Integrand size = 27, antiderivative size = 81 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc ^3(c+d x)}{3 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \] Output:
-1/8*a*cot(d*x+c)^8/d+1/3*a*csc(d*x+c)^3/d-3/5*a*csc(d*x+c)^5/d+3/7*a*csc( d*x+c)^7/d-1/9*a*csc(d*x+c)^9/d
Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc ^3(c+d x)}{3 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \] Input:
Integrate[Cot[c + d*x]^7*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
-1/8*(a*Cot[c + d*x]^8)/d + (a*Csc[c + d*x]^3)/(3*d) - (3*a*Csc[c + d*x]^5 )/(5*d) + (3*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)
Time = 0.45 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3313, 3042, 25, 3086, 25, 244, 2009, 3087, 15}
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.
| \(\displaystyle \int \cot ^7(c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7 (a \sin (c+d x)+a)}{\sin (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3313 |
| \(\displaystyle a \int \cot ^7(c+d x) \csc ^3(c+d x)dx+a \int \cot ^7(c+d x) \csc ^2(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx+a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -\frac {a \int -\csc ^2(c+d x) \left (1-\csc ^2(c+d x)\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \int \csc ^2(c+d x) \left (1-\csc ^2(c+d x)\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {a \int \left (-\csc ^8(c+d x)+3 \csc ^6(c+d x)-3 \csc ^4(c+d x)+\csc ^2(c+d x)\right )d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-\frac {a \left (\frac {1}{9} \csc ^9(c+d x)-\frac {3}{7} \csc ^7(c+d x)+\frac {3}{5} \csc ^5(c+d x)-\frac {1}{3} \csc ^3(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle -\frac {a \int -\cot ^7(c+d x)d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{9} \csc ^9(c+d x)-\frac {3}{7} \csc ^7(c+d x)+\frac {3}{5} \csc ^5(c+d x)-\frac {1}{3} \csc ^3(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \left (\frac {1}{9} \csc ^9(c+d x)-\frac {3}{7} \csc ^7(c+d x)+\frac {3}{5} \csc ^5(c+d x)-\frac {1}{3} \csc ^3(c+d x)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^7*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
-1/8*(a*Cot[c + d*x]^8)/d - (a*(-1/3*Csc[c + d*x]^3 + (3*Csc[c + d*x]^5)/5 - (3*Csc[c + d*x]^7)/7 + Csc[c + d*x]^9/9))/d
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[Cos[e + f*x]^ p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[Cos[e + f*x]^p*(d*Sin[e + f*x ])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2 ] && IntegerQ[n] && ((LtQ[p, 0] && NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] | | LtQ[p + 1, -n, 2*p + 1])
Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{9}}{9}+\frac {\csc \left (d x +c \right )^{8}}{8}-\frac {3 \csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{2}+\frac {3 \csc \left (d x +c \right )^{5}}{5}+\frac {3 \csc \left (d x +c \right )^{4}}{4}-\frac {\csc \left (d x +c \right )^{3}}{3}-\frac {\csc \left (d x +c \right )^{2}}{2}\right )}{d}\) | \(88\) |
| default | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{9}}{9}+\frac {\csc \left (d x +c \right )^{8}}{8}-\frac {3 \csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{2}+\frac {3 \csc \left (d x +c \right )^{5}}{5}+\frac {3 \csc \left (d x +c \right )^{4}}{4}-\frac {\csc \left (d x +c \right )^{3}}{3}-\frac {\csc \left (d x +c \right )^{2}}{2}\right )}{d}\) | \(88\) |
| risch | \(-\frac {2 a \left (420 i {\mathrm e}^{15 i \left (d x +c \right )}+315 \,{\mathrm e}^{16 i \left (d x +c \right )}+504 i {\mathrm e}^{13 i \left (d x +c \right )}-315 \,{\mathrm e}^{14 i \left (d x +c \right )}+2844 i {\mathrm e}^{11 i \left (d x +c \right )}+2205 \,{\mathrm e}^{12 i \left (d x +c \right )}+1424 i {\mathrm e}^{9 i \left (d x +c \right )}-2205 \,{\mathrm e}^{10 i \left (d x +c \right )}+2844 i {\mathrm e}^{7 i \left (d x +c \right )}+2205 \,{\mathrm e}^{8 i \left (d x +c \right )}+504 i {\mathrm e}^{5 i \left (d x +c \right )}-2205 \,{\mathrm e}^{6 i \left (d x +c \right )}+420 i {\mathrm e}^{3 i \left (d x +c \right )}+315 \,{\mathrm e}^{4 i \left (d x +c \right )}-315 \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}\) | \(193\) |
Input:
int(cot(d*x+c)^7*csc(d*x+c)^3*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-a/d*(1/9*csc(d*x+c)^9+1/8*csc(d*x+c)^8-3/7*csc(d*x+c)^7-1/2*csc(d*x+c)^6+ 3/5*csc(d*x+c)^5+3/4*csc(d*x+c)^4-1/3*csc(d*x+c)^3-1/2*csc(d*x+c)^2)
Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.72 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {840 \, a \cos \left (d x + c\right )^{6} - 1008 \, a \cos \left (d x + c\right )^{4} + 576 \, a \cos \left (d x + c\right )^{2} + 315 \, {\left (4 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 128 \, a}{2520 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/2520*(840*a*cos(d*x + c)^6 - 1008*a*cos(d*x + c)^4 + 576*a*cos(d*x + c) ^2 + 315*(4*a*cos(d*x + c)^6 - 6*a*cos(d*x + c)^4 + 4*a*cos(d*x + c)^2 - a )*sin(d*x + c) - 128*a)/((d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos( d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**7*csc(d*x+c)**3*(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1260 \, a \sin \left (d x + c\right )^{7} + 840 \, a \sin \left (d x + c\right )^{6} - 1890 \, a \sin \left (d x + c\right )^{5} - 1512 \, a \sin \left (d x + c\right )^{4} + 1260 \, a \sin \left (d x + c\right )^{3} + 1080 \, a \sin \left (d x + c\right )^{2} - 315 \, a \sin \left (d x + c\right ) - 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/2520*(1260*a*sin(d*x + c)^7 + 840*a*sin(d*x + c)^6 - 1890*a*sin(d*x + c) ^5 - 1512*a*sin(d*x + c)^4 + 1260*a*sin(d*x + c)^3 + 1080*a*sin(d*x + c)^2 - 315*a*sin(d*x + c) - 280*a)/(d*sin(d*x + c)^9)
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1260 \, a \sin \left (d x + c\right )^{7} + 840 \, a \sin \left (d x + c\right )^{6} - 1890 \, a \sin \left (d x + c\right )^{5} - 1512 \, a \sin \left (d x + c\right )^{4} + 1260 \, a \sin \left (d x + c\right )^{3} + 1080 \, a \sin \left (d x + c\right )^{2} - 315 \, a \sin \left (d x + c\right ) - 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/2520*(1260*a*sin(d*x + c)^7 + 840*a*sin(d*x + c)^6 - 1890*a*sin(d*x + c) ^5 - 1512*a*sin(d*x + c)^4 + 1260*a*sin(d*x + c)^3 + 1080*a*sin(d*x + c)^2 - 315*a*sin(d*x + c) - 280*a)/(d*sin(d*x + c)^9)
Time = 32.95 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{2}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{4}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{2}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {a\,\sin \left (c+d\,x\right )}{8}+\frac {a}{9}}{d\,{\sin \left (c+d\,x\right )}^9} \] Input:
int((cot(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^3,x)
Output:
-(a/9 + (a*sin(c + d*x))/8 - (3*a*sin(c + d*x)^2)/7 - (a*sin(c + d*x)^3)/2 + (3*a*sin(c + d*x)^4)/5 + (3*a*sin(c + d*x)^5)/4 - (a*sin(c + d*x)^6)/3 - (a*sin(c + d*x)^7)/2)/(d*sin(c + d*x)^9)
Time = 0.15 (sec) , antiderivative size = 312, normalized size of antiderivative = 3.85 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-35 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{5} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}+70 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}-105 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}-280 \cot \left (d x +c \right )^{6} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}-280 \cot \left (d x +c \right )^{6} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}+245 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}+240 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}-210 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}-192 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}-105 \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}+128 \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}-210 \sin \left (d x +c \right )^{2}+420\right )}{2520 \sin \left (d x +c \right )^{2} d} \] Input:
int(cot(d*x+c)^7*csc(d*x+c)^3*(a+a*sin(d*x+c)),x)
Output:
(a*( - 35*cos(c + d*x)*cot(c + d*x)**5*csc(c + d*x)**3*sin(c + d*x)**2 + 7 0*cos(c + d*x)*cot(c + d*x)**3*csc(c + d*x)**3*sin(c + d*x)**2 - 105*cos(c + d*x)*cot(c + d*x)*csc(c + d*x)**3*sin(c + d*x)**2 - 280*cot(c + d*x)**6 *csc(c + d*x)**3*sin(c + d*x)**3 - 280*cot(c + d*x)**6*csc(c + d*x)**3*sin (c + d*x)**2 + 245*cot(c + d*x)**4*csc(c + d*x)**3*sin(c + d*x)**3 + 240*c ot(c + d*x)**4*csc(c + d*x)**3*sin(c + d*x)**2 - 210*cot(c + d*x)**2*csc(c + d*x)**3*sin(c + d*x)**3 - 192*cot(c + d*x)**2*csc(c + d*x)**3*sin(c + d *x)**2 - 105*csc(c + d*x)**3*sin(c + d*x)**3 + 128*csc(c + d*x)**3*sin(c + d*x)**2 - 210*sin(c + d*x)**2 + 420))/(2520*sin(c + d*x)**2*d)