4/d*sin(2*d*x+2*c)
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.98 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {6 \, a^{4} \cos \left (d x + c\right )^{5} - 70 \, a^{4} \cos \left (d x + c\right )^{3} + 75 \, a^{4} d x + 30 \, a^{4} \cos \left (d x + c\right ) - 15 \, a^{4} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, a^{4} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (2 \, a^{4} \cos \left (d x + c\right )^{3} - 5 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
1/30*(6*a^4*cos(d*x + c)^5 - 70*a^4*cos(d*x + c)^3 + 75*a^4*d*x + 30*a^4*c os(d*x + c) - 15*a^4*log(1/2*cos(d*x + c) + 1/2) + 15*a^4*log(-1/2*cos(d*x + c) + 1/2) - 15*(2*a^4*cos(d*x + c)^3 - 5*a^4*cos(d*x + c))*sin(d*x + c) )/d
\[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=a^{4} \left (\int \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 4 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))**4,x)
Output:
a**4*(Integral(cos(c + d*x)*cot(c + d*x), x) + Integral(4*sin(c + d*x)*cos (c + d*x)*cot(c + d*x), x) + Integral(6*sin(c + d*x)**2*cos(c + d*x)*cot(c + d*x), x) + Integral(4*sin(c + d*x)**3*cos(c + d*x)*cot(c + d*x), x) + I ntegral(sin(c + d*x)**4*cos(c + d*x)*cot(c + d*x), x))
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {240 \, a^{4} \cos \left (d x + c\right )^{3} - 8 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{4} - 15 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 60 \, a^{4} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
-1/120*(240*a^4*cos(d*x + c)^3 - 8*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a ^4 - 15*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^4 - 120*(2*d*x + 2*c + sin(2*d* x + 2*c))*a^4 - 60*a^4*(2*cos(d*x + c) - log(cos(d*x + c) + 1) + log(cos(d *x + c) - 1)))/d
Time = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.55 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {75 \, {\left (d x + c\right )} a^{4} + 30 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 150 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 210 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 300 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 40 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 210 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 34 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{30 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
1/30*(75*(d*x + c)*a^4 + 30*a^4*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(45*a^4 *tan(1/2*d*x + 1/2*c)^9 + 150*a^4*tan(1/2*d*x + 1/2*c)^8 + 210*a^4*tan(1/2 *d*x + 1/2*c)^7 + 300*a^4*tan(1/2*d*x + 1/2*c)^6 + 40*a^4*tan(1/2*d*x + 1/ 2*c)^4 - 210*a^4*tan(1/2*d*x + 1/2*c)^3 + 20*a^4*tan(1/2*d*x + 1/2*c)^2 - 45*a^4*tan(1/2*d*x + 1/2*c) + 34*a^4)/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d
Time = 18.92 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.52 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {5\,a^4\,\mathrm {atan}\left (\frac {25\,a^8}{10\,a^8-25\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {10\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{10\,a^8-25\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+10\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+14\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+20\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-14\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-3\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {34\,a^4}{15}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:
int(cos(c + d*x)*cot(c + d*x)*(a + a*sin(c + d*x))^4,x)
Output:
(a^4*log(tan(c/2 + (d*x)/2)))/d + (5*a^4*atan((25*a^8)/(10*a^8 - 25*a^8*ta n(c/2 + (d*x)/2)) + (10*a^8*tan(c/2 + (d*x)/2))/(10*a^8 - 25*a^8*tan(c/2 + (d*x)/2))))/d - ((4*a^4*tan(c/2 + (d*x)/2)^2)/3 - 14*a^4*tan(c/2 + (d*x)/ 2)^3 + (8*a^4*tan(c/2 + (d*x)/2)^4)/3 + 20*a^4*tan(c/2 + (d*x)/2)^6 + 14*a ^4*tan(c/2 + (d*x)/2)^7 + 10*a^4*tan(c/2 + (d*x)/2)^8 + 3*a^4*tan(c/2 + (d *x)/2)^9 + (34*a^4)/15 - 3*a^4*tan(c/2 + (d*x)/2))/(d*(5*tan(c/2 + (d*x)/2 )^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x )/2)^8 + tan(c/2 + (d*x)/2)^10 + 1))
Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \left (6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+58 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+45 \cos \left (d x +c \right ) \sin \left (d x +c \right )-34 \cos \left (d x +c \right )+30 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 c +75 d x +34\right )}{30 d} \] Input:
int(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^4,x)
Output:
(a**4*(6*cos(c + d*x)*sin(c + d*x)**4 + 30*cos(c + d*x)*sin(c + d*x)**3 + 58*cos(c + d*x)*sin(c + d*x)**2 + 45*cos(c + d*x)*sin(c + d*x) - 34*cos(c + d*x) + 30*log(tan((c + d*x)/2)) + 75*c + 75*d*x + 34))/(30*d)