\(\int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx\) [696]

Optimal result

Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^7(c+d x)}{7 a d}-\frac {\csc ^8(c+d x)}{8 a d}-\frac {2 \csc ^9(c+d x)}{9 a d}+\frac {\csc ^{10}(c+d x)}{5 a d}+\frac {\csc ^{11}(c+d x)}{11 a d}-\frac {\csc ^{12}(c+d x)}{12 a d} \]

[Out]

1/7*csc(d*x+c)^7/a/d-1/8*csc(d*x+c)^8/a/d-2/9*csc(d*x+c)^9/a/d+1/5*csc(d*x+c)^10/a/d+1/11*csc(d*x+c)^11/a/d-1/
12*csc(d*x+c)^12/a/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90} \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^{12}(c+d x)}{12 a d}+\frac {\csc ^{11}(c+d x)}{11 a d}+\frac {\csc ^{10}(c+d x)}{5 a d}-\frac {2 \csc ^9(c+d x)}{9 a d}-\frac {\csc ^8(c+d x)}{8 a d}+\frac {\csc ^7(c+d x)}{7 a d} \]

[In]

Int[(Cot[c + d*x]^7*Csc[c + d*x]^6)/(a + a*Sin[c + d*x]),x]

[Out]

Csc[c + d*x]^7/(7*a*d) - Csc[c + d*x]^8/(8*a*d) - (2*Csc[c + d*x]^9)/(9*a*d) + Csc[c + d*x]^10/(5*a*d) + Csc[c
+ d*x]^11/(11*a*d) - Csc[c + d*x]^12/(12*a*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^{13} (a-x)^3 (a+x)^2}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {a^6 \text {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^6 \text {Subst}\left (\int \left (\frac {a^5}{x^{13}}-\frac {a^4}{x^{12}}-\frac {2 a^3}{x^{11}}+\frac {2 a^2}{x^{10}}+\frac {a}{x^9}-\frac {1}{x^8}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\csc ^7(c+d x)}{7 a d}-\frac {\csc ^8(c+d x)}{8 a d}-\frac {2 \csc ^9(c+d x)}{9 a d}+\frac {\csc ^{10}(c+d x)}{5 a d}+\frac {\csc ^{11}(c+d x)}{11 a d}-\frac {\csc ^{12}(c+d x)}{12 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^7(c+d x) \left (3960-3465 \csc (c+d x)-6160 \csc ^2(c+d x)+5544 \csc ^3(c+d x)+2520 \csc ^4(c+d x)-2310 \csc ^5(c+d x)\right )}{27720 a d} \]

[In]

Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^6)/(a + a*Sin[c + d*x]),x]

[Out]

(Csc[c + d*x]^7*(3960 - 3465*Csc[c + d*x] - 6160*Csc[c + d*x]^2 + 5544*Csc[c + d*x]^3 + 2520*Csc[c + d*x]^4 -
2310*Csc[c + d*x]^5))/(27720*a*d)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{12}-\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{5}+\frac {2 \left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}}{d a}\)	\(70\)
default	\(-\frac {\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{12}-\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{5}+\frac {2 \left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}}{d a}\)	\(70\)
parallelrisch	\(\frac {\left (-853398546-1132256664 \cos \left (2 d x +2 c \right )+3521826 \cos \left (8 d x +8 c \right )[phone] \sin \left (5 d x +5 c \right )-11739420 \cos \left (6 d x +6 c \right )[phone] \sin \left (d x +c \right )[phone] \sin \left (3 d x +3 c \right )-427750785 \cos \left (4 d x +4 c \right )+53361 \cos \left (12 d x +12 c \right )-640332 \cos \left (10 d x +10 c \right )\right ) \left (\sec ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{119056493445120 a d}\)	\(129\)
risch	\(-\frac {32 i \left (-3465 i {\mathrm e}^{16 i \left (d x +c \right )}+1980 \,{\mathrm e}^{17 i \left (d x +c \right )}-8316 i {\mathrm e}^{14 i \left (d x +c \right )}+2420 \,{\mathrm e}^{15 i \left (d x +c \right )}-13398 i {\mathrm e}^{12 i \left (d x +c \right )}+3000 \,{\mathrm e}^{13 i \left (d x +c \right )}-8316 i {\mathrm e}^{10 i \left (d x +c \right )}-3000 \,{\mathrm e}^{11 i \left (d x +c \right )}-3465 i {\mathrm e}^{8 i \left (d x +c \right )}-2420 \,{\mathrm e}^{9 i \left (d x +c \right )}-1980 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{3465 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}\)	\(150\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^13/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/d/a*(1/12*csc(d*x+c)^12-1/11*csc(d*x+c)^11-1/5*csc(d*x+c)^10+2/9*csc(d*x+c)^9+1/8*csc(d*x+c)^8-1/7*csc(d*x+
c)^7)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.20 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3465 \, \cos \left (d x + c\right )^{4} - 1386 \, \cos \left (d x + c\right )^{2} - 40 \, {\left (99 \, \cos \left (d x + c\right )^{4} - 44 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 231}{27720 \, {\left (a d \cos \left (d x + c\right )^{12} - 6 \, a d \cos \left (d x + c\right )^{10} + 15 \, a d \cos \left (d x + c\right )^{8} - 20 \, a d \cos \left (d x + c\right )^{6} + 15 \, a d \cos \left (d x + c\right )^{4} - 6 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^13/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/27720*(3465*cos(d*x + c)^4 - 1386*cos(d*x + c)^2 - 40*(99*cos(d*x + c)^4 - 44*cos(d*x + c)^2 + 8)*sin(d*x +
c) + 231)/(a*d*cos(d*x + c)^12 - 6*a*d*cos(d*x + c)^10 + 15*a*d*cos(d*x + c)^8 - 20*a*d*cos(d*x + c)^6 + 15*a
*d*cos(d*x + c)^4 - 6*a*d*cos(d*x + c)^2 + a*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**13/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3960 \, \sin \left (d x + c\right )^{5} - 3465 \, \sin \left (d x + c\right )^{4} - 6160 \, \sin \left (d x + c\right )^{3} + 5544 \, \sin \left (d x + c\right )^{2} + 2520 \, \sin \left (d x + c\right ) - 2310}{27720 \, a d \sin \left (d x + c\right )^{12}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^13/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/27720*(3960*sin(d*x + c)^5 - 3465*sin(d*x + c)^4 - 6160*sin(d*x + c)^3 + 5544*sin(d*x + c)^2 + 2520*sin(d*x
+ c) - 2310)/(a*d*sin(d*x + c)^12)

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3960 \, \sin \left (d x + c\right )^{5} - 3465 \, \sin \left (d x + c\right )^{4} - 6160 \, \sin \left (d x + c\right )^{3} + 5544 \, \sin \left (d x + c\right )^{2} + 2520 \, \sin \left (d x + c\right ) - 2310}{27720 \, a d \sin \left (d x + c\right )^{12}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^13/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/27720*(3960*sin(d*x + c)^5 - 3465*sin(d*x + c)^4 - 6160*sin(d*x + c)^3 + 5544*sin(d*x + c)^2 + 2520*sin(d*x
+ c) - 2310)/(a*d*sin(d*x + c)^12)

Mupad [B] (verification not implemented)

Time = 9.96 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^5}{7}-\frac {{\sin \left (c+d\,x\right )}^4}{8}-\frac {2\,{\sin \left (c+d\,x\right )}^3}{9}+\frac {{\sin \left (c+d\,x\right )}^2}{5}+\frac {\sin \left (c+d\,x\right )}{11}-\frac {1}{12}}{a\,d\,{\sin \left (c+d\,x\right )}^{12}} \]

[In]

int(cos(c + d*x)^7/(sin(c + d*x)^13*(a + a*sin(c + d*x))),x)

[Out]

(sin(c + d*x)/11 + sin(c + d*x)^2/5 - (2*sin(c + d*x)^3)/9 - sin(c + d*x)^4/8 + sin(c + d*x)^5/7 - 1/12)/(a*d*
sin(c + d*x)^12)