\(\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx\) [287]

Optimal result

Integrand size = 25, antiderivative size = 99 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13 a^3 x}{8}-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{d}+\frac {13 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d} \] Output:

13/8*a^3*x-a^3*arctanh(cos(d*x+c))/d+a^3*cos(d*x+c)/d-a^3*cos(d*x+c)^3/d+1 
3/8*a^3*cos(d*x+c)*sin(d*x+c)/d-1/4*a^3*cos(d*x+c)^3*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 6.63 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (52 c+52 d x+8 \cos (c+d x)-8 \cos (3 (c+d x))-32 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+32 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \sin (2 (c+d x))-\sin (4 (c+d x))\right )}{32 d} \] Input:

Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]
 

Output:

(a^3*(52*c + 52*d*x + 8*Cos[c + d*x] - 8*Cos[3*(c + d*x)] - 32*Log[Cos[(c 
+ d*x)/2]] + 32*Log[Sin[(c + d*x)/2]] + 24*Sin[2*(c + d*x)] - Sin[4*(c + d 
*x)]))/(32*d)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 99, 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.

\(\displaystyle \int \cos (c+d x) \cot (c+d x) (a \sin (c+d x)+a)^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^2 (a \sin (c+d x)+a)^3}{\sin (c+d x)}dx\)

\(\Big \downarrow \) 3352

\(\displaystyle \int \left (3 a^3 \cos ^2(c+d x)+a^3 \sin ^2(c+d x) \cos ^2(c+d x)+3 a^3 \sin (c+d x) \cos ^2(c+d x)+a^3 \cos (c+d x) \cot (c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^3 \cos ^3(c+d x)}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {13 a^3 x}{8}\)

Input:

Int[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]
 

Output:

(13*a^3*x)/8 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d - (a^3 
*Cos[c + d*x]^3)/d + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^3*Cos[c 
 + d*x]^3*Sin[c + d*x])/(4*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3352

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]
 

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-a^{3} \cos \left (d x +c \right )^{3}+3 a^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(115\)
default	\(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-a^{3} \cos \left (d x +c \right )^{3}+3 a^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(115\)
risch	\(\frac {13 a^{3} x}{8}+\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {a^{3} \sin \left (4 d x +4 c \right )}{32 d}-\frac {a^{3} \cos \left (3 d x +3 c \right )}{4 d}+\frac {3 a^{3} \sin \left (2 d x +2 c \right )}{4 d}\)	\(132\)

Input:

int(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^3*(-1/4*sin(d*x+c)*cos(d*x+c)^3+1/8*sin(d*x+c)*cos(d*x+c)+1/8*d*x+1 
/8*c)-a^3*cos(d*x+c)^3+3*a^3*(1/2*sin(d*x+c)*cos(d*x+c)+1/2*d*x+1/2*c)+a^3 
*(cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.02 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {8 \, a^{3} \cos \left (d x + c\right )^{3} - 13 \, a^{3} d x - 8 \, a^{3} \cos \left (d x + c\right ) + 4 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 4 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (2 \, a^{3} \cos \left (d x + c\right )^{3} - 13 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:

integrate(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")
 

Output:

-1/8*(8*a^3*cos(d*x + c)^3 - 13*a^3*d*x - 8*a^3*cos(d*x + c) + 4*a^3*log(1 
/2*cos(d*x + c) + 1/2) - 4*a^3*log(-1/2*cos(d*x + c) + 1/2) + (2*a^3*cos(d 
*x + c)^3 - 13*a^3*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [F]

\[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx\right ) \] Input:

integrate(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))**3,x)
 

Output:

a**3*(Integral(cos(c + d*x)*cot(c + d*x), x) + Integral(3*sin(c + d*x)*cos 
(c + d*x)*cot(c + d*x), x) + Integral(3*sin(c + d*x)**2*cos(c + d*x)*cot(c 
 + d*x), x) + Integral(sin(c + d*x)**3*cos(c + d*x)*cot(c + d*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {32 \, a^{3} \cos \left (d x + c\right )^{3} - {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 16 \, a^{3} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{32 \, d} \] Input:

integrate(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima")
 

Output:

-1/32*(32*a^3*cos(d*x + c)^3 - (4*d*x + 4*c - sin(4*d*x + 4*c))*a^3 - 24*( 
2*d*x + 2*c + sin(2*d*x + 2*c))*a^3 - 16*a^3*(2*cos(d*x + c) - log(cos(d*x 
 + c) + 1) + log(cos(d*x + c) - 1)))/d
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.45 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13 \, {\left (d x + c\right )} a^{3} + 8 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 19 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 19 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \] Input:

integrate(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")
 

Output:

1/8*(13*(d*x + c)*a^3 + 8*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(11*a^3*t 
an(1/2*d*x + 1/2*c)^7 + 16*a^3*tan(1/2*d*x + 1/2*c)^6 + 19*a^3*tan(1/2*d*x 
 + 1/2*c)^5 - 19*a^3*tan(1/2*d*x + 1/2*c)^3 - 16*a^3*tan(1/2*d*x + 1/2*c)^ 
2 - 11*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
 

Mupad [B] (verification not implemented)

Time = 18.48 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.46 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {13\,a^3\,\mathrm {atan}\left (\frac {169\,a^6}{16\,\left (\frac {13\,a^6}{2}-\frac {169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {13\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {13\,a^6}{2}-\frac {169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}+\frac {-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}-4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {11\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:

int(cos(c + d*x)*cot(c + d*x)*(a + a*sin(c + d*x))^3,x)
 

Output:

(a^3*log(tan(c/2 + (d*x)/2)))/d + (13*a^3*atan((169*a^6)/(16*((13*a^6)/2 - 
 (169*a^6*tan(c/2 + (d*x)/2))/16)) + (13*a^6*tan(c/2 + (d*x)/2))/(2*((13*a 
^6)/2 - (169*a^6*tan(c/2 + (d*x)/2))/16))))/(4*d) + (4*a^3*tan(c/2 + (d*x) 
/2)^2 + (19*a^3*tan(c/2 + (d*x)/2)^3)/4 - (19*a^3*tan(c/2 + (d*x)/2)^5)/4 
- 4*a^3*tan(c/2 + (d*x)/2)^6 - (11*a^3*tan(c/2 + (d*x)/2)^7)/4 + (11*a^3*t 
an(c/2 + (d*x)/2))/4)/(d*(4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 
+ 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.75 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+11 \cos \left (d x +c \right ) \sin \left (d x +c \right )+8 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+13 c +13 d x \right )}{8 d} \] Input:

int(cos(d*x+c)*cot(d*x+c)*(a+a*sin(d*x+c))^3,x)
 

Output:

(a**3*(2*cos(c + d*x)*sin(c + d*x)**3 + 8*cos(c + d*x)*sin(c + d*x)**2 + 1 
1*cos(c + d*x)*sin(c + d*x) + 8*log(tan((c + d*x)/2)) + 13*c + 13*d*x))/(8 
*d)