Integrand size = 21, antiderivative size = 84 \[ \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cot ^8(c+d x)}{8 a d}-\frac {\csc (c+d x)}{a d}+\frac {\csc ^3(c+d x)}{a d}-\frac {3 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^7(c+d x)}{7 a d} \] Output:
-1/8*cot(d*x+c)^8/a/d-csc(d*x+c)/a/d+csc(d*x+c)^3/a/d-3/5*csc(d*x+c)^5/a/d +1/7*csc(d*x+c)^7/a/d
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^8(c+d x) (-245 \cos (2 (c+d x))-35 \cos (6 (c+d x))-513 \sin (c+d x)+371 \sin (3 (c+d x))-105 \sin (5 (c+d x))+35 \sin (7 (c+d x)))}{2240 a d} \] Input:
Integrate[Cot[c + d*x]^9/(a + a*Sin[c + d*x]),x]
Output:
(Csc[c + d*x]^8*(-245*Cos[2*(c + d*x)] - 35*Cos[6*(c + d*x)] - 513*Sin[c + d*x] + 371*Sin[3*(c + d*x)] - 105*Sin[5*(c + d*x)] + 35*Sin[7*(c + d*x)]) )/(2240*a*d)
Time = 0.42 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.80, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3185, 3042, 25, 3086, 210, 2009, 3087, 15}
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.
\(\displaystyle \int \frac {\cot ^9(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^9 (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3185 |
\(\displaystyle \frac {\int \cot ^7(c+d x) \csc ^2(c+d x)dx}{a}-\frac {\int \cot ^7(c+d x) \csc (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx}{a}-\frac {\int -\sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^7dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx}{a}-\frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx}{a}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {\int \left (\csc ^2(c+d x)-1\right )^3d\csc (c+d x)}{a d}-\frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx}{a}\) |
\(\Big \downarrow \) 210 |
\(\displaystyle \frac {\int \left (\csc ^6(c+d x)-3 \csc ^4(c+d x)+3 \csc ^2(c+d x)-1\right )d\csc (c+d x)}{a d}-\frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{7} \csc ^7(c+d x)-\frac {3}{5} \csc ^5(c+d x)+\csc ^3(c+d x)-\csc (c+d x)}{a d}-\frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx}{a}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle \frac {\frac {1}{7} \csc ^7(c+d x)-\frac {3}{5} \csc ^5(c+d x)+\csc ^3(c+d x)-\csc (c+d x)}{a d}-\frac {\int -\cot ^7(c+d x)d(-\cot (c+d x))}{a d}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\frac {1}{7} \csc ^7(c+d x)-\frac {3}{5} \csc ^5(c+d x)+\csc ^3(c+d x)-\csc (c+d x)}{a d}-\frac {\cot ^8(c+d x)}{8 a d}\) |
Input:
Int[Cot[c + d*x]^9/(a + a*Sin[c + d*x]),x]
Output:
-1/8*Cot[c + d*x]^8/(a*d) + (-Csc[c + d*x] + Csc[c + d*x]^3 - (3*Csc[c + d *x]^5)/5 + Csc[c + d*x]^7/7)/(a*d)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, 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[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[1/a Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x ] - Simp[1/(b*g) Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; Fre eQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
Time = 4.90 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {1}{7 \sin \left (d x +c \right )^{7}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )^{3}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {3}{5 \sin \left (d x +c \right )^{5}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{\sin \left (d x +c \right )}+\frac {1}{2 \sin \left (d x +c \right )^{6}}}{d a}\) | \(87\) |
default | \(\frac {\frac {1}{7 \sin \left (d x +c \right )^{7}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )^{3}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {3}{5 \sin \left (d x +c \right )^{5}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{\sin \left (d x +c \right )}+\frac {1}{2 \sin \left (d x +c \right )^{6}}}{d a}\) | \(87\) |
risch | \(-\frac {2 i \left (-35 i {\mathrm e}^{14 i \left (d x +c \right )}+35 \,{\mathrm e}^{15 i \left (d x +c \right )}-105 \,{\mathrm e}^{13 i \left (d x +c \right )}-245 i {\mathrm e}^{10 i \left (d x +c \right )}+371 \,{\mathrm e}^{11 i \left (d x +c \right )}-513 \,{\mathrm e}^{9 i \left (d x +c \right )}-245 i {\mathrm e}^{6 i \left (d x +c \right )}+513 \,{\mathrm e}^{7 i \left (d x +c \right )}-371 \,{\mathrm e}^{5 i \left (d x +c \right )}-35 i {\mathrm e}^{2 i \left (d x +c \right )}+105 \,{\mathrm e}^{3 i \left (d x +c \right )}-35 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{35 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\) | \(160\) |
Input:
int(cot(d*x+c)^9/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(1/7/sin(d*x+c)^7+1/2/sin(d*x+c)^2+1/sin(d*x+c)^3-1/8/sin(d*x+c)^8-3 /5/sin(d*x+c)^5-3/4/sin(d*x+c)^4-1/sin(d*x+c)+1/2/sin(d*x+c)^6)
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.51 \[ \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {140 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{4} + 140 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{6} - 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) - 35}{280 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \] Input:
integrate(cot(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
-1/280*(140*cos(d*x + c)^6 - 210*cos(d*x + c)^4 + 140*cos(d*x + c)^2 - 8*( 35*cos(d*x + c)^6 - 70*cos(d*x + c)^4 + 56*cos(d*x + c)^2 - 16)*sin(d*x + c) - 35)/(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)
\[ \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cot ^{9}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(cot(d*x+c)**9/(a+a*sin(d*x+c)),x)
Output:
Integral(cot(c + d*x)**9/(sin(c + d*x) + 1), x)/a
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {280 \, \sin \left (d x + c\right )^{7} - 140 \, \sin \left (d x + c\right )^{6} - 280 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} + 168 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) + 35}{280 \, a d \sin \left (d x + c\right )^{8}} \] Input:
integrate(cot(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/280*(280*sin(d*x + c)^7 - 140*sin(d*x + c)^6 - 280*sin(d*x + c)^5 + 210 *sin(d*x + c)^4 + 168*sin(d*x + c)^3 - 140*sin(d*x + c)^2 - 40*sin(d*x + c ) + 35)/(a*d*sin(d*x + c)^8)
Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {280 \, \sin \left (d x + c\right )^{7} - 140 \, \sin \left (d x + c\right )^{6} - 280 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} + 168 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) + 35}{280 \, a d \sin \left (d x + c\right )^{8}} \] Input:
integrate(cot(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/280*(280*sin(d*x + c)^7 - 140*sin(d*x + c)^6 - 280*sin(d*x + c)^5 + 210 *sin(d*x + c)^4 + 168*sin(d*x + c)^3 - 140*sin(d*x + c)^2 - 40*sin(d*x + c ) + 35)/(a*d*sin(d*x + c)^8)
Time = 17.89 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-{\sin \left (c+d\,x\right )}^7+\frac {{\sin \left (c+d\,x\right )}^6}{2}+{\sin \left (c+d\,x\right )}^5-\frac {3\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {3\,{\sin \left (c+d\,x\right )}^3}{5}+\frac {{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{7}-\frac {1}{8}}{a\,d\,{\sin \left (c+d\,x\right )}^8} \] Input:
int(cot(c + d*x)^9/(a + a*sin(c + d*x)),x)
Output:
(sin(c + d*x)/7 + sin(c + d*x)^2/2 - (3*sin(c + d*x)^3)/5 - (3*sin(c + d*x )^4)/4 + sin(c + d*x)^5 + sin(c + d*x)^6/2 - sin(c + d*x)^7 - 1/8)/(a*d*si n(c + d*x)^8)
Time = 0.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.14 \[ \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-3255 \sin \left (d x +c \right )^{8}-35840 \sin \left (d x +c \right )^{7}+17920 \sin \left (d x +c \right )^{6}+35840 \sin \left (d x +c \right )^{5}-26880 \sin \left (d x +c \right )^{4}-21504 \sin \left (d x +c \right )^{3}+17920 \sin \left (d x +c \right )^{2}+5120 \sin \left (d x +c \right )-4480}{35840 \sin \left (d x +c \right )^{8} a d} \] Input:
int(cot(d*x+c)^9/(a+a*sin(d*x+c)),x)
Output:
( - 3255*sin(c + d*x)**8 - 35840*sin(c + d*x)**7 + 17920*sin(c + d*x)**6 + 35840*sin(c + d*x)**5 - 26880*sin(c + d*x)**4 - 21504*sin(c + d*x)**3 + 1 7920*sin(c + d*x)**2 + 5120*sin(c + d*x) - 4480)/(35840*sin(c + d*x)**8*a* d)