Integrand size = 27, antiderivative size = 113 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^{12}(c+d x)}{12 d}+\frac {a \csc ^5(c+d x)}{5 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]
-1/8*a*cot(d*x+c)^8/d-1/5*a*cot(d*x+c)^10/d-1/12*a*cot(d*x+c)^12/d+1/5*a*c sc(d*x+c)^5/d-3/7*a*csc(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 = 1.14 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^6(c+d x)}{6 d}-\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^8(c+d x)}{8 d}+\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{12}(c+d x)}{12 d} \]
(a*Csc[c + d*x]^5)/(5*d) + (a*Csc[c + d*x]^6)/(6*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) + (3*a*Csc[ c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d) - (a*Csc[c + d*x]^12)/(12 *d)
Time = 0.44 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3313, 3042, 25, 3086, 25, 244, 2009, 3087, 243, 49, 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.
| \(\displaystyle \int \cot ^7(c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7 (a \sin (c+d x)+a)}{\sin (c+d x)^{13}}dx\) |
| \(\Big \downarrow \) 3313 |
| \(\displaystyle a \int \cot ^7(c+d x) \csc ^6(c+d x)dx+a \int \cot ^7(c+d x) \csc ^5(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx+a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^6 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^5 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -\frac {a \int -\csc ^4(c+d x) \left (1-\csc ^2(c+d x)\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \int \csc ^4(c+d x) \left (1-\csc ^2(c+d x)\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {a \int \left (-\csc ^{10}(c+d x)+3 \csc ^8(c+d x)-3 \csc ^6(c+d x)+\csc ^4(c+d x)\right )d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle -\frac {a \int -\cot ^7(c+d x) \left (\cot ^2(c+d x)+1\right )^2d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {a \int -\cot ^3(c+d x) \left (\cot ^2(c+d x)+1\right )^2d\cot ^2(c+d x)}{2 d}-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {a \int \left (-\cot ^5(c+d x)+2 \cot ^4(c+d x)-\cot ^3(c+d x)\right )d\cot ^2(c+d x)}{2 d}-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (\frac {1}{6} \cot ^6(c+d x)-\frac {2}{5} \cot ^5(c+d x)+\frac {1}{4} \cot ^4(c+d x)\right )}{2 d}-\frac {a \left (\frac {1}{11} \csc ^{11}(c+d x)-\frac {1}{3} \csc ^9(c+d x)+\frac {3}{7} \csc ^7(c+d x)-\frac {1}{5} \csc ^5(c+d x)\right )}{d}\) |
-1/2*(a*(Cot[c + d*x]^4/4 - (2*Cot[c + d*x]^5)/5 + Cot[c + d*x]^6/6))/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
3.7.74.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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 = 0.56 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\) | \(88\) |
| default | \(-\frac {a \left (\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\) | \(88\) |
| parallelrisch | \(\frac {a \left (\sec ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-221939718-376472712 \cos \left (2 d x +2 c \right )-1428042 \cos \left (8 d x +8 c \right )-30277632 \sin \left (7 d x +7 c \right )-47579136 \sin \left (5 d x +5 c \right )-45702580 \cos \left (6 d x +6 c \right )+5898240 \sin \left (d x +c \right )-145620992 \sin \left (3 d x +3 c \right )-162098475 \cos \left (4 d x +4 c \right )-21637 \cos \left (12 d x +12 c \right )+259644 \cos \left (10 d x +10 c \right )\right )}{39685497815040 d}\) | \(138\) |
| risch | \(\frac {32 i a \left (385 i {\mathrm e}^{18 i \left (d x +c \right )}+231 \,{\mathrm e}^{19 i \left (d x +c \right )}+1155 i {\mathrm e}^{16 i \left (d x +c \right )}+363 \,{\mathrm e}^{17 i \left (d x +c \right )}+3003 i {\mathrm e}^{14 i \left (d x +c \right )}+1111 \,{\mathrm e}^{15 i \left (d x +c \right )}+3234 i {\mathrm e}^{12 i \left (d x +c \right )}-45 \,{\mathrm e}^{13 i \left (d x +c \right )}+3003 i {\mathrm e}^{10 i \left (d x +c \right )}+45 \,{\mathrm e}^{11 i \left (d x +c \right )}+1155 i {\mathrm e}^{8 i \left (d x +c \right )}-1111 \,{\mathrm e}^{9 i \left (d x +c \right )}+385 i {\mathrm e}^{6 i \left (d x +c \right )}-363 \,{\mathrm e}^{7 i \left (d x +c \right )}-231 \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{1155 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}\) | \(194\) |
-a/d*(1/12*csc(d*x+c)^12+1/11*csc(d*x+c)^11-3/10*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)^7-1/6*csc(d*x+c)^6-1/5*csc(d*x+c)^5)
Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.35 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {1540 \, a \cos \left (d x + c\right )^{6} - 1155 \, a \cos \left (d x + c\right )^{4} + 462 \, a \cos \left (d x + c\right )^{2} + 8 \, {\left (231 \, a \cos \left (d x + c\right )^{6} - 198 \, a \cos \left (d x + c\right )^{4} + 88 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 77 \, a}{9240 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
-1/9240*(1540*a*cos(d*x + c)^6 - 1155*a*cos(d*x + c)^4 + 462*a*cos(d*x + c )^2 + 8*(231*a*cos(d*x + c)^6 - 198*a*cos(d*x + c)^4 + 88*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c) - 77*a)/(d*cos(d*x + c)^12 - 6*d*cos(d*x + c)^10 + 1 5*d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*d*cos(d *x + c)^2 + d)
Timed out. \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1848 \, a \sin \left (d x + c\right )^{7} + 1540 \, a \sin \left (d x + c\right )^{6} - 3960 \, a \sin \left (d x + c\right )^{5} - 3465 \, a \sin \left (d x + c\right )^{4} + 3080 \, a \sin \left (d x + c\right )^{3} + 2772 \, a \sin \left (d x + c\right )^{2} - 840 \, a \sin \left (d x + c\right ) - 770 \, a}{9240 \, d \sin \left (d x + c\right )^{12}} \]
1/9240*(1848*a*sin(d*x + c)^7 + 1540*a*sin(d*x + c)^6 - 3960*a*sin(d*x + c )^5 - 3465*a*sin(d*x + c)^4 + 3080*a*sin(d*x + c)^3 + 2772*a*sin(d*x + c)^ 2 - 840*a*sin(d*x + c) - 770*a)/(d*sin(d*x + c)^12)
Time = 0.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1848 \, a \sin \left (d x + c\right )^{7} + 1540 \, a \sin \left (d x + c\right )^{6} - 3960 \, a \sin \left (d x + c\right )^{5} - 3465 \, a \sin \left (d x + c\right )^{4} + 3080 \, a \sin \left (d x + c\right )^{3} + 2772 \, a \sin \left (d x + c\right )^{2} - 840 \, a \sin \left (d x + c\right ) - 770 \, a}{9240 \, d \sin \left (d x + c\right )^{12}} \]
1/9240*(1848*a*sin(d*x + c)^7 + 1540*a*sin(d*x + c)^6 - 3960*a*sin(d*x + c )^5 - 3465*a*sin(d*x + c)^4 + 3080*a*sin(d*x + c)^3 + 2772*a*sin(d*x + c)^ 2 - 840*a*sin(d*x + c) - 770*a)/(d*sin(d*x + c)^12)
Time = 10.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{7}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{10}+\frac {a\,\sin \left (c+d\,x\right )}{11}+\frac {a}{12}}{d\,{\sin \left (c+d\,x\right )}^{12}} \]