Integrand size = 21, antiderivative size = 109 \[ \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\csc ^3(c+d x)}{3 a^3 d}+\frac {3 \csc ^4(c+d x)}{4 a^3 d}-\frac {2 \csc ^5(c+d x)}{5 a^3 d}-\frac {\csc ^6(c+d x)}{3 a^3 d}+\frac {3 \csc ^7(c+d x)}{7 a^3 d}-\frac {\csc ^8(c+d x)}{8 a^3 d} \] Output:
-1/3*csc(d*x+c)^3/a^3/d+3/4*csc(d*x+c)^4/a^3/d-2/5*csc(d*x+c)^5/a^3/d-1/3* csc(d*x+c)^6/a^3/d+3/7*csc(d*x+c)^7/a^3/d-1/8*csc(d*x+c)^8/a^3/d
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\csc ^3(c+d x) \left (280-630 \csc (c+d x)+336 \csc ^2(c+d x)+280 \csc ^3(c+d x)-360 \csc ^4(c+d x)+105 \csc ^5(c+d x)\right )}{840 a^3 d} \] Input:
Integrate[Cot[c + d*x]^9/(a + a*Sin[c + d*x])^3,x]
Output:
-1/840*(Csc[c + d*x]^3*(280 - 630*Csc[c + d*x] + 336*Csc[c + d*x]^2 + 280* Csc[c + d*x]^3 - 360*Csc[c + d*x]^4 + 105*Csc[c + d*x]^5))/(a^3*d)
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3186, 84, 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.
\(\displaystyle \int \frac {\cot ^9(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^9 (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3186 |
\(\displaystyle \frac {\int \frac {\csc ^9(c+d x) (a-a \sin (c+d x))^4 (\sin (c+d x) a+a)}{a^9}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 84 |
\(\displaystyle \frac {\int \left (\frac {\csc ^9(c+d x)}{a^4}-\frac {3 \csc ^8(c+d x)}{a^4}+\frac {2 \csc ^7(c+d x)}{a^4}+\frac {2 \csc ^6(c+d x)}{a^4}-\frac {3 \csc ^5(c+d x)}{a^4}+\frac {\csc ^4(c+d x)}{a^4}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\csc ^8(c+d x)}{8 a^3}+\frac {3 \csc ^7(c+d x)}{7 a^3}-\frac {\csc ^6(c+d x)}{3 a^3}-\frac {2 \csc ^5(c+d x)}{5 a^3}+\frac {3 \csc ^4(c+d x)}{4 a^3}-\frac {\csc ^3(c+d x)}{3 a^3}}{d}\) |
Input:
Int[Cot[c + d*x]^9/(a + a*Sin[c + d*x])^3,x]
Output:
(-1/3*Csc[c + d*x]^3/a^3 + (3*Csc[c + d*x]^4)/(4*a^3) - (2*Csc[c + d*x]^5) /(5*a^3) - Csc[c + d*x]^6/(3*a^3) + (3*Csc[c + d*x]^7)/(7*a^3) - Csc[c + d *x]^8/(8*a^3))/d
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x) ^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
Time = 41.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {2}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {3}{7 \sin \left (d x +c \right )^{7}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}}{d \,a^{3}}\) | \(69\) |
default | \(\frac {-\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {2}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {3}{7 \sin \left (d x +c \right )^{7}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}}{d \,a^{3}}\) | \(69\) |
risch | \(\frac {4 i \left (-315 i {\mathrm e}^{12 i \left (d x +c \right )}+70 \,{\mathrm e}^{13 i \left (d x +c \right )}+700 i {\mathrm e}^{10 i \left (d x +c \right )}-686 \,{\mathrm e}^{11 i \left (d x +c \right )}+70 i {\mathrm e}^{8 i \left (d x +c \right )}+268 \,{\mathrm e}^{9 i \left (d x +c \right )}+700 i {\mathrm e}^{6 i \left (d x +c \right )}-268 \,{\mathrm e}^{7 i \left (d x +c \right )}-315 i {\mathrm e}^{4 i \left (d x +c \right )}+686 \,{\mathrm e}^{5 i \left (d x +c \right )}-70 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{105 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\) | \(150\) |
Input:
int(cot(d*x+c)^9/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(-1/3/sin(d*x+c)^6-2/5/sin(d*x+c)^5-1/3/sin(d*x+c)^3-1/8/sin(d*x+c )^8+3/7/sin(d*x+c)^7+3/4/sin(d*x+c)^4)
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {630 \, \cos \left (d x + c\right )^{4} - 980 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 112 \, \cos \left (d x + c\right )^{2} + 32\right )} \sin \left (d x + c\right ) + 245}{840 \, {\left (a^{3} d \cos \left (d x + c\right )^{8} - 4 \, a^{3} d \cos \left (d x + c\right )^{6} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \] Input:
integrate(cot(d*x+c)^9/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
1/840*(630*cos(d*x + c)^4 - 980*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^4 - 11 2*cos(d*x + c)^2 + 32)*sin(d*x + c) + 245)/(a^3*d*cos(d*x + c)^8 - 4*a^3*d *cos(d*x + c)^6 + 6*a^3*d*cos(d*x + c)^4 - 4*a^3*d*cos(d*x + c)^2 + a^3*d)
\[ \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{9}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cot(d*x+c)**9/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)**9/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {280 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4} + 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} - 360 \, \sin \left (d x + c\right ) + 105}{840 \, a^{3} d \sin \left (d x + c\right )^{8}} \] Input:
integrate(cot(d*x+c)^9/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
-1/840*(280*sin(d*x + c)^5 - 630*sin(d*x + c)^4 + 336*sin(d*x + c)^3 + 280 *sin(d*x + c)^2 - 360*sin(d*x + c) + 105)/(a^3*d*sin(d*x + c)^8)
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {280 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4} + 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} - 360 \, \sin \left (d x + c\right ) + 105}{840 \, a^{3} d \sin \left (d x + c\right )^{8}} \] Input:
integrate(cot(d*x+c)^9/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/840*(280*sin(d*x + c)^5 - 630*sin(d*x + c)^4 + 336*sin(d*x + c)^3 + 280 *sin(d*x + c)^2 - 360*sin(d*x + c) + 105)/(a^3*d*sin(d*x + c)^8)
Time = 17.91 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {280\,{\sin \left (c+d\,x\right )}^5-630\,{\sin \left (c+d\,x\right )}^4+336\,{\sin \left (c+d\,x\right )}^3+280\,{\sin \left (c+d\,x\right )}^2-360\,\sin \left (c+d\,x\right )+105}{840\,a^3\,d\,{\sin \left (c+d\,x\right )}^8} \] Input:
int(cot(c + d*x)^9/(a + a*sin(c + d*x))^3,x)
Output:
-(280*sin(c + d*x)^2 - 360*sin(c + d*x) + 336*sin(c + d*x)^3 - 630*sin(c + d*x)^4 + 280*sin(c + d*x)^5 + 105)/(840*a^3*d*sin(c + d*x)^8)
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-15365 \sin \left (d x +c \right )^{8}-35840 \sin \left (d x +c \right )^{5}+80640 \sin \left (d x +c \right )^{4}-43008 \sin \left (d x +c \right )^{3}-35840 \sin \left (d x +c \right )^{2}+46080 \sin \left (d x +c \right )-13440}{107520 \sin \left (d x +c \right )^{8} a^{3} d} \] Input:
int(cot(d*x+c)^9/(a+a*sin(d*x+c))^3,x)
Output:
( - 15365*sin(c + d*x)**8 - 35840*sin(c + d*x)**5 + 80640*sin(c + d*x)**4 - 43008*sin(c + d*x)**3 - 35840*sin(c + d*x)**2 + 46080*sin(c + d*x) - 134 40)/(107520*sin(c + d*x)**8*a**3*d)