Integrand size = 27, antiderivative size = 81 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \] Output:
-1/6*a*cot(d*x+c)^6/d-1/8*a*cot(d*x+c)^8/d-1/5*a*csc(d*x+c)^5/d+2/7*a*csc( d*x+c)^7/d-1/9*a*csc(d*x+c)^9/d
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^6(c+d x)}{3 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d} \] Input:
Integrate[Cot[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]
Output:
-1/4*(a*Csc[c + d*x]^4)/d - (a*Csc[c + d*x]^5)/(5*d) + (a*Csc[c + d*x]^6)/ (3*d) + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^9)/(9*d)
Time = 0.46 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3313, 3042, 25, 3086, 244, 2009, 3087, 244, 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.
\(\displaystyle \int \cot ^5(c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)}{\sin (c+d x)^{10}}dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cot ^5(c+d x) \csc ^5(c+d x)dx+a \int \cot ^5(c+d x) \csc ^4(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^4 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx+a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^5 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {a \int \csc ^4(c+d x) \left (1-\csc ^2(c+d x)\right )^2d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {a \int \left (\csc ^8(c+d x)-2 \csc ^6(c+d x)+\csc ^4(c+d x)\right )d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx-\frac {a \left (\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle -\frac {a \int -\cot ^5(c+d x) \left (\cot ^2(c+d x)+1\right )d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {a \int \left (-\cot ^7(c+d x)-\cot ^5(c+d x)\right )d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \left (\frac {1}{8} \cot ^8(c+d x)+\frac {1}{6} \cot ^6(c+d x)\right )}{d}-\frac {a \left (\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]
Output:
-((a*(Cot[c + d*x]^6/6 + Cot[c + d*x]^8/8))/d) - (a*(Csc[c + d*x]^5/5 - (2 *Csc[c + d*x]^7)/7 + Csc[c + d*x]^9/9))/d
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[Cos[e + f*x]^ p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[Cos[e + f*x]^p*(d*Sin[e + f*x ])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2 ] && IntegerQ[n] && ((LtQ[p, 0] && NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] | | LtQ[p + 1, -n, 2*p + 1])
Time = 0.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{9}}{9}+\frac {\csc \left (d x +c \right )^{8}}{8}-\frac {2 \csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{3}+\frac {\csc \left (d x +c \right )^{5}}{5}+\frac {\csc \left (d x +c \right )^{4}}{4}\right )}{d}\) | \(68\) |
default | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{9}}{9}+\frac {\csc \left (d x +c \right )^{8}}{8}-\frac {2 \csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{3}+\frac {\csc \left (d x +c \right )^{5}}{5}+\frac {\csc \left (d x +c \right )^{4}}{4}\right )}{d}\) | \(68\) |
risch | \(-\frac {4 a \left (504 i {\mathrm e}^{13 i \left (d x +c \right )}+315 \,{\mathrm e}^{14 i \left (d x +c \right )}+864 i {\mathrm e}^{11 i \left (d x +c \right )}+105 \,{\mathrm e}^{12 i \left (d x +c \right )}+1744 i {\mathrm e}^{9 i \left (d x +c \right )}+630 \,{\mathrm e}^{10 i \left (d x +c \right )}+864 i {\mathrm e}^{7 i \left (d x +c \right )}-630 \,{\mathrm e}^{8 i \left (d x +c \right )}+504 i {\mathrm e}^{5 i \left (d x +c \right )}-105 \,{\mathrm e}^{6 i \left (d x +c \right )}-315 \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}\) | \(147\) |
Input:
int(cot(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-a/d*(1/9*csc(d*x+c)^9+1/8*csc(d*x+c)^8-2/7*csc(d*x+c)^7-1/3*csc(d*x+c)^6+ 1/5*csc(d*x+c)^5+1/4*csc(d*x+c)^4)
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.42 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {504 \, a \cos \left (d x + c\right )^{4} - 288 \, a \cos \left (d x + c\right )^{2} + 105 \, {\left (6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 64 \, a}{2520 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/2520*(504*a*cos(d*x + c)^4 - 288*a*cos(d*x + c)^2 + 105*(6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^2 + a)*sin(d*x + c) + 64*a)/((d*cos(d*x + c)^8 - 4 *d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**5*csc(d*x+c)**5*(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {630 \, a \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, a \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/2520*(630*a*sin(d*x + c)^5 + 504*a*sin(d*x + c)^4 - 840*a*sin(d*x + c)^ 3 - 720*a*sin(d*x + c)^2 + 315*a*sin(d*x + c) + 280*a)/(d*sin(d*x + c)^9)
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {630 \, a \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, a \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/2520*(630*a*sin(d*x + c)^5 + 504*a*sin(d*x + c)^4 - 840*a*sin(d*x + c)^ 3 - 720*a*sin(d*x + c)^2 + 315*a*sin(d*x + c) + 280*a)/(d*sin(d*x + c)^9)
Time = 18.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a\,{\sin \left (c+d\,x\right )}^5}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {a\,\sin \left (c+d\,x\right )}{8}+\frac {a}{9}}{d\,{\sin \left (c+d\,x\right )}^9} \] Input:
int((cot(c + d*x)^5*(a + a*sin(c + d*x)))/sin(c + d*x)^5,x)
Output:
-(a/9 + (a*sin(c + d*x))/8 - (2*a*sin(c + d*x)^2)/7 - (a*sin(c + d*x)^3)/3 + (a*sin(c + d*x)^4)/5 + (a*sin(c + d*x)^5)/4)/(d*sin(c + d*x)^9)
Time = 0.14 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.81 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-140 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{4}+180 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{4}-1120 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{5}-1120 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{4}+660 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{5}+640 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{4}+180 \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{5}-256 \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{4}+225 \sin \left (d x +c \right )^{4}-600\right )}{10080 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c)),x)
Output:
(a*( - 140*cos(c + d*x)*cot(c + d*x)**3*csc(c + d*x)**5*sin(c + d*x)**4 + 180*cos(c + d*x)*cot(c + d*x)*csc(c + d*x)**5*sin(c + d*x)**4 - 1120*cot(c + d*x)**4*csc(c + d*x)**5*sin(c + d*x)**5 - 1120*cot(c + d*x)**4*csc(c + d*x)**5*sin(c + d*x)**4 + 660*cot(c + d*x)**2*csc(c + d*x)**5*sin(c + d*x) **5 + 640*cot(c + d*x)**2*csc(c + d*x)**5*sin(c + d*x)**4 + 180*csc(c + d* x)**5*sin(c + d*x)**5 - 256*csc(c + d*x)**5*sin(c + d*x)**4 + 225*sin(c + d*x)**4 - 600))/(10080*sin(c + d*x)**4*d)