\(\int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\) [529]

Optimal result

Integrand size = 29, antiderivative size = 133 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc (c+d x)}{d}+\frac {5 a^3 \csc ^2(c+d x)}{2 d}+\frac {5 a^3 \csc ^3(c+d x)}{3 d}-\frac {a^3 \csc ^4(c+d x)}{4 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^6(c+d x)}{6 d}+\frac {3 a^3 \log (\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d} \] Output:

-a^3*csc(d*x+c)/d+5/2*a^3*csc(d*x+c)^2/d+5/3*a^3*csc(d*x+c)^3/d-1/4*a^3*cs 
c(d*x+c)^4/d-3/5*a^3*csc(d*x+c)^5/d-1/6*a^3*csc(d*x+c)^6/d+3*a^3*ln(sin(d* 
x+c))/d+a^3*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.85 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (-\frac {\csc (c+d x)}{d}+\frac {5 \csc ^2(c+d x)}{2 d}+\frac {5 \csc ^3(c+d x)}{3 d}-\frac {\csc ^4(c+d x)}{4 d}-\frac {3 \csc ^5(c+d x)}{5 d}-\frac {\csc ^6(c+d x)}{6 d}+\frac {3 \log (\sin (c+d x))}{d}+\frac {\sin (c+d x)}{d}\right ) \] Input:

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
 

Output:

a^3*(-(Csc[c + d*x]/d) + (5*Csc[c + d*x]^2)/(2*d) + (5*Csc[c + d*x]^3)/(3* 
d) - Csc[c + d*x]^4/(4*d) - (3*Csc[c + d*x]^5)/(5*d) - Csc[c + d*x]^6/(6*d 
) + (3*Log[Sin[c + d*x]])/d + Sin[c + d*x]/d)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 27, 99, 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.

\(\displaystyle \int \cot ^5(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)^3}{\sin (c+d x)^7}dx\)

\(\Big \downarrow \) 3315

\(\displaystyle \frac {\int \csc ^7(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5d(a \sin (c+d x))}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^2 \int \frac {\csc ^7(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5}{a^7}d(a \sin (c+d x))}{d}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {a^2 \int \left (\csc ^7(c+d x)+3 \csc ^6(c+d x)+\csc ^5(c+d x)-5 \csc ^4(c+d x)-5 \csc ^3(c+d x)+\csc ^2(c+d x)+3 \csc (c+d x)+1\right )d(a \sin (c+d x))}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^2 \left (a \sin (c+d x)-\frac {1}{6} a \csc ^6(c+d x)-\frac {3}{5} a \csc ^5(c+d x)-\frac {1}{4} a \csc ^4(c+d x)+\frac {5}{3} a \csc ^3(c+d x)+\frac {5}{2} a \csc ^2(c+d x)-a \csc (c+d x)+3 a \log (a \sin (c+d x))\right )}{d}\)

Input:

Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
 

Output:

(a^2*(-(a*Csc[c + d*x]) + (5*a*Csc[c + d*x]^2)/2 + (5*a*Csc[c + d*x]^3)/3 
- (a*Csc[c + d*x]^4)/4 - (3*a*Csc[c + d*x]^5)/5 - (a*Csc[c + d*x]^6)/6 + 3 
*a*Log[a*Sin[c + d*x]] + a*Sin[c + d*x]))/d
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

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 3315

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ 
.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* 
f)   Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, 
 x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege 
rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
 

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.64

method	result	size
derivativedivides	\(-\frac {a^{3} \left (\frac {\csc \left (d x +c \right )^{6}}{6}+\frac {3 \csc \left (d x +c \right )^{5}}{5}+\frac {\csc \left (d x +c \right )^{4}}{4}-\frac {5 \csc \left (d x +c \right )^{3}}{3}-\frac {5 \csc \left (d x +c \right )^{2}}{2}+\csc \left (d x +c \right )-\frac {1}{\csc \left (d x +c \right )}+3 \ln \left (\csc \left (d x +c \right )\right )\right )}{d}\)	\(85\)
default	\(-\frac {a^{3} \left (\frac {\csc \left (d x +c \right )^{6}}{6}+\frac {3 \csc \left (d x +c \right )^{5}}{5}+\frac {\csc \left (d x +c \right )^{4}}{4}-\frac {5 \csc \left (d x +c \right )^{3}}{3}-\frac {5 \csc \left (d x +c \right )^{2}}{2}+\csc \left (d x +c \right )-\frac {1}{\csc \left (d x +c \right )}+3 \ln \left (\csc \left (d x +c \right )\right )\right )}{d}\)	\(85\)
risch	\(-3 i a^{3} x -\frac {i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {6 i a^{3} c}{d}-\frac {2 i a^{3} \left (-75 i {\mathrm e}^{10 i \left (d x +c \right )}+15 \,{\mathrm e}^{11 i \left (d x +c \right )}+270 i {\mathrm e}^{8 i \left (d x +c \right )}+25 \,{\mathrm e}^{9 i \left (d x +c \right )}-310 i {\mathrm e}^{6 i \left (d x +c \right )}-6 \,{\mathrm e}^{7 i \left (d x +c \right )}+270 i {\mathrm e}^{4 i \left (d x +c \right )}+6 \,{\mathrm e}^{5 i \left (d x +c \right )}-75 i {\mathrm e}^{2 i \left (d x +c \right )}-25 \,{\mathrm e}^{3 i \left (d x +c \right )}-15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(224\)

Input:

int(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

-a^3/d*(1/6*csc(d*x+c)^6+3/5*csc(d*x+c)^5+1/4*csc(d*x+c)^4-5/3*csc(d*x+c)^ 
3-5/2*csc(d*x+c)^2+csc(d*x+c)-1/csc(d*x+c)+3*ln(csc(d*x+c)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.35 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {150 \, a^{3} \cos \left (d x + c\right )^{4} - 285 \, a^{3} \cos \left (d x + c\right )^{2} + 125 \, a^{3} - 180 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{6} - 30 \, a^{3} \cos \left (d x + c\right )^{4} + 40 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:

integrate(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="frica 
s")
 

Output:

-1/60*(150*a^3*cos(d*x + c)^4 - 285*a^3*cos(d*x + c)^2 + 125*a^3 - 180*(a^ 
3*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log( 
1/2*sin(d*x + c)) - 4*(15*a^3*cos(d*x + c)^6 - 30*a^3*cos(d*x + c)^4 + 40* 
a^3*cos(d*x + c)^2 - 16*a^3)*sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x 
 + c)^4 + 3*d*cos(d*x + c)^2 - d)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**5*csc(d*x+c)**2*(a+a*sin(d*x+c))**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {180 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac {60 \, a^{3} \sin \left (d x + c\right )^{5} - 150 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} + 15 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{\sin \left (d x + c\right )^{6}}}{60 \, d} \] Input:

integrate(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxim 
a")
 

Output:

1/60*(180*a^3*log(sin(d*x + c)) + 60*a^3*sin(d*x + c) - (60*a^3*sin(d*x + 
c)^5 - 150*a^3*sin(d*x + c)^4 - 100*a^3*sin(d*x + c)^3 + 15*a^3*sin(d*x + 
c)^2 + 36*a^3*sin(d*x + c) + 10*a^3)/sin(d*x + c)^6)/d
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.82 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {180 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac {60 \, a^{3} \sin \left (d x + c\right )^{5} - 150 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} + 15 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{\sin \left (d x + c\right )^{6}}}{60 \, d} \] Input:

integrate(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac" 
)
 

Output:

1/60*(180*a^3*log(abs(sin(d*x + c))) + 60*a^3*sin(d*x + c) - (60*a^3*sin(d 
*x + c)^5 - 150*a^3*sin(d*x + c)^4 - 100*a^3*sin(d*x + c)^3 + 15*a^3*sin(d 
*x + c)^2 + 36*a^3*sin(d*x + c) + 10*a^3)/sin(d*x + c)^6)/d
 

Mupad [B] (verification not implemented)

Time = 33.40 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.23 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {67\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {a^3\,\left (5760\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-5760\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{1920\,d}-\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {67\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{96}+\frac {63\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128}+\frac {23\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{240}-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{384}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{160}-\frac {a^3}{384}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:

int((cot(c + d*x)^5*(a + a*sin(c + d*x))^3)/sin(c + d*x)^2,x)
 

Output:

(67*a^3*tan(c/2 + (d*x)/2)^2)/(128*d) + (11*a^3*tan(c/2 + (d*x)/2)^3)/(96* 
d) - (a^3*tan(c/2 + (d*x)/2)^4)/(32*d) - (3*a^3*tan(c/2 + (d*x)/2)^5)/(160 
*d) - (a^3*tan(c/2 + (d*x)/2)^6)/(384*d) + (a^3*(5760*log(tan(c/2 + (d*x)/ 
2)) - 5760*log(tan(c/2 + (d*x)/2)^2 + 1)))/(1920*d) - (a^3*tan(c/2 + (d*x) 
/2))/(16*d) + (cot(c/2 + (d*x)/2)^6*((23*a^3*tan(c/2 + (d*x)/2)^3)/240 - ( 
13*a^3*tan(c/2 + (d*x)/2)^2)/384 + (63*a^3*tan(c/2 + (d*x)/2)^4)/128 + (5* 
a^3*tan(c/2 + (d*x)/2)^5)/96 + (67*a^3*tan(c/2 + (d*x)/2)^6)/128 + (31*a^3 
*tan(c/2 + (d*x)/2)^7)/16 - a^3/384 - (3*a^3*tan(c/2 + (d*x)/2))/160))/(d* 
(tan(c/2 + (d*x)/2)^2 + 1))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-5760 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{6}+5760 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}+1920 \sin \left (d x +c \right )^{7}-3125 \sin \left (d x +c \right )^{6}-1920 \sin \left (d x +c \right )^{5}+4800 \sin \left (d x +c \right )^{4}+3200 \sin \left (d x +c \right )^{3}-480 \sin \left (d x +c \right )^{2}-1152 \sin \left (d x +c \right )-320\right )}{1920 \sin \left (d x +c \right )^{6} d} \] Input:

int(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x)
 

Output:

(a**3*( - 5760*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**6 + 5760*log(tan 
((c + d*x)/2))*sin(c + d*x)**6 + 1920*sin(c + d*x)**7 - 3125*sin(c + d*x)* 
*6 - 1920*sin(c + d*x)**5 + 4800*sin(c + d*x)**4 + 3200*sin(c + d*x)**3 - 
480*sin(c + d*x)**2 - 1152*sin(c + d*x) - 320))/(1920*sin(c + d*x)**6*d)