Integrand size = 27, antiderivative size = 65 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \cot ^6(c+d x)}{6 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \] Output:
-1/6*b*cot(d*x+c)^6/d-1/3*a*csc(d*x+c)^3/d+2/5*a*csc(d*x+c)^5/d-1/7*a*csc( d*x+c)^7/d
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \cot ^6(c+d x)}{6 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \] Input:
Integrate[Cot[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x]),x]
Output:
-1/6*(b*Cot[c + d*x]^6)/d - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5 )/(5*d) - (a*Csc[c + d*x]^7)/(7*d)
Time = 0.43 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92, 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, 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.
\(\displaystyle \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5 (a+b \sin (c+d x))}{\sin (c+d x)^8}dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cot ^5(c+d x) \csc ^3(c+d x)dx+b \int \cot ^5(c+d x) \csc ^2(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx+b \int -\sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx-b \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {a \int \csc ^2(c+d x) \left (1-\csc ^2(c+d x)\right )^2d\csc (c+d x)}{d}-b \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {a \int \left (\csc ^6(c+d x)-2 \csc ^4(c+d x)+\csc ^2(c+d x)\right )d\csc (c+d x)}{d}-b \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -b \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx-\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {2}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle -\frac {b \int -\cot ^5(c+d x)d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {2}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {2}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)\right )}{d}-\frac {b \cot ^6(c+d x)}{6 d}\) |
Input:
Int[Cot[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x]),x]
Output:
-1/6*(b*Cot[c + d*x]^6)/d - (a*(Csc[c + d*x]^3/3 - (2*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[((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 = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(-\frac {\frac {a \csc \left (d x +c \right )^{7}}{7}+\frac {b \csc \left (d x +c \right )^{6}}{6}-\frac {2 a \csc \left (d x +c \right )^{5}}{5}-\frac {b \csc \left (d x +c \right )^{4}}{2}+\frac {a \csc \left (d x +c \right )^{3}}{3}+\frac {b \csc \left (d x +c \right )^{2}}{2}}{d}\) | \(73\) |
default | \(-\frac {\frac {a \csc \left (d x +c \right )^{7}}{7}+\frac {b \csc \left (d x +c \right )^{6}}{6}-\frac {2 a \csc \left (d x +c \right )^{5}}{5}-\frac {b \csc \left (d x +c \right )^{4}}{2}+\frac {a \csc \left (d x +c \right )^{3}}{3}+\frac {b \csc \left (d x +c \right )^{2}}{2}}{d}\) | \(73\) |
risch | \(\frac {\frac {8 i a \,{\mathrm e}^{11 i \left (d x +c \right )}}{3}+2 b \,{\mathrm e}^{12 i \left (d x +c \right )}+\frac {32 i a \,{\mathrm e}^{9 i \left (d x +c \right )}}{15}-2 b \,{\mathrm e}^{10 i \left (d x +c \right )}+\frac {304 i a \,{\mathrm e}^{7 i \left (d x +c \right )}}{35}+\frac {20 b \,{\mathrm e}^{8 i \left (d x +c \right )}}{3}+\frac {32 i a \,{\mathrm e}^{5 i \left (d x +c \right )}}{15}-\frac {20 b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}+2 b \,{\mathrm e}^{4 i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\) | \(157\) |
Input:
int(cot(d*x+c)^5*csc(d*x+c)^3*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/d*(1/7*a*csc(d*x+c)^7+1/6*b*csc(d*x+c)^6-2/5*a*csc(d*x+c)^5-1/2*b*csc(d *x+c)^4+1/3*a*csc(d*x+c)^3+1/2*b*csc(d*x+c)^2)
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.63 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {70 \, a \cos \left (d x + c\right )^{4} - 56 \, a \cos \left (d x + c\right )^{2} + 35 \, {\left (3 \, b \cos \left (d x + c\right )^{4} - 3 \, b \cos \left (d x + c\right )^{2} + b\right )} \sin \left (d x + c\right ) + 16 \, a}{210 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/210*(70*a*cos(d*x + c)^4 - 56*a*cos(d*x + c)^2 + 35*(3*b*cos(d*x + c)^4 - 3*b*cos(d*x + c)^2 + b)*sin(d*x + c) + 16*a)/((d*cos(d*x + c)^6 - 3*d*co s(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**5*csc(d*x+c)**3*(a+b*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.08 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {105 \, b \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, b \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, b \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/210*(105*b*sin(d*x + c)^5 + 70*a*sin(d*x + c)^4 - 105*b*sin(d*x + c)^3 - 84*a*sin(d*x + c)^2 + 35*b*sin(d*x + c) + 30*a)/(d*sin(d*x + c)^7)
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.08 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {105 \, b \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, b \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, b \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
-1/210*(105*b*sin(d*x + c)^5 + 70*a*sin(d*x + c)^4 - 105*b*sin(d*x + c)^3 - 84*a*sin(d*x + c)^2 + 35*b*sin(d*x + c) + 30*a)/(d*sin(d*x + c)^7)
Time = 21.80 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.08 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {105\,b\,{\sin \left (c+d\,x\right )}^5+70\,a\,{\sin \left (c+d\,x\right )}^4-105\,b\,{\sin \left (c+d\,x\right )}^3-84\,a\,{\sin \left (c+d\,x\right )}^2+35\,b\,\sin \left (c+d\,x\right )+30\,a}{210\,d\,{\sin \left (c+d\,x\right )}^7} \] Input:
int((cot(c + d*x)^5*(a + b*sin(c + d*x)))/sin(c + d*x)^3,x)
Output:
-(30*a + 35*b*sin(c + d*x) - 84*a*sin(c + d*x)^2 + 70*a*sin(c + d*x)^4 - 1 05*b*sin(c + d*x)^3 + 105*b*sin(c + d*x)^5)/(210*d*sin(c + d*x)^7)
Time = 0.17 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.66 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {-10 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} b +20 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} b -60 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3} b -60 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} a +50 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3} b +48 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} a +20 \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3} b -32 \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} a +45 \sin \left (d x +c \right )^{2} b -90 b}{420 \sin \left (d x +c \right )^{2} d} \] Input:
int(cot(d*x+c)^5*csc(d*x+c)^3*(a+b*sin(d*x+c)),x)
Output:
( - 10*cos(c + d*x)*cot(c + d*x)**3*csc(c + d*x)**3*sin(c + d*x)**2*b + 20 *cos(c + d*x)*cot(c + d*x)*csc(c + d*x)**3*sin(c + d*x)**2*b - 60*cot(c + d*x)**4*csc(c + d*x)**3*sin(c + d*x)**3*b - 60*cot(c + d*x)**4*csc(c + d*x )**3*sin(c + d*x)**2*a + 50*cot(c + d*x)**2*csc(c + d*x)**3*sin(c + d*x)** 3*b + 48*cot(c + d*x)**2*csc(c + d*x)**3*sin(c + d*x)**2*a + 20*csc(c + d* x)**3*sin(c + d*x)**3*b - 32*csc(c + d*x)**3*sin(c + d*x)**2*a + 45*sin(c + d*x)**2*b - 90*b)/(420*sin(c + d*x)**2*d)