Integrand size = 27, antiderivative size = 81 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \]
-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 = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^6(c+d x)}{3 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d} \]
-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 = 0.43 (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 |
\(\displaystyle -\frac {a \left (\frac {1}{8} \cot ^8(c+d x)+\frac {1}{6} \cot ^6(c+d x)\right )}{d}-\frac {a \left (\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
-((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
3.6.8.3.1 Defintions of rubi rules used
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 = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\) | \(68\) |
default | \(-\frac {a \left (\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\) | \(68\) |
parallelrisch | \(-\frac {a \left (3538944 \cos \left (2 d x +2 c \right )-7665 \sin \left (9 d x +9 c \right )+68985 \sin \left (7 d x +7 c \right )+1014300 \sin \left (5 d x +5 c \right )+1614690 \sin \left (d x +c \right )+1073940 \sin \left (3 d x +3 c \right )+2064384 \cos \left (4 d x +4 c \right )+3571712\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42278584320 d}\) | \(105\) |
risch | \(-\frac {4 a \left (504 i {\mathrm e}^{13 i \left (d x +c \right )}+315 \,{\mathrm e}^{14 i \left (d x +c \right )}+864 i {\mathrm e}^{11 i \left (d x +c \right )}+105 \,{\mathrm e}^{12 i \left (d x +c \right )}+1744 i {\mathrm e}^{9 i \left (d x +c \right )}+630 \,{\mathrm e}^{10 i \left (d x +c \right )}+864 i {\mathrm e}^{7 i \left (d x +c \right )}-630 \,{\mathrm e}^{8 i \left (d x +c \right )}+504 i {\mathrm e}^{5 i \left (d x +c \right )}-105 \,{\mathrm e}^{6 i \left (d x +c \right )}-315 \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}\) | \(147\) |
-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 = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.42 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {504 \, a \cos \left (d x + c\right )^{4} - 288 \, a \cos \left (d x + c\right )^{2} + 105 \, {\left (6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 64 \, a}{2520 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
-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. \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {630 \, a \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, a \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]
-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 = 0.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {630 \, a \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, a \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]
-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 = 10.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a\,{\sin \left (c+d\,x\right )}^5}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {a\,\sin \left (c+d\,x\right )}{8}+\frac {a}{9}}{d\,{\sin \left (c+d\,x\right )}^9} \]