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

Optimal. Leaf size=109 \[ -\frac {\csc ^{12}(c+d x)}{12 a d}+\frac {\csc ^{11}(c+d x)}{11 a d}+\frac {\csc ^{10}(c+d x)}{5 a d}-\frac {2 \csc ^9(c+d x)}{9 a d}-\frac {\csc ^8(c+d x)}{8 a d}+\frac {\csc ^7(c+d x)}{7 a d} \]

[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

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Rubi [A]  time = 0.12, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac {\csc ^{12}(c+d x)}{12 a d}+\frac {\csc ^{11}(c+d x)}{11 a d}+\frac {\csc ^{10}(c+d x)}{5 a d}-\frac {2 \csc ^9(c+d x)}{9 a d}-\frac {\csc ^8(c+d x)}{8 a d}+\frac {\csc ^7(c+d x)}{7 a d} \]

Antiderivative was successfully verified.

[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 88

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 2836

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*x)/b
)^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*} \int \frac {\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {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 \operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {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*}

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Mathematica [A]  time = 0.13, size = 68, normalized size = 0.62 \[ \frac {\csc ^7(c+d x) \left (-2310 \csc ^5(c+d x)+2520 \csc ^4(c+d x)+5544 \csc ^3(c+d x)-6160 \csc ^2(c+d x)-3465 \csc (c+d x)+3960\right )}{27720 a d} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.50, size = 131, normalized size = 1.20 \[ -\frac {3465 \, \cos \left (d x + c\right )^{4} - 1386 \, \cos \left (d x + c\right )^{2} - 40 \, {\left (99 \, \cos \left (d x + c\right )^{4} - 44 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 231}{27720 \, {\left (a d \cos \left (d x + c\right )^{12} - 6 \, a d \cos \left (d x + c\right )^{10} + 15 \, a d \cos \left (d x + c\right )^{8} - 20 \, a d \cos \left (d x + c\right )^{6} + 15 \, a d \cos \left (d x + c\right )^{4} - 6 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.27, size = 66, normalized size = 0.61 \[ \frac {3960 \, \sin \left (d x + c\right )^{5} - 3465 \, \sin \left (d x + c\right )^{4} - 6160 \, \sin \left (d x + c\right )^{3} + 5544 \, \sin \left (d x + c\right )^{2} + 2520 \, \sin \left (d x + c\right ) - 2310}{27720 \, a d \sin \left (d x + c\right )^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.66, size = 69, normalized size = 0.63 \[ \frac {\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {2}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {1}{11 \sin \left (d x +c \right )^{11}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}+\frac {1}{5 \sin \left (d x +c \right )^{10}}}{d a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 66, normalized size = 0.61 \[ \frac {3960 \, \sin \left (d x + c\right )^{5} - 3465 \, \sin \left (d x + c\right )^{4} - 6160 \, \sin \left (d x + c\right )^{3} + 5544 \, \sin \left (d x + c\right )^{2} + 2520 \, \sin \left (d x + c\right ) - 2310}{27720 \, a d \sin \left (d x + c\right )^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 9.14, size = 65, normalized size = 0.60 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^5}{7}-\frac {{\sin \left (c+d\,x\right )}^4}{8}-\frac {2\,{\sin \left (c+d\,x\right )}^3}{9}+\frac {{\sin \left (c+d\,x\right )}^2}{5}+\frac {\sin \left (c+d\,x\right )}{11}-\frac {1}{12}}{a\,d\,{\sin \left (c+d\,x\right )}^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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