\(\int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx\) [509]

Optimal result

Integrand size = 27, antiderivative size = 97 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {2 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \]

[Out]

-1/5*a*csc(d*x+c)^5/d-1/6*a*csc(d*x+c)^6/d+2/7*a*csc(d*x+c)^7/d+1/4*a*csc(d*x+c)^8/d-1/9*a*csc(d*x+c)^9/d-1/10
*a*csc(d*x+c)^10/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 90} \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^5(c+d x)}{5 d} \]

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]

[Out]

-1/5*(a*Csc[c + d*x]^5)/d - (a*Csc[c + d*x]^6)/(6*d) + (2*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) -
(a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^{11} (a-x)^2 (a+x)^3}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {a^6 \text {Subst}\left (\int \frac {(a-x)^2 (a+x)^3}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^6 \text {Subst}\left (\int \left (\frac {a^5}{x^{11}}+\frac {a^4}{x^{10}}-\frac {2 a^3}{x^9}-\frac {2 a^2}{x^8}+\frac {a}{x^7}+\frac {1}{x^6}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {2 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {2 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \]

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]

[Out]

-1/5*(a*Csc[c + d*x]^5)/d - (a*Csc[c + d*x]^6)/(6*d) + (2*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) -
(a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{4}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\)	\(68\)
default	\(-\frac {a \left (\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{4}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\)	\(68\)
risch	\(-\frac {32 i a \left (105 i {\mathrm e}^{14 i \left (d x +c \right )}+63 \,{\mathrm e}^{15 i \left (d x +c \right )}+210 i {\mathrm e}^{12 i \left (d x +c \right )}+45 \,{\mathrm e}^{13 i \left (d x +c \right )}+378 i {\mathrm e}^{10 i \left (d x +c \right )}+110 \,{\mathrm e}^{11 i \left (d x +c \right )}+210 i {\mathrm e}^{8 i \left (d x +c \right )}-110 \,{\mathrm e}^{9 i \left (d x +c \right )}+105 i {\mathrm e}^{6 i \left (d x +c \right )}-45 \,{\mathrm e}^{7 i \left (d x +c \right )}-63 \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}\)	\(148\)
parallelrisch	\(-\frac {\left (\tan ^{20}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {20 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {20 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {25 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-16 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {80 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+50 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+50 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {80 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {25 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}+1\right ) a}{10240 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}\)	\(211\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^11*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/10*csc(d*x+c)^10+1/9*csc(d*x+c)^9-1/4*csc(d*x+c)^8-2/7*csc(d*x+c)^7+1/6*csc(d*x+c)^6+1/5*csc(d*x+c)^5)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {210 \, a \cos \left (d x + c\right )^{4} - 105 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (63 \, a \cos \left (d x + c\right )^{4} - 36 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) + 21 \, a}{1260 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/1260*(210*a*cos(d*x + c)^4 - 105*a*cos(d*x + c)^2 + 4*(63*a*cos(d*x + c)^4 - 36*a*cos(d*x + c)^2 + 8*a)*sin(
d*x + c) + 21*a)/(d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos
(d*x + c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**11*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {252 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, a \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, a \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/1260*(252*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*a*sin(d*x + c)^3 - 315*a*sin(d*x + c)^2 + 140*a*sin
(d*x + c) + 126*a)/(d*sin(d*x + c)^10)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {252 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, a \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, a \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/1260*(252*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*a*sin(d*x + c)^3 - 315*a*sin(d*x + c)^2 + 140*a*sin
(d*x + c) + 126*a)/(d*sin(d*x + c)^10)

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {252\,a\,{\sin \left (c+d\,x\right )}^5+210\,a\,{\sin \left (c+d\,x\right )}^4-360\,a\,{\sin \left (c+d\,x\right )}^3-315\,a\,{\sin \left (c+d\,x\right )}^2+140\,a\,\sin \left (c+d\,x\right )+126\,a}{1260\,d\,{\sin \left (c+d\,x\right )}^{10}} \]

[In]

int((cos(c + d*x)^5*(a + a*sin(c + d*x)))/sin(c + d*x)^11,x)

[Out]

-(126*a + 140*a*sin(c + d*x) - 315*a*sin(c + d*x)^2 - 360*a*sin(c + d*x)^3 + 210*a*sin(c + d*x)^4 + 252*a*sin(
c + d*x)^5)/(1260*d*sin(c + d*x)^10)