Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}-\frac {\csc ^8(c+d x)}{4 a d}+\frac {2 \csc ^9(c+d x)}{9 a d}+\frac {\csc ^{10}(c+d x)}{10 a d}-\frac {\csc ^{11}(c+d x)}{11 a d} \] Output:
1/6*csc(d*x+c)^6/a/d-1/7*csc(d*x+c)^7/a/d-1/4*csc(d*x+c)^8/a/d+2/9*csc(d*x +c)^9/a/d+1/10*csc(d*x+c)^10/a/d-1/11*csc(d*x+c)^11/a/d
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^6(c+d x) \left (2310-1980 \csc (c+d x)-3465 \csc ^2(c+d x)+3080 \csc ^3(c+d x)+1386 \csc ^4(c+d x)-1260 \csc ^5(c+d x)\right )}{13860 a d} \] Input:
Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^5)/(a + a*Sin[c + d*x]),x]
Output:
(Csc[c + d*x]^6*(2310 - 1980*Csc[c + d*x] - 3465*Csc[c + d*x]^2 + 3080*Csc [c + d*x]^3 + 1386*Csc[c + d*x]^4 - 1260*Csc[c + d*x]^5))/(13860*a*d)
Time = 0.34 (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.
\(\displaystyle \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^7}{\sin (c+d x)^{12} (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \csc ^{12}(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^5 \int \frac {\csc ^{12}(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2}{a^{12}}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {a^5 \int \left (\frac {\csc ^{12}(c+d x)}{a^7}-\frac {\csc ^{11}(c+d x)}{a^7}-\frac {2 \csc ^{10}(c+d x)}{a^7}+\frac {2 \csc ^9(c+d x)}{a^7}+\frac {\csc ^8(c+d x)}{a^7}-\frac {\csc ^7(c+d x)}{a^7}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^5 \left (-\frac {\csc ^{11}(c+d x)}{11 a^6}+\frac {\csc ^{10}(c+d x)}{10 a^6}+\frac {2 \csc ^9(c+d x)}{9 a^6}-\frac {\csc ^8(c+d x)}{4 a^6}-\frac {\csc ^7(c+d x)}{7 a^6}+\frac {\csc ^6(c+d x)}{6 a^6}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]^7*Csc[c + d*x]^5)/(a + a*Sin[c + d*x]),x]
Output:
(a^5*(Csc[c + d*x]^6/(6*a^6) - Csc[c + d*x]^7/(7*a^6) - Csc[c + d*x]^8/(4* a^6) + (2*Csc[c + d*x]^9)/(9*a^6) + Csc[c + d*x]^10/(10*a^6) - Csc[c + d*x ]^11/(11*a^6)))/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 = 3.82 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {\csc \left (d x +c \right )^{11}}{11}+\frac {\csc \left (d x +c \right )^{10}}{10}+\frac {2 \csc \left (d x +c \right )^{9}}{9}-\frac {\csc \left (d x +c \right )^{8}}{4}-\frac {\csc \left (d x +c \right )^{7}}{7}+\frac {\csc \left (d x +c \right )^{6}}{6}}{d a}\) | \(69\) |
default | \(\frac {-\frac {\csc \left (d x +c \right )^{11}}{11}+\frac {\csc \left (d x +c \right )^{10}}{10}+\frac {2 \csc \left (d x +c \right )^{9}}{9}-\frac {\csc \left (d x +c \right )^{8}}{4}-\frac {\csc \left (d x +c \right )^{7}}{7}+\frac {\csc \left (d x +c \right )^{6}}{6}}{d a}\) | \(69\) |
risch | \(-\frac {32 \left (-1980 i {\mathrm e}^{15 i \left (d x +c \right )}+1155 \,{\mathrm e}^{16 i \left (d x +c \right )}-4400 i {\mathrm e}^{13 i \left (d x +c \right )}+1155 \,{\mathrm e}^{14 i \left (d x +c \right )}-7400 i {\mathrm e}^{11 i \left (d x +c \right )}+1848 \,{\mathrm e}^{12 i \left (d x +c \right )}-4400 i {\mathrm e}^{9 i \left (d x +c \right )}-1848 \,{\mathrm e}^{10 i \left (d x +c \right )}-1980 i {\mathrm e}^{7 i \left (d x +c \right )}-1155 \,{\mathrm e}^{8 i \left (d x +c \right )}-1155 \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{3465 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}\) | \(149\) |
Input:
int(cot(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(-1/11*csc(d*x+c)^11+1/10*csc(d*x+c)^10+2/9*csc(d*x+c)^9-1/4*csc(d*x +c)^8-1/7*csc(d*x+c)^7+1/6*csc(d*x+c)^6)
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1980 \, \cos \left (d x + c\right )^{4} - 880 \, \cos \left (d x + c\right )^{2} - 231 \, {\left (10 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 160}{13860 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/13860*(1980*cos(d*x + c)^4 - 880*cos(d*x + c)^2 - 231*(10*cos(d*x + c)^4 - 5*cos(d*x + c)^2 + 1)*sin(d*x + c) + 160)/((a*d*cos(d*x + c)^10 - 5*a*d *cos(d*x + c)^8 + 10*a*d*cos(d*x + c)^6 - 10*a*d*cos(d*x + c)^4 + 5*a*d*co s(d*x + c)^2 - a*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**7*csc(d*x+c)**5/(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2310 \, \sin \left (d x + c\right )^{5} - 1980 \, \sin \left (d x + c\right )^{4} - 3465 \, \sin \left (d x + c\right )^{3} + 3080 \, \sin \left (d x + c\right )^{2} + 1386 \, \sin \left (d x + c\right ) - 1260}{13860 \, a d \sin \left (d x + c\right )^{11}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/13860*(2310*sin(d*x + c)^5 - 1980*sin(d*x + c)^4 - 3465*sin(d*x + c)^3 + 3080*sin(d*x + c)^2 + 1386*sin(d*x + c) - 1260)/(a*d*sin(d*x + c)^11)
Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2310 \, \sin \left (d x + c\right )^{5} - 1980 \, \sin \left (d x + c\right )^{4} - 3465 \, \sin \left (d x + c\right )^{3} + 3080 \, \sin \left (d x + c\right )^{2} + 1386 \, \sin \left (d x + c\right ) - 1260}{13860 \, a d \sin \left (d x + c\right )^{11}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/13860*(2310*sin(d*x + c)^5 - 1980*sin(d*x + c)^4 - 3465*sin(d*x + c)^3 + 3080*sin(d*x + c)^2 + 1386*sin(d*x + c) - 1260)/(a*d*sin(d*x + c)^11)
Time = 32.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^5}{6}-\frac {{\sin \left (c+d\,x\right )}^4}{7}-\frac {{\sin \left (c+d\,x\right )}^3}{4}+\frac {2\,{\sin \left (c+d\,x\right )}^2}{9}+\frac {\sin \left (c+d\,x\right )}{10}-\frac {1}{11}}{a\,d\,{\sin \left (c+d\,x\right )}^{11}} \] Input:
int(cot(c + d*x)^7/(sin(c + d*x)^5*(a + a*sin(c + d*x))),x)
Output:
(sin(c + d*x)/10 + (2*sin(c + d*x)^2)/9 - sin(c + d*x)^3/4 - sin(c + d*x)^ 4/7 + sin(c + d*x)^5/6 - 1/11)/(a*d*sin(c + d*x)^11)
Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-231 \sin \left (d x +c \right )^{11}+4620 \sin \left (d x +c \right )^{5}-3960 \sin \left (d x +c \right )^{4}-6930 \sin \left (d x +c \right )^{3}+6160 \sin \left (d x +c \right )^{2}+2772 \sin \left (d x +c \right )-2520}{27720 \sin \left (d x +c \right )^{11} a d} \] Input:
int(cot(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x)
Output:
( - 231*sin(c + d*x)**11 + 4620*sin(c + d*x)**5 - 3960*sin(c + d*x)**4 - 6 930*sin(c + d*x)**3 + 6160*sin(c + d*x)**2 + 2772*sin(c + d*x) - 2520)/(27 720*sin(c + d*x)**11*a*d)