Integrand size = 21, antiderivative size = 145 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\csc ^3(c+d x)}{3 a^3 d}+\frac {3 \csc ^4(c+d x)}{4 a^3 d}-\frac {4 \csc ^6(c+d x)}{3 a^3 d}+\frac {6 \csc ^7(c+d x)}{7 a^3 d}+\frac {3 \csc ^8(c+d x)}{4 a^3 d}-\frac {8 \csc ^9(c+d x)}{9 a^3 d}+\frac {3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {\csc ^{12}(c+d x)}{12 a^3 d} \] Output:
-1/3*csc(d*x+c)^3/a^3/d+3/4*csc(d*x+c)^4/a^3/d-4/3*csc(d*x+c)^6/a^3/d+6/7* csc(d*x+c)^7/a^3/d+3/4*csc(d*x+c)^8/a^3/d-8/9*csc(d*x+c)^9/a^3/d+3/11*csc( d*x+c)^11/a^3/d-1/12*csc(d*x+c)^12/a^3/d
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\csc ^3(c+d x) \left (-924+2079 \csc (c+d x)-3696 \csc ^3(c+d x)+2376 \csc ^4(c+d x)+2079 \csc ^5(c+d x)-2464 \csc ^6(c+d x)+756 \csc ^8(c+d x)-231 \csc ^9(c+d x)\right )}{2772 a^3 d} \] Input:
Integrate[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^3,x]
Output:
(Csc[c + d*x]^3*(-924 + 2079*Csc[c + d*x] - 3696*Csc[c + d*x]^3 + 2376*Csc [c + d*x]^4 + 2079*Csc[c + d*x]^5 - 2464*Csc[c + d*x]^6 + 756*Csc[c + d*x] ^8 - 231*Csc[c + d*x]^9))/(2772*a^3*d)
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3186, 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.
\(\displaystyle \int \frac {\cot ^{13}(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^{13} (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3186 |
\(\displaystyle \frac {\int \frac {\csc ^{13}(c+d x) (a-a \sin (c+d x))^6 (\sin (c+d x) a+a)^3}{a^{13}}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (\frac {\csc ^{13}(c+d x)}{a^4}-\frac {3 \csc ^{12}(c+d x)}{a^4}+\frac {8 \csc ^{10}(c+d x)}{a^4}-\frac {6 \csc ^9(c+d x)}{a^4}-\frac {6 \csc ^8(c+d x)}{a^4}+\frac {8 \csc ^7(c+d x)}{a^4}-\frac {3 \csc ^5(c+d x)}{a^4}+\frac {\csc ^4(c+d x)}{a^4}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\csc ^{12}(c+d x)}{12 a^3}+\frac {3 \csc ^{11}(c+d x)}{11 a^3}-\frac {8 \csc ^9(c+d x)}{9 a^3}+\frac {3 \csc ^8(c+d x)}{4 a^3}+\frac {6 \csc ^7(c+d x)}{7 a^3}-\frac {4 \csc ^6(c+d x)}{3 a^3}+\frac {3 \csc ^4(c+d x)}{4 a^3}-\frac {\csc ^3(c+d x)}{3 a^3}}{d}\) |
Input:
Int[Cot[c + d*x]^13/(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) - (4*Csc[c + d*x]^6) /(3*a^3) + (6*Csc[c + d*x]^7)/(7*a^3) + (3*Csc[c + d*x]^8)/(4*a^3) - (8*Cs c[c + d*x]^9)/(9*a^3) + (3*Csc[c + d*x]^11)/(11*a^3) - Csc[c + d*x]^12/(12 *a^3))/d
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[((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 = 113.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(\frac {\frac {3}{11 \sin \left (d x +c \right )^{11}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}-\frac {4}{3 \sin \left (d x +c \right )^{6}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}+\frac {3}{4 \sin \left (d x +c \right )^{8}}-\frac {8}{9 \sin \left (d x +c \right )^{9}}+\frac {6}{7 \sin \left (d x +c \right )^{7}}}{d \,a^{3}}\) | \(89\) |
default | \(\frac {\frac {3}{11 \sin \left (d x +c \right )^{11}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}-\frac {4}{3 \sin \left (d x +c \right )^{6}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}+\frac {3}{4 \sin \left (d x +c \right )^{8}}-\frac {8}{9 \sin \left (d x +c \right )^{9}}+\frac {6}{7 \sin \left (d x +c \right )^{7}}}{d \,a^{3}}\) | \(89\) |
risch | \(\frac {4 i \left (27720 i {\mathrm e}^{10 i \left (d x +c \right )}+462 \,{\mathrm e}^{21 i \left (d x +c \right )}+1848 i {\mathrm e}^{6 i \left (d x +c \right )}-4158 \,{\mathrm e}^{19 i \left (d x +c \right )}+1848 i {\mathrm e}^{18 i \left (d x +c \right )}-2376 \,{\mathrm e}^{17 i \left (d x +c \right )}-2079 i {\mathrm e}^{4 i \left (d x +c \right )}-22616 \,{\mathrm e}^{15 i \left (d x +c \right )}-2772 i {\mathrm e}^{16 i \left (d x +c \right )}+7908 \,{\mathrm e}^{13 i \left (d x +c \right )}+27720 i {\mathrm e}^{14 i \left (d x +c \right )}-7908 \,{\mathrm e}^{11 i \left (d x +c \right )}-2079 i {\mathrm e}^{20 i \left (d x +c \right )}+22616 \,{\mathrm e}^{9 i \left (d x +c \right )}-2772 i {\mathrm e}^{8 i \left (d x +c \right )}+2376 \,{\mathrm e}^{7 i \left (d x +c \right )}+9702 i {\mathrm e}^{12 i \left (d x +c \right )}+4158 \,{\mathrm e}^{5 i \left (d x +c \right )}-462 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{693 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}\) | \(242\) |
Input:
int(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(3/11/sin(d*x+c)^11-1/12/sin(d*x+c)^12-4/3/sin(d*x+c)^6+3/4/sin(d* x+c)^4-1/3/sin(d*x+c)^3+3/4/sin(d*x+c)^8-8/9/sin(d*x+c)^9+6/7/sin(d*x+c)^7 )
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2079 \, \cos \left (d x + c\right )^{8} - 4620 \, \cos \left (d x + c\right )^{6} + 3465 \, \cos \left (d x + c\right )^{4} - 1386 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (231 \, \cos \left (d x + c\right )^{8} - 924 \, \cos \left (d x + c\right )^{6} + 792 \, \cos \left (d x + c\right )^{4} - 352 \, \cos \left (d x + c\right )^{2} + 64\right )} \sin \left (d x + c\right ) + 231}{2772 \, {\left (a^{3} d \cos \left (d x + c\right )^{12} - 6 \, a^{3} d \cos \left (d x + c\right )^{10} + 15 \, a^{3} d \cos \left (d x + c\right )^{8} - 20 \, a^{3} d \cos \left (d x + c\right )^{6} + 15 \, a^{3} d \cos \left (d x + c\right )^{4} - 6 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \] Input:
integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
1/2772*(2079*cos(d*x + c)^8 - 4620*cos(d*x + c)^6 + 3465*cos(d*x + c)^4 - 1386*cos(d*x + c)^2 - 4*(231*cos(d*x + c)^8 - 924*cos(d*x + c)^6 + 792*cos (d*x + c)^4 - 352*cos(d*x + c)^2 + 64)*sin(d*x + c) + 231)/(a^3*d*cos(d*x + c)^12 - 6*a^3*d*cos(d*x + c)^10 + 15*a^3*d*cos(d*x + c)^8 - 20*a^3*d*cos (d*x + c)^6 + 15*a^3*d*cos(d*x + c)^4 - 6*a^3*d*cos(d*x + c)^2 + a^3*d)
Timed out. \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**13/(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.59 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {924 \, \sin \left (d x + c\right )^{9} - 2079 \, \sin \left (d x + c\right )^{8} + 3696 \, \sin \left (d x + c\right )^{6} - 2376 \, \sin \left (d x + c\right )^{5} - 2079 \, \sin \left (d x + c\right )^{4} + 2464 \, \sin \left (d x + c\right )^{3} - 756 \, \sin \left (d x + c\right ) + 231}{2772 \, a^{3} d \sin \left (d x + c\right )^{12}} \] Input:
integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
-1/2772*(924*sin(d*x + c)^9 - 2079*sin(d*x + c)^8 + 3696*sin(d*x + c)^6 - 2376*sin(d*x + c)^5 - 2079*sin(d*x + c)^4 + 2464*sin(d*x + c)^3 - 756*sin( d*x + c) + 231)/(a^3*d*sin(d*x + c)^12)
Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.59 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {924 \, \sin \left (d x + c\right )^{9} - 2079 \, \sin \left (d x + c\right )^{8} + 3696 \, \sin \left (d x + c\right )^{6} - 2376 \, \sin \left (d x + c\right )^{5} - 2079 \, \sin \left (d x + c\right )^{4} + 2464 \, \sin \left (d x + c\right )^{3} - 756 \, \sin \left (d x + c\right ) + 231}{2772 \, a^{3} d \sin \left (d x + c\right )^{12}} \] Input:
integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/2772*(924*sin(d*x + c)^9 - 2079*sin(d*x + c)^8 + 3696*sin(d*x + c)^6 - 2376*sin(d*x + c)^5 - 2079*sin(d*x + c)^4 + 2464*sin(d*x + c)^3 - 756*sin( d*x + c) + 231)/(a^3*d*sin(d*x + c)^12)
Time = 18.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-\frac {{\sin \left (c+d\,x\right )}^9}{3}+\frac {3\,{\sin \left (c+d\,x\right )}^8}{4}-\frac {4\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {6\,{\sin \left (c+d\,x\right )}^5}{7}+\frac {3\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {8\,{\sin \left (c+d\,x\right )}^3}{9}+\frac {3\,\sin \left (c+d\,x\right )}{11}-\frac {1}{12}}{a^3\,d\,{\sin \left (c+d\,x\right )}^{12}} \] Input:
int(cot(c + d*x)^13/(a + a*sin(c + d*x))^3,x)
Output:
((3*sin(c + d*x))/11 - (8*sin(c + d*x)^3)/9 + (3*sin(c + d*x)^4)/4 + (6*si n(c + d*x)^5)/7 - (4*sin(c + d*x)^6)/3 + (3*sin(c + d*x)^8)/4 - sin(c + d* x)^9/3 - 1/12)/(a^3*d*sin(c + d*x)^12)
Time = 0.36 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-144375 \sin \left (d x +c \right )^{12}-946176 \sin \left (d x +c \right )^{9}+2128896 \sin \left (d x +c \right )^{8}-3784704 \sin \left (d x +c \right )^{6}+2433024 \sin \left (d x +c \right )^{5}+2128896 \sin \left (d x +c \right )^{4}-2523136 \sin \left (d x +c \right )^{3}+774144 \sin \left (d x +c \right )-236544}{2838528 \sin \left (d x +c \right )^{12} a^{3} d} \] Input:
int(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x)
Output:
( - 144375*sin(c + d*x)**12 - 946176*sin(c + d*x)**9 + 2128896*sin(c + d*x )**8 - 3784704*sin(c + d*x)**6 + 2433024*sin(c + d*x)**5 + 2128896*sin(c + d*x)**4 - 2523136*sin(c + d*x)**3 + 774144*sin(c + d*x) - 236544)/(283852 8*sin(c + d*x)**12*a**3*d)