Integrand size = 29, antiderivative size = 91 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cot ^6(c+d x)}{6 a d}+\frac {\cot ^8(c+d x)}{8 a d}-\frac {\csc ^5(c+d x)}{5 a d}+\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^9(c+d x)}{9 a d} \]
1/6*cot(d*x+c)^6/a/d+1/8*cot(d*x+c)^8/a/d-1/5*csc(d*x+c)^5/a/d+2/7*csc(d*x +c)^7/a/d-1/9*csc(d*x+c)^9/a/d
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^4(c+d x) \left (630-504 \csc (c+d x)-840 \csc ^2(c+d x)+720 \csc ^3(c+d x)+315 \csc ^4(c+d x)-280 \csc ^5(c+d x)\right )}{2520 a d} \]
(Csc[c + d*x]^4*(630 - 504*Csc[c + d*x] - 840*Csc[c + d*x]^2 + 720*Csc[c + d*x]^3 + 315*Csc[c + d*x]^4 - 280*Csc[c + d*x]^5))/(2520*a*d)
Time = 0.47 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 3314, 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.
\(\displaystyle \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^7}{\sin (c+d x)^{10} (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3314 |
\(\displaystyle \frac {\int \cot ^5(c+d x) \csc ^5(c+d x)dx}{a}-\frac {\int \cot ^5(c+d x) \csc ^4(c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx}{a}-\frac {\int -\sec \left (c+d x-\frac {\pi }{2}\right )^4 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx}{a}-\frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^5 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx}{a}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx}{a}-\frac {\int \csc ^4(c+d x) \left (1-\csc ^2(c+d x)\right )^2d\csc (c+d x)}{a d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx}{a}-\frac {\int \left (\csc ^8(c+d x)-2 \csc ^6(c+d x)+\csc ^4(c+d x)\right )d\csc (c+d x)}{a d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx}{a}-\frac {\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)}{a d}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle \frac {\int -\cot ^5(c+d x) \left (\cot ^2(c+d x)+1\right )d(-\cot (c+d x))}{a d}-\frac {\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)}{a d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\int \left (-\cot ^7(c+d x)-\cot ^5(c+d x)\right )d(-\cot (c+d x))}{a d}-\frac {\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)}{a d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{8} \cot ^8(c+d x)+\frac {1}{6} \cot ^6(c+d x)}{a d}-\frac {\frac {1}{9} \csc ^9(c+d x)-\frac {2}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)}{a d}\) |
(Cot[c + d*x]^6/6 + Cot[c + d*x]^8/8)/(a*d) - (Csc[c + d*x]^5/5 - (2*Csc[c + d*x]^7)/7 + Csc[c + d*x]^9/9)/(a*d)
3.7.93.3.1 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[1/a Int[Cos[e + f *x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[1/(b*d) Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] & & IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n, -p]))
Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}}{d a}\) | \(70\) |
default | \(-\frac {\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}}{d a}\) | \(70\) |
parallelrisch | \(-\frac {\left (3571712+3538944 \cos \left (2 d x +2 c \right )+7665 \sin \left (9 d x +9 c \right )-68985 \sin \left (7 d x +7 c \right )-1014300 \sin \left (5 d x +5 c \right )-1614690 \sin \left (d x +c \right )-1073940 \sin \left (3 d x +3 c \right )+2064384 \cos \left (4 d x +4 c \right )\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42278584320 a d}\) | \(107\) |
risch | \(\frac {-\frac {32 i {\mathrm e}^{13 i \left (d x +c \right )}}{5}+4 \,{\mathrm e}^{14 i \left (d x +c \right )}-\frac {384 i {\mathrm e}^{11 i \left (d x +c \right )}}{35}+\frac {4 \,{\mathrm e}^{12 i \left (d x +c \right )}}{3}-\frac {6976 i {\mathrm e}^{9 i \left (d x +c \right )}}{315}+8 \,{\mathrm e}^{10 i \left (d x +c \right )}-\frac {384 i {\mathrm e}^{7 i \left (d x +c \right )}}{35}-8 \,{\mathrm e}^{8 i \left (d x +c \right )}-\frac {32 i {\mathrm e}^{5 i \left (d x +c \right )}}{5}-\frac {4 \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}-4 \,{\mathrm e}^{4 i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}\) | \(149\) |
-1/d/a*(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 = 0.25 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.26 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {504 \, \cos \left (d x + c\right )^{4} - 288 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 64}{2520 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \]
-1/2520*(504*cos(d*x + c)^4 - 288*cos(d*x + c)^2 - 105*(6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*sin(d*x + c) + 64)/((a*d*cos(d*x + c)^8 - 4*a*d*cos (d*x + c)^6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 + a*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {630 \, \sin \left (d x + c\right )^{5} - 504 \, \sin \left (d x + c\right )^{4} - 840 \, \sin \left (d x + c\right )^{3} + 720 \, \sin \left (d x + c\right )^{2} + 315 \, \sin \left (d x + c\right ) - 280}{2520 \, a d \sin \left (d x + c\right )^{9}} \]
1/2520*(630*sin(d*x + c)^5 - 504*sin(d*x + c)^4 - 840*sin(d*x + c)^3 + 720 *sin(d*x + c)^2 + 315*sin(d*x + c) - 280)/(a*d*sin(d*x + c)^9)
Time = 0.54 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {630 \, \sin \left (d x + c\right )^{5} - 504 \, \sin \left (d x + c\right )^{4} - 840 \, \sin \left (d x + c\right )^{3} + 720 \, \sin \left (d x + c\right )^{2} + 315 \, \sin \left (d x + c\right ) - 280}{2520 \, a d \sin \left (d x + c\right )^{9}} \]
1/2520*(630*sin(d*x + c)^5 - 504*sin(d*x + c)^4 - 840*sin(d*x + c)^3 + 720 *sin(d*x + c)^2 + 315*sin(d*x + c) - 280)/(a*d*sin(d*x + c)^9)
Time = 10.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^5}{4}-\frac {{\sin \left (c+d\,x\right )}^4}{5}-\frac {{\sin \left (c+d\,x\right )}^3}{3}+\frac {2\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {\sin \left (c+d\,x\right )}{8}-\frac {1}{9}}{a\,d\,{\sin \left (c+d\,x\right )}^9} \]