Integrand size = 27, antiderivative size = 97 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \] Output:
-1/6*a*csc(d*x+c)^6/d-1/7*a*csc(d*x+c)^7/d+1/4*a*csc(d*x+c)^8/d+2/9*a*csc( d*x+c)^9/d-1/10*a*csc(d*x+c)^10/d-1/11*a*csc(d*x+c)^11/d
Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \] Input:
Integrate[Cot[c + d*x]^5*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]
Output:
-1/6*(a*Csc[c + d*x]^6)/d - (a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/ (4*d) + (2*a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d)
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
\(\displaystyle \int \cot ^5(c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)}{\sin (c+d x)^{12}}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \csc ^{12}(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^3d(a \sin (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^7 \int \frac {\csc ^{12}(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^3}{a^{12}}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {a^7 \int \left (\frac {\csc ^{12}(c+d x)}{a^7}+\frac {\csc ^{11}(c+d x)}{a^7}-\frac {2 \csc ^{10}(c+d x)}{a^7}-\frac {2 \csc ^9(c+d x)}{a^7}+\frac {\csc ^8(c+d x)}{a^7}+\frac {\csc ^7(c+d x)}{a^7}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^7 \left (-\frac {\csc ^{11}(c+d x)}{11 a^6}-\frac {\csc ^{10}(c+d x)}{10 a^6}+\frac {2 \csc ^9(c+d x)}{9 a^6}+\frac {\csc ^8(c+d x)}{4 a^6}-\frac {\csc ^7(c+d x)}{7 a^6}-\frac {\csc ^6(c+d x)}{6 a^6}\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]
Output:
(a^7*(-1/6*Csc[c + d*x]^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 = 0.78 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{11}}{11}+\frac {\csc \left (d x +c \right )^{10}}{10}-\frac {2 \csc \left (d x +c \right )^{9}}{9}-\frac {\csc \left (d x +c \right )^{8}}{4}+\frac {\csc \left (d x +c \right )^{7}}{7}+\frac {\csc \left (d x +c \right )^{6}}{6}\right )}{d}\) | \(68\) |
default | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{11}}{11}+\frac {\csc \left (d x +c \right )^{10}}{10}-\frac {2 \csc \left (d x +c \right )^{9}}{9}-\frac {\csc \left (d x +c \right )^{8}}{4}+\frac {\csc \left (d x +c \right )^{7}}{7}+\frac {\csc \left (d x +c \right )^{6}}{6}\right )}{d}\) | \(68\) |
risch | \(\frac {32 a \left (1980 i {\mathrm e}^{15 i \left (d x +c \right )}+1155 \,{\mathrm e}^{16 i \left (d x +c \right )}+4400 i {\mathrm e}^{13 i \left (d x +c \right )}+1155 \,{\mathrm e}^{14 i \left (d x +c \right )}+7400 i {\mathrm e}^{11 i \left (d x +c \right )}+1848 \,{\mathrm e}^{12 i \left (d x +c \right )}+4400 i {\mathrm e}^{9 i \left (d x +c \right )}-1848 \,{\mathrm e}^{10 i \left (d x +c \right )}+1980 i {\mathrm e}^{7 i \left (d x +c \right )}-1155 \,{\mathrm e}^{8 i \left (d x +c \right )}-1155 \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{3465 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}\) | \(147\) |
Input:
int(cot(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-a/d*(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 = 1.32 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1980 \, a \cos \left (d x + c\right )^{4} - 880 \, a \cos \left (d x + c\right )^{2} + 231 \, {\left (10 \, a \cos \left (d x + c\right )^{4} - 5 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 160 \, a}{13860 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/13860*(1980*a*cos(d*x + c)^4 - 880*a*cos(d*x + c)^2 + 231*(10*a*cos(d*x + c)^4 - 5*a*cos(d*x + c)^2 + a)*sin(d*x + c) + 160*a)/((d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*co s(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**5*csc(d*x+c)**7*(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {2310 \, a \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, a \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, a \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/13860*(2310*a*sin(d*x + c)^5 + 1980*a*sin(d*x + c)^4 - 3465*a*sin(d*x + c)^3 - 3080*a*sin(d*x + c)^2 + 1386*a*sin(d*x + c) + 1260*a)/(d*sin(d*x + c)^11)
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {2310 \, a \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, a \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, a \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/13860*(2310*a*sin(d*x + c)^5 + 1980*a*sin(d*x + c)^4 - 3465*a*sin(d*x + c)^3 - 3080*a*sin(d*x + c)^2 + 1386*a*sin(d*x + c) + 1260*a)/(d*sin(d*x + c)^11)
Time = 18.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a\,{\sin \left (c+d\,x\right )}^5}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{4}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^2}{9}+\frac {a\,\sin \left (c+d\,x\right )}{10}+\frac {a}{11}}{d\,{\sin \left (c+d\,x\right )}^{11}} \] Input:
int((cot(c + d*x)^5*(a + a*sin(c + d*x)))/sin(c + d*x)^7,x)
Output:
-(a/11 + (a*sin(c + d*x))/10 - (2*a*sin(c + d*x)^2)/9 - (a*sin(c + d*x)^3) /4 + (a*sin(c + d*x)^4)/7 + (a*sin(c + d*x)^5)/6)/(d*sin(c + d*x)^11)
Time = 0.15 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.35 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-1008 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}+952 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}-10080 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{7}-10080 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}+4592 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{7}+4480 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}+952 \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{7}-1280 \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}+875 \sin \left (d x +c \right )^{6}-2800\right )}{110880 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x)
Output:
(a*( - 1008*cos(c + d*x)*cot(c + d*x)**3*csc(c + d*x)**7*sin(c + d*x)**6 + 952*cos(c + d*x)*cot(c + d*x)*csc(c + d*x)**7*sin(c + d*x)**6 - 10080*cot (c + d*x)**4*csc(c + d*x)**7*sin(c + d*x)**7 - 10080*cot(c + d*x)**4*csc(c + d*x)**7*sin(c + d*x)**6 + 4592*cot(c + d*x)**2*csc(c + d*x)**7*sin(c + d*x)**7 + 4480*cot(c + d*x)**2*csc(c + d*x)**7*sin(c + d*x)**6 + 952*csc(c + d*x)**7*sin(c + d*x)**7 - 1280*csc(c + d*x)**7*sin(c + d*x)**6 + 875*si n(c + d*x)**6 - 2800))/(110880*sin(c + d*x)**6*d)