Integrand size = 27, antiderivative size = 98 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-3 a^2 x+\frac {a^2 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^2 \cos ^3(c+d x)}{3 d}-\frac {2 a^2 \cot (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^2 \cos (c+d x) \sin (c+d x)}{d} \] Output:
-3*a^2*x+1/2*a^2*arctanh(cos(d*x+c))/d+1/3*a^2*cos(d*x+c)^3/d-2*a^2*cot(d* x+c)/d-1/2*a^2*cot(d*x+c)*csc(d*x+c)/d-a^2*cos(d*x+c)*sin(d*x+c)/d
Time = 6.91 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.61 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (1+\sin (c+d x))^2 \left (6 \cos (c+d x)+2 \cos (3 (c+d x))+3 \left (-24 c-24 d x-8 \cot \left (\frac {1}{2} (c+d x)\right )-\csc ^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{2} (c+d x)\right )-4 \sin (2 (c+d x))+8 \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}{24 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \] Input:
Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]
Output:
(a^2*(1 + Sin[c + d*x])^2*(6*Cos[c + d*x] + 2*Cos[3*(c + d*x)] + 3*(-24*c - 24*d*x - 8*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 + 4*Log[Cos[(c + d*x)/2 ]] - 4*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 - 4*Sin[2*(c + d*x)] + 8 *Tan[(c + d*x)/2])))/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)
Time = 0.39 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3351, 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.
| \(\displaystyle \int \cos (c+d x) \cot ^3(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^2}{\sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\csc ^3(c+d x) a^6+\sin ^3(c+d x) a^6+2 \csc ^2(c+d x) a^6+2 \sin ^2(c+d x) a^6-\csc (c+d x) a^6-\sin (c+d x) a^6-4 a^6\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^6 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^6 \cos ^3(c+d x)}{3 d}-\frac {2 a^6 \cot (c+d x)}{d}-\frac {a^6 \sin (c+d x) \cos (c+d x)}{d}-\frac {a^6 \cot (c+d x) \csc (c+d x)}{2 d}-3 a^6 x}{a^4}\) |
Input:
Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]
Output:
(-3*a^6*x + (a^6*ArcTanh[Cos[c + d*x]])/(2*d) + (a^6*Cos[c + d*x]^3)/(3*d) - (2*a^6*Cot[c + d*x])/d - (a^6*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (a^6*C os[c + d*x]*Sin[c + d*x])/d)/a^4
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Time = 1.76 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.61
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\cos \left (d x +c \right )^{3}}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{3}}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(158\) |
| default | \(\frac {a^{2} \left (\frac {\cos \left (d x +c \right )^{3}}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{3}}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(158\) |
| risch | \(-3 a^{2} x +\frac {a^{2} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}+\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}+\frac {a^{2} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {a^{2} \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-4 i {\mathrm e}^{2 i \left (d x +c \right )}+4 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(205\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^3*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(1/3*cos(d*x+c)^3+cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c)))+2*a^2*(-1 /sin(d*x+c)*cos(d*x+c)^5-(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)-3/2*d*x- 3/2*c)+a^2*(-1/2/sin(d*x+c)^2*cos(d*x+c)^5-1/2*cos(d*x+c)^3-3/2*cos(d*x+c) -3/2*ln(csc(d*x+c)-cot(d*x+c))))
Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.76 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {4 \, a^{2} \cos \left (d x + c\right )^{5} - 36 \, a^{2} d x \cos \left (d x + c\right )^{2} - 4 \, a^{2} \cos \left (d x + c\right )^{3} + 36 \, a^{2} d x + 6 \, a^{2} \cos \left (d x + c\right ) + 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, {\left (a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
1/12*(4*a^2*cos(d*x + c)^5 - 36*a^2*d*x*cos(d*x + c)^2 - 4*a^2*cos(d*x + c )^3 + 36*a^2*d*x + 6*a^2*cos(d*x + c) + 3*(a^2*cos(d*x + c)^2 - a^2)*log(1 /2*cos(d*x + c) + 1/2) - 3*(a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + c ) + 1/2) - 12*(a^2*cos(d*x + c)^3 - 3*a^2*cos(d*x + c))*sin(d*x + c))/(d*c os(d*x + c)^2 - d)
\[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int \cos {\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int 2 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**3*(a+a*sin(d*x+c))**2,x)
Output:
a**2*(Integral(cos(c + d*x)*cot(c + d*x)**3, x) + Integral(2*sin(c + d*x)* cos(c + d*x)*cot(c + d*x)**3, x) + Integral(sin(c + d*x)**2*cos(c + d*x)*c ot(c + d*x)**3, x))
Time = 0.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.54 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {2 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} - 12 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{2} + 3 \, a^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
1/12*(2*(2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3*l og(cos(d*x + c) - 1))*a^2 - 12*(3*d*x + 3*c + (3*tan(d*x + c)^2 + 2)/(tan( d*x + c)^3 + tan(d*x + c)))*a^2 + 3*a^2*(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)))/ d
Time = 0.20 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.82 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 72 \, {\left (d x + c\right )} a^{2} - 12 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3 \, {\left (6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {16 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{24 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/24*(3*a^2*tan(1/2*d*x + 1/2*c)^2 - 72*(d*x + c)*a^2 - 12*a^2*log(abs(tan (1/2*d*x + 1/2*c))) + 24*a^2*tan(1/2*d*x + 1/2*c) + 3*(6*a^2*tan(1/2*d*x + 1/2*c)^2 - 8*a^2*tan(1/2*d*x + 1/2*c) - a^2)/tan(1/2*d*x + 1/2*c)^2 + 16* (3*a^2*tan(1/2*d*x + 1/2*c)^5 + 3*a^2*tan(1/2*d*x + 1/2*c)^4 - 3*a^2*tan(1 /2*d*x + 1/2*c) + a^2)/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 17.68 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.09 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {6\,a^2\,\mathrm {atan}\left (\frac {36\,a^4}{6\,a^4-36\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {6\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6\,a^4-36\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {-4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+12\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+20\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}+4\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^2}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \] Input:
int(cos(c + d*x)*cot(c + d*x)^3*(a + a*sin(c + d*x))^2,x)
Output:
(a^2*tan(c/2 + (d*x)/2)^2)/(8*d) - (a^2*log(tan(c/2 + (d*x)/2)))/(2*d) - ( 6*a^2*atan((36*a^4)/(6*a^4 - 36*a^4*tan(c/2 + (d*x)/2)) + (6*a^4*tan(c/2 + (d*x)/2))/(6*a^4 - 36*a^4*tan(c/2 + (d*x)/2))))/d - (20*a^2*tan(c/2 + (d* x)/2)^3 - (7*a^2*tan(c/2 + (d*x)/2)^2)/6 + (3*a^2*tan(c/2 + (d*x)/2)^4)/2 + 12*a^2*tan(c/2 + (d*x)/2)^5 - (15*a^2*tan(c/2 + (d*x)/2)^6)/2 - 4*a^2*ta n(c/2 + (d*x)/2)^7 + a^2/2 + 4*a^2*tan(c/2 + (d*x)/2))/(d*(4*tan(c/2 + (d* x)/2)^2 + 12*tan(c/2 + (d*x)/2)^4 + 12*tan(c/2 + (d*x)/2)^6 + 4*tan(c/2 + (d*x)/2)^8)) + (a^2*tan(c/2 + (d*x)/2))/d
Time = 0.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.30 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-48 \cos \left (d x +c \right ) \sin \left (d x +c \right )-12 \cos \left (d x +c \right )-12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}-72 \sin \left (d x +c \right )^{2} d x +\sin \left (d x +c \right )^{2}\right )}{24 \sin \left (d x +c \right )^{2} d} \] Input:
int(cos(d*x+c)*cot(d*x+c)^3*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*( - 8*cos(c + d*x)*sin(c + d*x)**4 - 24*cos(c + d*x)*sin(c + d*x)**3 + 8*cos(c + d*x)*sin(c + d*x)**2 - 48*cos(c + d*x)*sin(c + d*x) - 12*cos( c + d*x) - 12*log(tan((c + d*x)/2))*sin(c + d*x)**2 - 72*sin(c + d*x)**2*d *x + sin(c + d*x)**2))/(24*sin(c + d*x)**2*d)