Integrand size = 27, antiderivative size = 97 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{10 d}+\frac {a \csc ^5(c+d x)}{5 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]
-1/8*a*cot(d*x+c)^8/d-1/10*a*cot(d*x+c)^10/d+1/5*a*csc(d*x+c)^5/d-3/7*a*cs c(d*x+c)^7/d+1/3*a*csc(d*x+c)^9/d-1/11*a*csc(d*x+c)^11/d
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.33 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^4(c+d x)}{4 d}+\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{2 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {3 a \csc ^8(c+d x)}{8 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]
(a*Csc[c + d*x]^4)/(4*d) + (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/( 2*d) - (3*a*Csc[c + d*x]^7)/(7*d) + (3*a*Csc[c + d*x]^8)/(8*d) + (a*Csc[c + d*x]^9)/(3*d) - (a*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d)
Time = 0.44 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 3313, 3042, 25, 3086, 25, 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.
| \(\displaystyle \int \cot ^7(c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7 (a \sin (c+d x)+a)}{\sin (c+d x)^{12}}dx\) |
| \(\Big \downarrow \) 3313 |
| \(\displaystyle a \int \cot ^7(c+d x) \csc ^5(c+d x)dx+a \int \cot ^7(c+d x) \csc ^4(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^4 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx+a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^5 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -\frac {a \int -\csc ^4(c+d x) \left (1-\csc ^2(c+d x)\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \int \csc ^4(c+d x) \left (1-\csc ^2(c+d x)\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {a \int \left (-\csc ^{10}(c+d x)+3 \csc ^8(c+d x)-3 \csc ^6(c+d x)+\csc ^4(c+d x)\right )d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle -\frac {a \int -\cot ^7(c+d x) \left (\cot ^2(c+d x)+1\right )d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {a \int \left (-\cot ^9(c+d x)-\cot ^7(c+d x)\right )d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (\frac {1}{10} \cot ^{10}(c+d x)+\frac {1}{8} \cot ^8(c+d x)\right )}{d}-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
-((a*(Cot[c + d*x]^8/8 + Cot[c + d*x]^10/10))/d) - (a*(-1/5*Csc[c + d*x]^5 + (3*Csc[c + d*x]^7)/7 - Csc[c + d*x]^9/3 + Csc[c + d*x]^11/11))/d
3.7.73.3.1 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 = 0.48 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}-\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\) | \(88\) |
| default | \(-\frac {a \left (\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}-\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\) | \(88\) |
| parallelrisch | \(\frac {a \left (\sec ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-27197440-55869440 \cos \left (2 d x +2 c \right )-200739 \sin \left (9 d x +9 c \right )-3727185 \sin \left (7 d x +7 c \right )-7741965 \sin \left (5 d x +5 c \right )-7569408 \cos \left (6 d x +6 c \right )-1807806 \sin \left (d x +c \right )-17632230 \sin \left (3 d x +3 c \right )-19464192 \cos \left (4 d x +4 c \right )+18249 \sin \left (11 d x +11 c \right )\right )}{2480343613440 d}\) | \(127\) |
| risch | \(\frac {4 a \left (1848 i {\mathrm e}^{17 i \left (d x +c \right )}+1155 \,{\mathrm e}^{18 i \left (d x +c \right )}+4752 i {\mathrm e}^{15 i \left (d x +c \right )}+1155 \,{\mathrm e}^{16 i \left (d x +c \right )}+13640 i {\mathrm e}^{13 i \left (d x +c \right )}+5775 \,{\mathrm e}^{14 i \left (d x +c \right )}+13280 i {\mathrm e}^{11 i \left (d x +c \right )}-1617 \,{\mathrm e}^{12 i \left (d x +c \right )}+13640 i {\mathrm e}^{9 i \left (d x +c \right )}+1617 \,{\mathrm e}^{10 i \left (d x +c \right )}+4752 i {\mathrm e}^{7 i \left (d x +c \right )}-5775 \,{\mathrm e}^{8 i \left (d x +c \right )}+1848 i {\mathrm e}^{5 i \left (d x +c \right )}-1155 \,{\mathrm e}^{6 i \left (d x +c \right )}-1155 \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{1155 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}\) | \(193\) |
-a/d*(1/11*csc(d*x+c)^11+1/10*csc(d*x+c)^10-1/3*csc(d*x+c)^9-3/8*csc(d*x+c )^8+3/7*csc(d*x+c)^7+1/2*csc(d*x+c)^6-1/5*csc(d*x+c)^5-1/4*csc(d*x+c)^4)
Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1848 \, a \cos \left (d x + c\right )^{6} - 1584 \, a \cos \left (d x + c\right )^{4} + 704 \, a \cos \left (d x + c\right )^{2} + 231 \, {\left (10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 128 \, a}{9240 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
1/9240*(1848*a*cos(d*x + c)^6 - 1584*a*cos(d*x + c)^4 + 704*a*cos(d*x + c) ^2 + 231*(10*a*cos(d*x + c)^6 - 10*a*cos(d*x + c)^4 + 5*a*cos(d*x + c)^2 - a)*sin(d*x + c) - 128*a)/((d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d* cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)*sin(d*x + c ))
Timed out. \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2310 \, a \sin \left (d x + c\right )^{7} + 1848 \, a \sin \left (d x + c\right )^{6} - 4620 \, a \sin \left (d x + c\right )^{5} - 3960 \, a \sin \left (d x + c\right )^{4} + 3465 \, a \sin \left (d x + c\right )^{3} + 3080 \, a \sin \left (d x + c\right )^{2} - 924 \, a \sin \left (d x + c\right ) - 840 \, a}{9240 \, d \sin \left (d x + c\right )^{11}} \]
1/9240*(2310*a*sin(d*x + c)^7 + 1848*a*sin(d*x + c)^6 - 4620*a*sin(d*x + c )^5 - 3960*a*sin(d*x + c)^4 + 3465*a*sin(d*x + c)^3 + 3080*a*sin(d*x + c)^ 2 - 924*a*sin(d*x + c) - 840*a)/(d*sin(d*x + c)^11)
Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2310 \, a \sin \left (d x + c\right )^{7} + 1848 \, a \sin \left (d x + c\right )^{6} - 4620 \, a \sin \left (d x + c\right )^{5} - 3960 \, a \sin \left (d x + c\right )^{4} + 3465 \, a \sin \left (d x + c\right )^{3} + 3080 \, a \sin \left (d x + c\right )^{2} - 924 \, a \sin \left (d x + c\right ) - 840 \, a}{9240 \, d \sin \left (d x + c\right )^{11}} \]
1/9240*(2310*a*sin(d*x + c)^7 + 1848*a*sin(d*x + c)^6 - 4620*a*sin(d*x + c )^5 - 3960*a*sin(d*x + c)^4 + 3465*a*sin(d*x + c)^3 + 3080*a*sin(d*x + c)^ 2 - 924*a*sin(d*x + c) - 840*a)/(d*sin(d*x + c)^11)
Time = 10.37 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{4}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{2}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{7}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{3}+\frac {a\,\sin \left (c+d\,x\right )}{10}+\frac {a}{11}}{d\,{\sin \left (c+d\,x\right )}^{11}} \]