Integrand size = 27, antiderivative size = 61 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc (c+d x)}{d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d} \] Output:
-1/6*a*cot(d*x+c)^6/d-a*csc(d*x+c)/d+2/3*a*csc(d*x+c)^3/d-1/5*a*csc(d*x+c) ^5/d
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc (c+d x)}{d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d} \] Input:
Integrate[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
-1/6*(a*Cot[c + d*x]^6)/d - (a*Csc[c + d*x])/d + (2*a*Csc[c + d*x]^3)/(3*d ) - (a*Csc[c + d*x]^5)/(5*d)
Time = 0.43 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89, 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.
\(\displaystyle \int \cot ^5(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)}{\sin (c+d x)^7}dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cot ^5(c+d x) \csc ^2(c+d x)dx+a \int \cot ^5(c+d x) \csc (c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^5dx+a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {a \int \left (\csc ^2(c+d x)-1\right )^2d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 210 |
\(\displaystyle -\frac {a \int \left (\csc ^4(c+d x)-2 \csc ^2(c+d x)+1\right )d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx-\frac {a \left (\frac {1}{5} \csc ^5(c+d x)-\frac {2}{3} \csc ^3(c+d x)+\csc (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle -\frac {a \int -\cot ^5(c+d x)d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{5} \csc ^5(c+d x)-\frac {2}{3} \csc ^3(c+d x)+\csc (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \left (\frac {1}{5} \csc ^5(c+d x)-\frac {2}{3} \csc ^3(c+d x)+\csc (c+d x)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
-1/6*(a*Cot[c + d*x]^6)/d - (a*(Csc[c + d*x] - (2*Csc[c + d*x]^3)/3 + Csc[ c + d*x]^5/5))/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 = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{6}}{6}+\frac {\csc \left (d x +c \right )^{5}}{5}-\frac {\csc \left (d x +c \right )^{4}}{2}-\frac {2 \csc \left (d x +c \right )^{3}}{3}+\frac {\csc \left (d x +c \right )^{2}}{2}+\csc \left (d x +c \right )\right )}{d}\) | \(64\) |
default | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{6}}{6}+\frac {\csc \left (d x +c \right )^{5}}{5}-\frac {\csc \left (d x +c \right )^{4}}{2}-\frac {2 \csc \left (d x +c \right )^{3}}{3}+\frac {\csc \left (d x +c \right )^{2}}{2}+\csc \left (d x +c \right )\right )}{d}\) | \(64\) |
risch | \(-\frac {2 i a \left (15 i {\mathrm e}^{10 i \left (d x +c \right )}+15 \,{\mathrm e}^{11 i \left (d x +c \right )}-35 \,{\mathrm e}^{9 i \left (d x +c \right )}+50 i {\mathrm e}^{6 i \left (d x +c \right )}+78 \,{\mathrm e}^{7 i \left (d x +c \right )}-78 \,{\mathrm e}^{5 i \left (d x +c \right )}+15 i {\mathrm e}^{2 i \left (d x +c \right )}+35 \,{\mathrm e}^{3 i \left (d x +c \right )}-15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}\) | \(124\) |
Input:
int(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-a/d*(1/6*csc(d*x+c)^6+1/5*csc(d*x+c)^5-1/2*csc(d*x+c)^4-2/3*csc(d*x+c)^3+ 1/2*csc(d*x+c)^2+csc(d*x+c))
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.64 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {15 \, a \cos \left (d x + c\right )^{4} - 15 \, a \cos \left (d x + c\right )^{2} + 2 \, {\left (15 \, a \cos \left (d x + c\right )^{4} - 20 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) + 5 \, a}{30 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/30*(15*a*cos(d*x + c)^4 - 15*a*cos(d*x + c)^2 + 2*(15*a*cos(d*x + c)^4 - 20*a*cos(d*x + c)^2 + 8*a)*sin(d*x + c) + 5*a)/(d*cos(d*x + c)^6 - 3*d*co s(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**5*csc(d*x+c)**2*(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {30 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} - 15 \, a \sin \left (d x + c\right )^{2} + 6 \, a \sin \left (d x + c\right ) + 5 \, a}{30 \, d \sin \left (d x + c\right )^{6}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/30*(30*a*sin(d*x + c)^5 + 15*a*sin(d*x + c)^4 - 20*a*sin(d*x + c)^3 - 1 5*a*sin(d*x + c)^2 + 6*a*sin(d*x + c) + 5*a)/(d*sin(d*x + c)^6)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {30 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} - 15 \, a \sin \left (d x + c\right )^{2} + 6 \, a \sin \left (d x + c\right ) + 5 \, a}{30 \, d \sin \left (d x + c\right )^{6}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/30*(30*a*sin(d*x + c)^5 + 15*a*sin(d*x + c)^4 - 20*a*sin(d*x + c)^3 - 1 5*a*sin(d*x + c)^2 + 6*a*sin(d*x + c) + 5*a)/(d*sin(d*x + c)^6)
Time = 18.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a\,{\sin \left (c+d\,x\right )}^5+\frac {a\,{\sin \left (c+d\,x\right )}^4}{2}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {a\,\sin \left (c+d\,x\right )}{5}+\frac {a}{6}}{d\,{\sin \left (c+d\,x\right )}^6} \] Input:
int((cot(c + d*x)^5*(a + a*sin(c + d*x)))/sin(c + d*x)^2,x)
Output:
-(a/6 + (a*sin(c + d*x))/5 - (a*sin(c + d*x)^2)/2 - (2*a*sin(c + d*x)^3)/3 + (a*sin(c + d*x)^4)/2 + a*sin(c + d*x)^5)/(d*sin(c + d*x)^6)
Time = 0.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 3.41 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-4 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )+11 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )-20 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}-20 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )+21 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}+20 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )+11 \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}-20 \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )-75\right )}{120 \sin \left (d x +c \right ) d} \] Input:
int(cot(d*x+c)^5*csc(d*x+c)^2*(a+a*sin(d*x+c)),x)
Output:
(a*( - 4*cos(c + d*x)*cot(c + d*x)**3*csc(c + d*x)**2*sin(c + d*x) + 11*co s(c + d*x)*cot(c + d*x)*csc(c + d*x)**2*sin(c + d*x) - 20*cot(c + d*x)**4* csc(c + d*x)**2*sin(c + d*x)**2 - 20*cot(c + d*x)**4*csc(c + d*x)**2*sin(c + d*x) + 21*cot(c + d*x)**2*csc(c + d*x)**2*sin(c + d*x)**2 + 20*cot(c + d*x)**2*csc(c + d*x)**2*sin(c + d*x) + 11*csc(c + d*x)**2*sin(c + d*x)**2 - 20*csc(c + d*x)**2*sin(c + d*x) - 75))/(120*sin(c + d*x)*d)