Integrand size = 27, antiderivative size = 74 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \] Output:
-1/8*a*cot(d*x+c)^8/d+a*csc(d*x+c)/d-a*csc(d*x+c)^3/d+3/5*a*csc(d*x+c)^5/d -1/7*a*csc(d*x+c)^7/d
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \] Input:
Integrate[Cot[c + d*x]^7*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
-1/8*(a*Cot[c + d*x]^8)/d + (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/d + (3 *a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)
Time = 0.48 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3313, 3042, 25, 3086, 210, 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.
| \(\displaystyle \int \cot ^7(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7 (a \sin (c+d x)+a)}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3313 |
| \(\displaystyle a \int \cot ^7(c+d x) \csc ^2(c+d x)dx+a \int \cot ^7(c+d x) \csc (c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^7dx+a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -\frac {a \int \left (\csc ^2(c+d x)-1\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle -\frac {a \int \left (\csc ^6(c+d x)-3 \csc ^4(c+d x)+3 \csc ^2(c+d x)-1\right )d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {3}{5} \csc ^5(c+d x)+\csc ^3(c+d x)-\csc (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle -\frac {a \int -\cot ^7(c+d x)d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {3}{5} \csc ^5(c+d x)+\csc ^3(c+d x)-\csc (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {3}{5} \csc ^5(c+d x)+\csc ^3(c+d x)-\csc (c+d x)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^7*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
-1/8*(a*Cot[c + d*x]^8)/d - (a*(-Csc[c + d*x] + Csc[c + d*x]^3 - (3*Csc[c + d*x]^5)/5 + Csc[c + d*x]^7/7))/d
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, 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 = 0.34 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{8}}{8}+\frac {\csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{2}-\frac {3 \csc \left (d x +c \right )^{5}}{5}+\frac {3 \csc \left (d x +c \right )^{4}}{4}+\csc \left (d x +c \right )^{3}-\frac {\csc \left (d x +c \right )^{2}}{2}-\csc \left (d x +c \right )\right )}{d}\) | \(84\) |
| default | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{8}}{8}+\frac {\csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{2}-\frac {3 \csc \left (d x +c \right )^{5}}{5}+\frac {3 \csc \left (d x +c \right )^{4}}{4}+\csc \left (d x +c \right )^{3}-\frac {\csc \left (d x +c \right )^{2}}{2}-\csc \left (d x +c \right )\right )}{d}\) | \(84\) |
| risch | \(\frac {2 i a \left (35 i {\mathrm e}^{14 i \left (d x +c \right )}+35 \,{\mathrm e}^{15 i \left (d x +c \right )}-105 \,{\mathrm e}^{13 i \left (d x +c \right )}+245 i {\mathrm e}^{10 i \left (d x +c \right )}+371 \,{\mathrm e}^{11 i \left (d x +c \right )}-513 \,{\mathrm e}^{9 i \left (d x +c \right )}+245 i {\mathrm e}^{6 i \left (d x +c \right )}+513 \,{\mathrm e}^{7 i \left (d x +c \right )}-371 \,{\mathrm e}^{5 i \left (d x +c \right )}+35 i {\mathrm e}^{2 i \left (d x +c \right )}+105 \,{\mathrm e}^{3 i \left (d x +c \right )}-35 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{35 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\) | \(158\) |
Input:
int(cot(d*x+c)^7*csc(d*x+c)^2*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-a/d*(1/8*csc(d*x+c)^8+1/7*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+csc(d*x+c)^3-1/2*csc(d*x+c)^2-csc(d*x+c))
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.77 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {140 \, a \cos \left (d x + c\right )^{6} - 210 \, a \cos \left (d x + c\right )^{4} + 140 \, a \cos \left (d x + c\right )^{2} + 8 \, {\left (35 \, a \cos \left (d x + c\right )^{6} - 70 \, a \cos \left (d x + c\right )^{4} + 56 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 35 \, a}{280 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/280*(140*a*cos(d*x + c)^6 - 210*a*cos(d*x + c)^4 + 140*a*cos(d*x + c)^2 + 8*(35*a*cos(d*x + c)^6 - 70*a*cos(d*x + c)^4 + 56*a*cos(d*x + c)^2 - 16 *a)*sin(d*x + c) - 35*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)
Timed out. \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**7*csc(d*x+c)**2*(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {280 \, a \sin \left (d x + c\right )^{7} + 140 \, a \sin \left (d x + c\right )^{6} - 280 \, a \sin \left (d x + c\right )^{5} - 210 \, a \sin \left (d x + c\right )^{4} + 168 \, a \sin \left (d x + c\right )^{3} + 140 \, a \sin \left (d x + c\right )^{2} - 40 \, a \sin \left (d x + c\right ) - 35 \, a}{280 \, d \sin \left (d x + c\right )^{8}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/280*(280*a*sin(d*x + c)^7 + 140*a*sin(d*x + c)^6 - 280*a*sin(d*x + c)^5 - 210*a*sin(d*x + c)^4 + 168*a*sin(d*x + c)^3 + 140*a*sin(d*x + c)^2 - 40* a*sin(d*x + c) - 35*a)/(d*sin(d*x + c)^8)
Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {280 \, a \sin \left (d x + c\right )^{7} + 140 \, a \sin \left (d x + c\right )^{6} - 280 \, a \sin \left (d x + c\right )^{5} - 210 \, a \sin \left (d x + c\right )^{4} + 168 \, a \sin \left (d x + c\right )^{3} + 140 \, a \sin \left (d x + c\right )^{2} - 40 \, a \sin \left (d x + c\right ) - 35 \, a}{280 \, d \sin \left (d x + c\right )^{8}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/280*(280*a*sin(d*x + c)^7 + 140*a*sin(d*x + c)^6 - 280*a*sin(d*x + c)^5 - 210*a*sin(d*x + c)^4 + 168*a*sin(d*x + c)^3 + 140*a*sin(d*x + c)^2 - 40* a*sin(d*x + c) - 35*a)/(d*sin(d*x + c)^8)
Time = 33.88 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-a\,{\sin \left (c+d\,x\right )}^7-\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}+a\,{\sin \left (c+d\,x\right )}^5+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {a\,\sin \left (c+d\,x\right )}{7}+\frac {a}{8}}{d\,{\sin \left (c+d\,x\right )}^8} \] Input:
int((cot(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^2,x)
Output:
-(a/8 + (a*sin(c + d*x))/7 - (a*sin(c + d*x)^2)/2 - (3*a*sin(c + d*x)^3)/5 + (3*a*sin(c + d*x)^4)/4 + a*sin(c + d*x)^5 - (a*sin(c + d*x)^6)/2 - a*si n(c + d*x)^7)/(d*sin(c + d*x)^8)
Time = 0.17 (sec) , antiderivative size = 288, normalized size of antiderivative = 3.89 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-120 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{5} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )+292 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )-603 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )-840 \cot \left (d x +c \right )^{6} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}-840 \cot \left (d x +c \right )^{6} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )+860 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}+840 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )-933 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}-840 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )-603 \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}+840 \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )+3675\right )}{6720 \sin \left (d x +c \right ) d} \] Input:
int(cot(d*x+c)^7*csc(d*x+c)^2*(a+a*sin(d*x+c)),x)
Output:
(a*( - 120*cos(c + d*x)*cot(c + d*x)**5*csc(c + d*x)**2*sin(c + d*x) + 292 *cos(c + d*x)*cot(c + d*x)**3*csc(c + d*x)**2*sin(c + d*x) - 603*cos(c + d *x)*cot(c + d*x)*csc(c + d*x)**2*sin(c + d*x) - 840*cot(c + d*x)**6*csc(c + d*x)**2*sin(c + d*x)**2 - 840*cot(c + d*x)**6*csc(c + d*x)**2*sin(c + d* x) + 860*cot(c + d*x)**4*csc(c + d*x)**2*sin(c + d*x)**2 + 840*cot(c + d*x )**4*csc(c + d*x)**2*sin(c + d*x) - 933*cot(c + d*x)**2*csc(c + d*x)**2*si n(c + d*x)**2 - 840*cot(c + d*x)**2*csc(c + d*x)**2*sin(c + d*x) - 603*csc (c + d*x)**2*sin(c + d*x)**2 + 840*csc(c + d*x)**2*sin(c + d*x) + 3675))/( 6720*sin(c + d*x)*d)