Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^7(c+d x)}{7 a d}-\frac {\csc ^8(c+d x)}{8 a d}-\frac {2 \csc ^9(c+d x)}{9 a d}+\frac {\csc ^{10}(c+d x)}{5 a d}+\frac {\csc ^{11}(c+d x)}{11 a d}-\frac {\csc ^{12}(c+d x)}{12 a d} \] Output:
1/7*csc(d*x+c)^7/a/d-1/8*csc(d*x+c)^8/a/d-2/9*csc(d*x+c)^9/a/d+1/5*csc(d*x +c)^10/a/d+1/11*csc(d*x+c)^11/a/d-1/12*csc(d*x+c)^12/a/d
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^7(c+d x) \left (3960-3465 \csc (c+d x)-6160 \csc ^2(c+d x)+5544 \csc ^3(c+d x)+2520 \csc ^4(c+d x)-2310 \csc ^5(c+d x)\right )}{27720 a d} \] Input:
Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^6)/(a + a*Sin[c + d*x]),x]
Output:
(Csc[c + d*x]^7*(3960 - 3465*Csc[c + d*x] - 6160*Csc[c + d*x]^2 + 5544*Csc [c + d*x]^3 + 2520*Csc[c + d*x]^4 - 2310*Csc[c + d*x]^5))/(27720*a*d)
Time = 0.33 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, 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.
\(\displaystyle \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^7}{\sin (c+d x)^{13} (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \csc ^{13}(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^6 \int \frac {\csc ^{13}(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2}{a^{13}}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {a^6 \int \left (\frac {\csc ^{13}(c+d x)}{a^8}-\frac {\csc ^{12}(c+d x)}{a^8}-\frac {2 \csc ^{11}(c+d x)}{a^8}+\frac {2 \csc ^{10}(c+d x)}{a^8}+\frac {\csc ^9(c+d x)}{a^8}-\frac {\csc ^8(c+d x)}{a^8}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^6 \left (-\frac {\csc ^{12}(c+d x)}{12 a^7}+\frac {\csc ^{11}(c+d x)}{11 a^7}+\frac {\csc ^{10}(c+d x)}{5 a^7}-\frac {2 \csc ^9(c+d x)}{9 a^7}-\frac {\csc ^8(c+d x)}{8 a^7}+\frac {\csc ^7(c+d x)}{7 a^7}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]^7*Csc[c + d*x]^6)/(a + a*Sin[c + d*x]),x]
Output:
(a^6*(Csc[c + d*x]^7/(7*a^7) - Csc[c + d*x]^8/(8*a^7) - (2*Csc[c + d*x]^9) /(9*a^7) + Csc[c + d*x]^10/(5*a^7) + Csc[c + d*x]^11/(11*a^7) - Csc[c + d* x]^12/(12*a^7)))/d
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 = 5.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {\csc \left (d x +c \right )^{12}}{12}+\frac {\csc \left (d x +c \right )^{11}}{11}+\frac {\csc \left (d x +c \right )^{10}}{5}-\frac {2 \csc \left (d x +c \right )^{9}}{9}-\frac {\csc \left (d x +c \right )^{8}}{8}+\frac {\csc \left (d x +c \right )^{7}}{7}}{d a}\) | \(69\) |
default | \(\frac {-\frac {\csc \left (d x +c \right )^{12}}{12}+\frac {\csc \left (d x +c \right )^{11}}{11}+\frac {\csc \left (d x +c \right )^{10}}{5}-\frac {2 \csc \left (d x +c \right )^{9}}{9}-\frac {\csc \left (d x +c \right )^{8}}{8}+\frac {\csc \left (d x +c \right )^{7}}{7}}{d a}\) | \(69\) |
risch | \(-\frac {32 i \left (-3465 i {\mathrm e}^{16 i \left (d x +c \right )}+1980 \,{\mathrm e}^{17 i \left (d x +c \right )}-8316 i {\mathrm e}^{14 i \left (d x +c \right )}+2420 \,{\mathrm e}^{15 i \left (d x +c \right )}-13398 i {\mathrm e}^{12 i \left (d x +c \right )}+3000 \,{\mathrm e}^{13 i \left (d x +c \right )}-8316 i {\mathrm e}^{10 i \left (d x +c \right )}-3000 \,{\mathrm e}^{11 i \left (d x +c \right )}-3465 i {\mathrm e}^{8 i \left (d x +c \right )}-2420 \,{\mathrm e}^{9 i \left (d x +c \right )}-1980 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{3465 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}\) | \(150\) |
Input:
int(cot(d*x+c)^7*csc(d*x+c)^6/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(-1/12*csc(d*x+c)^12+1/11*csc(d*x+c)^11+1/5*csc(d*x+c)^10-2/9*csc(d* x+c)^9-1/8*csc(d*x+c)^8+1/7*csc(d*x+c)^7)
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.20 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3465 \, \cos \left (d x + c\right )^{4} - 1386 \, \cos \left (d x + c\right )^{2} - 40 \, {\left (99 \, \cos \left (d x + c\right )^{4} - 44 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 231}{27720 \, {\left (a d \cos \left (d x + c\right )^{12} - 6 \, a d \cos \left (d x + c\right )^{10} + 15 \, a d \cos \left (d x + c\right )^{8} - 20 \, a d \cos \left (d x + c\right )^{6} + 15 \, a d \cos \left (d x + c\right )^{4} - 6 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/27720*(3465*cos(d*x + c)^4 - 1386*cos(d*x + c)^2 - 40*(99*cos(d*x + c)^ 4 - 44*cos(d*x + c)^2 + 8)*sin(d*x + c) + 231)/(a*d*cos(d*x + c)^12 - 6*a* d*cos(d*x + c)^10 + 15*a*d*cos(d*x + c)^8 - 20*a*d*cos(d*x + c)^6 + 15*a*d *cos(d*x + c)^4 - 6*a*d*cos(d*x + c)^2 + a*d)
Timed out. \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**7*csc(d*x+c)**6/(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3960 \, \sin \left (d x + c\right )^{5} - 3465 \, \sin \left (d x + c\right )^{4} - 6160 \, \sin \left (d x + c\right )^{3} + 5544 \, \sin \left (d x + c\right )^{2} + 2520 \, \sin \left (d x + c\right ) - 2310}{27720 \, a d \sin \left (d x + c\right )^{12}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/27720*(3960*sin(d*x + c)^5 - 3465*sin(d*x + c)^4 - 6160*sin(d*x + c)^3 + 5544*sin(d*x + c)^2 + 2520*sin(d*x + c) - 2310)/(a*d*sin(d*x + c)^12)
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3960 \, \sin \left (d x + c\right )^{5} - 3465 \, \sin \left (d x + c\right )^{4} - 6160 \, \sin \left (d x + c\right )^{3} + 5544 \, \sin \left (d x + c\right )^{2} + 2520 \, \sin \left (d x + c\right ) - 2310}{27720 \, a d \sin \left (d x + c\right )^{12}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/27720*(3960*sin(d*x + c)^5 - 3465*sin(d*x + c)^4 - 6160*sin(d*x + c)^3 + 5544*sin(d*x + c)^2 + 2520*sin(d*x + c) - 2310)/(a*d*sin(d*x + c)^12)
Time = 31.97 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^5}{7}-\frac {{\sin \left (c+d\,x\right )}^4}{8}-\frac {2\,{\sin \left (c+d\,x\right )}^3}{9}+\frac {{\sin \left (c+d\,x\right )}^2}{5}+\frac {\sin \left (c+d\,x\right )}{11}-\frac {1}{12}}{a\,d\,{\sin \left (c+d\,x\right )}^{12}} \] Input:
int(cot(c + d*x)^7/(sin(c + d*x)^6*(a + a*sin(c + d*x))),x)
Output:
(sin(c + d*x)/11 + sin(c + d*x)^2/5 - (2*sin(c + d*x)^3)/9 - sin(c + d*x)^ 4/8 + sin(c + d*x)^5/7 - 1/12)/(a*d*sin(c + d*x)^12)
Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {53361 \sin \left (d x +c \right )^{12}+2027520 \sin \left (d x +c \right )^{5}-1774080 \sin \left (d x +c \right )^{4}-3153920 \sin \left (d x +c \right )^{3}+2838528 \sin \left (d x +c \right )^{2}+1290240 \sin \left (d x +c \right )-1182720}{14192640 \sin \left (d x +c \right )^{12} a d} \] Input:
int(cot(d*x+c)^7*csc(d*x+c)^6/(a+a*sin(d*x+c)),x)
Output:
(53361*sin(c + d*x)**12 + 2027520*sin(c + d*x)**5 - 1774080*sin(c + d*x)** 4 - 3153920*sin(c + d*x)**3 + 2838528*sin(c + d*x)**2 + 1290240*sin(c + d* x) - 1182720)/(14192640*sin(c + d*x)**12*a*d)