\(\int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx\) [508]

Optimal result

Integrand size = 27, antiderivative size = 81 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \]

[Out]

-1/6*a*cot(d*x+c)^6/d-1/8*a*cot(d*x+c)^8/d-1/5*a*csc(d*x+c)^5/d+2/7*a*csc(d*x+c)^7/d-1/9*a*csc(d*x+c)^9/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2913, 2686, 276, 2687, 14} \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d} \]

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-1/6*(a*Cot[c + d*x]^6)/d - (a*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^5)/(5*d) + (2*a*Csc[c + d*x]^7)/(7*d) -
(a*Csc[c + d*x]^9)/(9*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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])

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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])

Rule 2913

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[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])

Rubi steps \begin{align*} \text {integral}& = a \int \cot ^5(c+d x) \csc ^4(c+d x) \, dx+a \int \cot ^5(c+d x) \csc ^5(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \text {Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \text {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac {a \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^6(c+d x)}{3 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d} \]

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-1/4*(a*Csc[c + d*x]^4)/d - (a*Csc[c + d*x]^5)/(5*d) + (a*Csc[c + d*x]^6)/(3*d) + (2*a*Csc[c + d*x]^7)/(7*d) -
(a*Csc[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^9)/(9*d)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\)	\(68\)
default	\(-\frac {a \left (\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\)	\(68\)
parallelrisch	\(-\frac {a \left (3538944 \cos \left (2 d x +2 c \right )-7665 \sin \left (9 d x +9 c \right )+68985 \sin \left (7 d x +7 c \right )+1014300 \sin \left (5 d x +5 c \right )+1614690 \sin \left (d x +c \right )+1073940 \sin \left (3 d x +3 c \right )+2064384 \cos \left (4 d x +4 c \right )+3571712\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42278584320 d}\)	\(105\)
risch	\(-\frac {4 a \left (504 i {\mathrm e}^{13 i \left (d x +c \right )}+315 \,{\mathrm e}^{14 i \left (d x +c \right )}+864 i {\mathrm e}^{11 i \left (d x +c \right )}+105 \,{\mathrm e}^{12 i \left (d x +c \right )}+1744 i {\mathrm e}^{9 i \left (d x +c \right )}+630 \,{\mathrm e}^{10 i \left (d x +c \right )}+864 i {\mathrm e}^{7 i \left (d x +c \right )}-630 \,{\mathrm e}^{8 i \left (d x +c \right )}+504 i {\mathrm e}^{5 i \left (d x +c \right )}-105 \,{\mathrm e}^{6 i \left (d x +c \right )}-315 \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}\)	\(147\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^10*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/9*csc(d*x+c)^9+1/8*csc(d*x+c)^8-2/7*csc(d*x+c)^7-1/3*csc(d*x+c)^6+1/5*csc(d*x+c)^5+1/4*csc(d*x+c)^4)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.42 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {504 \, a \cos \left (d x + c\right )^{4} - 288 \, a \cos \left (d x + c\right )^{2} + 105 \, {\left (6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 64 \, a}{2520 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^10*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2520*(504*a*cos(d*x + c)^4 - 288*a*cos(d*x + c)^2 + 105*(6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^2 + a)*sin(d
*x + c) + 64*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))

Sympy [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**10*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {630 \, a \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, a \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^10*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2520*(630*a*sin(d*x + c)^5 + 504*a*sin(d*x + c)^4 - 840*a*sin(d*x + c)^3 - 720*a*sin(d*x + c)^2 + 315*a*sin
(d*x + c) + 280*a)/(d*sin(d*x + c)^9)

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {630 \, a \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, a \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^10*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/2520*(630*a*sin(d*x + c)^5 + 504*a*sin(d*x + c)^4 - 840*a*sin(d*x + c)^3 - 720*a*sin(d*x + c)^2 + 315*a*sin
(d*x + c) + 280*a)/(d*sin(d*x + c)^9)

Mupad [B] (verification not implemented)

Time = 10.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a\,{\sin \left (c+d\,x\right )}^5}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {a\,\sin \left (c+d\,x\right )}{8}+\frac {a}{9}}{d\,{\sin \left (c+d\,x\right )}^9} \]

[In]

int((cos(c + d*x)^5*(a + a*sin(c + d*x)))/sin(c + d*x)^10,x)

[Out]

-(a/9 + (a*sin(c + d*x))/8 - (2*a*sin(c + d*x)^2)/7 - (a*sin(c + d*x)^3)/3 + (a*sin(c + d*x)^4)/5 + (a*sin(c +
d*x)^5)/4)/(d*sin(c + d*x)^9)