Integrand size = 27, antiderivative size = 97 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {2 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \]
-1/5*a*csc(d*x+c)^5/d-1/6*a*csc(d*x+c)^6/d+2/7*a*csc(d*x+c)^7/d+1/4*a*csc( d*x+c)^8/d-1/9*a*csc(d*x+c)^9/d-1/10*a*csc(d*x+c)^10/d
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {2 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \]
-1/5*(a*Csc[c + d*x]^5)/d - (a*Csc[c + d*x]^6)/(6*d) + (2*a*Csc[c + d*x]^7 )/(7*d) + (a*Csc[c + d*x]^8)/(4*d) - (a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3315, 27, 99, 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.
\(\displaystyle \int \cot ^5(c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)}{\sin (c+d x)^{11}}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \csc ^{11}(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^3d(a \sin (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^6 \int \frac {\csc ^{11}(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^3}{a^{11}}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {a^6 \int \left (\frac {\csc ^{11}(c+d x)}{a^6}+\frac {\csc ^{10}(c+d x)}{a^6}-\frac {2 \csc ^9(c+d x)}{a^6}-\frac {2 \csc ^8(c+d x)}{a^6}+\frac {\csc ^7(c+d x)}{a^6}+\frac {\csc ^6(c+d x)}{a^6}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^6 \left (-\frac {\csc ^{10}(c+d x)}{10 a^5}-\frac {\csc ^9(c+d x)}{9 a^5}+\frac {\csc ^8(c+d x)}{4 a^5}+\frac {2 \csc ^7(c+d x)}{7 a^5}-\frac {\csc ^6(c+d x)}{6 a^5}-\frac {\csc ^5(c+d x)}{5 a^5}\right )}{d}\) |
(a^6*(-1/5*Csc[c + d*x]^5/a^5 - Csc[c + d*x]^6/(6*a^5) + (2*Csc[c + d*x]^7 )/(7*a^5) + Csc[c + d*x]^8/(4*a^5) - Csc[c + d*x]^9/(9*a^5) - Csc[c + d*x] ^10/(10*a^5)))/d
3.6.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{4}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\) | \(68\) |
default | \(-\frac {a \left (\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{4}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\) | \(68\) |
risch | \(-\frac {32 i a \left (105 i {\mathrm e}^{14 i \left (d x +c \right )}+63 \,{\mathrm e}^{15 i \left (d x +c \right )}+210 i {\mathrm e}^{12 i \left (d x +c \right )}+45 \,{\mathrm e}^{13 i \left (d x +c \right )}+378 i {\mathrm e}^{10 i \left (d x +c \right )}+110 \,{\mathrm e}^{11 i \left (d x +c \right )}+210 i {\mathrm e}^{8 i \left (d x +c \right )}-110 \,{\mathrm e}^{9 i \left (d x +c \right )}+105 i {\mathrm e}^{6 i \left (d x +c \right )}-45 \,{\mathrm e}^{7 i \left (d x +c \right )}-63 \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}\) | \(148\) |
parallelrisch | \(-\frac {\left (\tan ^{20}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {20 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {20 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {25 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-16 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {80 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+50 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+50 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {80 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {25 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}+1\right ) a}{10240 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}\) | \(211\) |
-a/d*(1/10*csc(d*x+c)^10+1/9*csc(d*x+c)^9-1/4*csc(d*x+c)^8-2/7*csc(d*x+c)^ 7+1/6*csc(d*x+c)^6+1/5*csc(d*x+c)^5)
Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {210 \, a \cos \left (d x + c\right )^{4} - 105 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (63 \, a \cos \left (d x + c\right )^{4} - 36 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) + 21 \, a}{1260 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
1/1260*(210*a*cos(d*x + c)^4 - 105*a*cos(d*x + c)^2 + 4*(63*a*cos(d*x + c) ^4 - 36*a*cos(d*x + c)^2 + 8*a)*sin(d*x + c) + 21*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)
Timed out. \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {252 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, a \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, a \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \]
-1/1260*(252*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*a*sin(d*x + c)^ 3 - 315*a*sin(d*x + c)^2 + 140*a*sin(d*x + c) + 126*a)/(d*sin(d*x + c)^10)
Time = 0.38 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {252 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, a \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, a \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \]
-1/1260*(252*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*a*sin(d*x + c)^ 3 - 315*a*sin(d*x + c)^2 + 140*a*sin(d*x + c) + 126*a)/(d*sin(d*x + c)^10)
Time = 10.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {252\,a\,{\sin \left (c+d\,x\right )}^5+210\,a\,{\sin \left (c+d\,x\right )}^4-360\,a\,{\sin \left (c+d\,x\right )}^3-315\,a\,{\sin \left (c+d\,x\right )}^2+140\,a\,\sin \left (c+d\,x\right )+126\,a}{1260\,d\,{\sin \left (c+d\,x\right )}^{10}} \]