Integrand size = 21, antiderivative size = 199 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^2(c+d x)}{2 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {3 \csc ^4(c+d x)}{4 a^2 d}-\frac {8 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{3 a^2 d}+\frac {12 \csc ^7(c+d x)}{7 a^2 d}-\frac {\csc ^8(c+d x)}{4 a^2 d}-\frac {8 \csc ^9(c+d x)}{9 a^2 d}+\frac {3 \csc ^{10}(c+d x)}{10 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac {\csc ^{12}(c+d x)}{12 a^2 d} \] Output:
-1/2*csc(d*x+c)^2/a^2/d+2/3*csc(d*x+c)^3/a^2/d+3/4*csc(d*x+c)^4/a^2/d-8/5* csc(d*x+c)^5/a^2/d-1/3*csc(d*x+c)^6/a^2/d+12/7*csc(d*x+c)^7/a^2/d-1/4*csc( d*x+c)^8/a^2/d-8/9*csc(d*x+c)^9/a^2/d+3/10*csc(d*x+c)^10/a^2/d+2/11*csc(d* x+c)^11/a^2/d-1/12*csc(d*x+c)^12/a^2/d
Time = 0.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.59 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^2(c+d x) \left (6930-9240 \csc (c+d x)-10395 \csc ^2(c+d x)+22176 \csc ^3(c+d x)+4620 \csc ^4(c+d x)-23760 \csc ^5(c+d x)+3465 \csc ^6(c+d x)+12320 \csc ^7(c+d x)-4158 \csc ^8(c+d x)-2520 \csc ^9(c+d x)+1155 \csc ^{10}(c+d x)\right )}{13860 a^2 d} \] Input:
Integrate[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^2,x]
Output:
-1/13860*(Csc[c + d*x]^2*(6930 - 9240*Csc[c + d*x] - 10395*Csc[c + d*x]^2 + 22176*Csc[c + d*x]^3 + 4620*Csc[c + d*x]^4 - 23760*Csc[c + d*x]^5 + 3465 *Csc[c + d*x]^6 + 12320*Csc[c + d*x]^7 - 4158*Csc[c + d*x]^8 - 2520*Csc[c + d*x]^9 + 1155*Csc[c + d*x]^10))/(a^2*d)
Time = 0.31 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.85, 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.
| \(\displaystyle \int \frac {\cot ^{13}(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^{13} (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3186 |
| \(\displaystyle \frac {\int \frac {\csc ^{13}(c+d x) (a-a \sin (c+d x))^6 (\sin (c+d x) a+a)^4}{a^{13}}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\frac {\csc ^{13}(c+d x)}{a^3}-\frac {2 \csc ^{12}(c+d x)}{a^3}-\frac {3 \csc ^{11}(c+d x)}{a^3}+\frac {8 \csc ^{10}(c+d x)}{a^3}+\frac {2 \csc ^9(c+d x)}{a^3}-\frac {12 \csc ^8(c+d x)}{a^3}+\frac {2 \csc ^7(c+d x)}{a^3}+\frac {8 \csc ^6(c+d x)}{a^3}-\frac {3 \csc ^5(c+d x)}{a^3}-\frac {2 \csc ^4(c+d x)}{a^3}+\frac {\csc ^3(c+d x)}{a^3}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\csc ^{12}(c+d x)}{12 a^2}+\frac {2 \csc ^{11}(c+d x)}{11 a^2}+\frac {3 \csc ^{10}(c+d x)}{10 a^2}-\frac {8 \csc ^9(c+d x)}{9 a^2}-\frac {\csc ^8(c+d x)}{4 a^2}+\frac {12 \csc ^7(c+d x)}{7 a^2}-\frac {\csc ^6(c+d x)}{3 a^2}-\frac {8 \csc ^5(c+d x)}{5 a^2}+\frac {3 \csc ^4(c+d x)}{4 a^2}+\frac {2 \csc ^3(c+d x)}{3 a^2}-\frac {\csc ^2(c+d x)}{2 a^2}}{d}\) |
Input:
Int[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^2,x]
Output:
(-1/2*Csc[c + d*x]^2/a^2 + (2*Csc[c + d*x]^3)/(3*a^2) + (3*Csc[c + d*x]^4) /(4*a^2) - (8*Csc[c + d*x]^5)/(5*a^2) - Csc[c + d*x]^6/(3*a^2) + (12*Csc[c + d*x]^7)/(7*a^2) - Csc[c + d*x]^8/(4*a^2) - (8*Csc[c + d*x]^9)/(9*a^2) + (3*Csc[c + d*x]^10)/(10*a^2) + (2*Csc[c + d*x]^11)/(11*a^2) - Csc[c + d*x ]^12/(12*a^2))/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 = 54.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.60
| method | result | size |
| derivativedivides | \(\frac {\frac {3}{10 \sin \left (d x +c \right )^{10}}-\frac {8}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{4 \sin \left (d x +c \right )^{8}}-\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {8}{5 \sin \left (d x +c \right )^{5}}+\frac {12}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}+\frac {2}{11 \sin \left (d x +c \right )^{11}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {2}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{2}}\) | \(119\) |
| default | \(\frac {\frac {3}{10 \sin \left (d x +c \right )^{10}}-\frac {8}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{4 \sin \left (d x +c \right )^{8}}-\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {8}{5 \sin \left (d x +c \right )^{5}}+\frac {12}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}+\frac {2}{11 \sin \left (d x +c \right )^{11}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {2}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{2}}\) | \(119\) |
| risch | \(\frac {2 \,{\mathrm e}^{22 i \left (d x +c \right )}-8 \,{\mathrm e}^{20 i \left (d x +c \right )}+\frac {4672 i {\mathrm e}^{15 i \left (d x +c \right )}}{315}+\frac {46 \,{\mathrm e}^{18 i \left (d x +c \right )}}{3}-\frac {16 i {\mathrm e}^{19 i \left (d x +c \right )}}{5}-96 \,{\mathrm e}^{16 i \left (d x +c \right )}+\frac {18784 i {\mathrm e}^{11 i \left (d x +c \right )}}{231}+\frac {84 \,{\mathrm e}^{14 i \left (d x +c \right )}}{5}-\frac {4672 i {\mathrm e}^{9 i \left (d x +c \right )}}{315}-\frac {1008 \,{\mathrm e}^{12 i \left (d x +c \right )}}{5}-\frac {1856 i {\mathrm e}^{17 i \left (d x +c \right )}}{35}+\frac {84 \,{\mathrm e}^{10 i \left (d x +c \right )}}{5}+\frac {16 i {\mathrm e}^{3 i \left (d x +c \right )}}{3}-96 \,{\mathrm e}^{8 i \left (d x +c \right )}-\frac {16 i {\mathrm e}^{21 i \left (d x +c \right )}}{3}+\frac {46 \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}-\frac {18784 i {\mathrm e}^{13 i \left (d x +c \right )}}{231}-8 \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {1856 i {\mathrm e}^{7 i \left (d x +c \right )}}{35}+2 \,{\mathrm e}^{2 i \left (d x +c \right )}+\frac {16 i {\mathrm e}^{5 i \left (d x +c \right )}}{5}}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}\) | \(264\) |
Input:
int(cot(d*x+c)^13/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d/a^2*(3/10/sin(d*x+c)^10-8/9/sin(d*x+c)^9-1/4/sin(d*x+c)^8-1/3/sin(d*x+ c)^6-8/5/sin(d*x+c)^5+12/7/sin(d*x+c)^7-1/2/sin(d*x+c)^2-1/12/sin(d*x+c)^1 2+2/11/sin(d*x+c)^11+3/4/sin(d*x+c)^4+2/3/sin(d*x+c)^3)
Time = 0.17 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {6930 \, \cos \left (d x + c\right )^{10} - 24255 \, \cos \left (d x + c\right )^{8} + 32340 \, \cos \left (d x + c\right )^{6} - 24255 \, \cos \left (d x + c\right )^{4} + 9702 \, \cos \left (d x + c\right )^{2} + 8 \, {\left (1155 \, \cos \left (d x + c\right )^{8} - 1848 \, \cos \left (d x + c\right )^{6} + 1584 \, \cos \left (d x + c\right )^{4} - 704 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) - 1617}{13860 \, {\left (a^{2} d \cos \left (d x + c\right )^{12} - 6 \, a^{2} d \cos \left (d x + c\right )^{10} + 15 \, a^{2} d \cos \left (d x + c\right )^{8} - 20 \, a^{2} d \cos \left (d x + c\right )^{6} + 15 \, a^{2} d \cos \left (d x + c\right )^{4} - 6 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}} \] Input:
integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
Output:
1/13860*(6930*cos(d*x + c)^10 - 24255*cos(d*x + c)^8 + 32340*cos(d*x + c)^ 6 - 24255*cos(d*x + c)^4 + 9702*cos(d*x + c)^2 + 8*(1155*cos(d*x + c)^8 - 1848*cos(d*x + c)^6 + 1584*cos(d*x + c)^4 - 704*cos(d*x + c)^2 + 128)*sin( d*x + c) - 1617)/(a^2*d*cos(d*x + c)^12 - 6*a^2*d*cos(d*x + c)^10 + 15*a^2 *d*cos(d*x + c)^8 - 20*a^2*d*cos(d*x + c)^6 + 15*a^2*d*cos(d*x + c)^4 - 6* a^2*d*cos(d*x + c)^2 + a^2*d)
Timed out. \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**13/(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.58 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {6930 \, \sin \left (d x + c\right )^{10} - 9240 \, \sin \left (d x + c\right )^{9} - 10395 \, \sin \left (d x + c\right )^{8} + 22176 \, \sin \left (d x + c\right )^{7} + 4620 \, \sin \left (d x + c\right )^{6} - 23760 \, \sin \left (d x + c\right )^{5} + 3465 \, \sin \left (d x + c\right )^{4} + 12320 \, \sin \left (d x + c\right )^{3} - 4158 \, \sin \left (d x + c\right )^{2} - 2520 \, \sin \left (d x + c\right ) + 1155}{13860 \, a^{2} d \sin \left (d x + c\right )^{12}} \] Input:
integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^2,x, algorithm="maxima")
Output:
-1/13860*(6930*sin(d*x + c)^10 - 9240*sin(d*x + c)^9 - 10395*sin(d*x + c)^ 8 + 22176*sin(d*x + c)^7 + 4620*sin(d*x + c)^6 - 23760*sin(d*x + c)^5 + 34 65*sin(d*x + c)^4 + 12320*sin(d*x + c)^3 - 4158*sin(d*x + c)^2 - 2520*sin( d*x + c) + 1155)/(a^2*d*sin(d*x + c)^12)
Time = 0.15 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.58 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {6930 \, \sin \left (d x + c\right )^{10} - 9240 \, \sin \left (d x + c\right )^{9} - 10395 \, \sin \left (d x + c\right )^{8} + 22176 \, \sin \left (d x + c\right )^{7} + 4620 \, \sin \left (d x + c\right )^{6} - 23760 \, \sin \left (d x + c\right )^{5} + 3465 \, \sin \left (d x + c\right )^{4} + 12320 \, \sin \left (d x + c\right )^{3} - 4158 \, \sin \left (d x + c\right )^{2} - 2520 \, \sin \left (d x + c\right ) + 1155}{13860 \, a^{2} d \sin \left (d x + c\right )^{12}} \] Input:
integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
-1/13860*(6930*sin(d*x + c)^10 - 9240*sin(d*x + c)^9 - 10395*sin(d*x + c)^ 8 + 22176*sin(d*x + c)^7 + 4620*sin(d*x + c)^6 - 23760*sin(d*x + c)^5 + 34 65*sin(d*x + c)^4 + 12320*sin(d*x + c)^3 - 4158*sin(d*x + c)^2 - 2520*sin( d*x + c) + 1155)/(a^2*d*sin(d*x + c)^12)
Time = 17.87 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.58 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {{\sin \left (c+d\,x\right )}^{10}}{2}-\frac {2\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,{\sin \left (c+d\,x\right )}^8}{4}+\frac {8\,{\sin \left (c+d\,x\right )}^7}{5}+\frac {{\sin \left (c+d\,x\right )}^6}{3}-\frac {12\,{\sin \left (c+d\,x\right )}^5}{7}+\frac {{\sin \left (c+d\,x\right )}^4}{4}+\frac {8\,{\sin \left (c+d\,x\right )}^3}{9}-\frac {3\,{\sin \left (c+d\,x\right )}^2}{10}-\frac {2\,\sin \left (c+d\,x\right )}{11}+\frac {1}{12}}{a^2\,d\,{\sin \left (c+d\,x\right )}^{12}} \] Input:
int(cot(c + d*x)^13/(a + a*sin(c + d*x))^2,x)
Output:
-((8*sin(c + d*x)^3)/9 - (3*sin(c + d*x)^2)/10 - (2*sin(c + d*x))/11 + sin (c + d*x)^4/4 - (12*sin(c + d*x)^5)/7 + sin(c + d*x)^6/3 + (8*sin(c + d*x) ^7)/5 - (3*sin(c + d*x)^8)/4 - (2*sin(c + d*x)^9)/3 + sin(c + d*x)^10/2 + 1/12)/(a^2*d*sin(c + d*x)^12)
Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.63 \[ \int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {1224069 \sin \left (d x +c \right )^{12}-7096320 \sin \left (d x +c \right )^{10}+9461760 \sin \left (d x +c \right )^{9}+10644480 \sin \left (d x +c \right )^{8}-22708224 \sin \left (d x +c \right )^{7}-4730880 \sin \left (d x +c \right )^{6}+24330240 \sin \left (d x +c \right )^{5}-3548160 \sin \left (d x +c \right )^{4}-12615680 \sin \left (d x +c \right )^{3}+4257792 \sin \left (d x +c \right )^{2}+2580480 \sin \left (d x +c \right )-1182720}{14192640 \sin \left (d x +c \right )^{12} a^{2} d} \] Input:
int(cot(d*x+c)^13/(a+a*sin(d*x+c))^2,x)
Output:
(1224069*sin(c + d*x)**12 - 7096320*sin(c + d*x)**10 + 9461760*sin(c + d*x )**9 + 10644480*sin(c + d*x)**8 - 22708224*sin(c + d*x)**7 - 4730880*sin(c + d*x)**6 + 24330240*sin(c + d*x)**5 - 3548160*sin(c + d*x)**4 - 12615680 *sin(c + d*x)**3 + 4257792*sin(c + d*x)**2 + 2580480*sin(c + d*x) - 118272 0)/(14192640*sin(c + d*x)**12*a**2*d)