3.70 \(\int \frac{\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=145 \[ -\frac{\csc ^{10}(c+d x)}{10 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}+\frac{\csc ^8(c+d x)}{4 a^2 d}-\frac{6 \csc ^7(c+d x)}{7 a^2 d}+\frac{6 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^4(c+d x)}{2 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d} \]

[Out]

Csc[c + d*x]^2/(2*a^2*d) - (2*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(2*a^2*d) + (6*Csc[c + d*x]^5)/(5*a^2
*d) - (6*Csc[c + d*x]^7)/(7*a^2*d) + Csc[c + d*x]^8/(4*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d) - Csc[c + d*x]^10
/(10*a^2*d)

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Rubi [A]  time = 0.0811955, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 88} \[ -\frac{\csc ^{10}(c+d x)}{10 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}+\frac{\csc ^8(c+d x)}{4 a^2 d}-\frac{6 \csc ^7(c+d x)}{7 a^2 d}+\frac{6 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^4(c+d x)}{2 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^2,x]

[Out]

Csc[c + d*x]^2/(2*a^2*d) - (2*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(2*a^2*d) + (6*Csc[c + d*x]^5)/(5*a^2
*d) - (6*Csc[c + d*x]^7)/(7*a^2*d) + Csc[c + d*x]^8/(4*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d) - Csc[c + d*x]^10
/(10*a^2*d)

Rule 2707

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

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]))

Rubi steps

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

Mathematica [A]  time = 0.205023, size = 88, normalized size = 0.61 \[ \frac{\csc ^2(c+d x) \left (-126 \csc ^8(c+d x)+280 \csc ^7(c+d x)+315 \csc ^6(c+d x)-1080 \csc ^5(c+d x)+1512 \csc ^3(c+d x)-630 \csc ^2(c+d x)-840 \csc (c+d x)+630\right )}{1260 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^2,x]

[Out]

(Csc[c + d*x]^2*(630 - 840*Csc[c + d*x] - 630*Csc[c + d*x]^2 + 1512*Csc[c + d*x]^3 - 1080*Csc[c + d*x]^5 + 315
*Csc[c + d*x]^6 + 280*Csc[c + d*x]^7 - 126*Csc[c + d*x]^8))/(1260*a^2*d)

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Maple [A]  time = 0.139, size = 89, normalized size = 0.6 \begin{align*}{\frac{1}{d{a}^{2}} \left ( -{\frac{1}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{6}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}+{\frac{6}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{2}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{2}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x)

[Out]

1/d/a^2*(-1/10/sin(d*x+c)^10-6/7/sin(d*x+c)^7+1/4/sin(d*x+c)^8+6/5/sin(d*x+c)^5-1/2/sin(d*x+c)^4+2/9/sin(d*x+c
)^9-2/3/sin(d*x+c)^3+1/2/sin(d*x+c)^2)

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Maxima [A]  time = 1.83669, size = 116, normalized size = 0.8 \begin{align*} \frac{630 \, \sin \left (d x + c\right )^{8} - 840 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 1512 \, \sin \left (d x + c\right )^{5} - 1080 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 126}{1260 \, a^{2} d \sin \left (d x + c\right )^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/1260*(630*sin(d*x + c)^8 - 840*sin(d*x + c)^7 - 630*sin(d*x + c)^6 + 1512*sin(d*x + c)^5 - 1080*sin(d*x + c)
^3 + 315*sin(d*x + c)^2 + 280*sin(d*x + c) - 126)/(a^2*d*sin(d*x + c)^10)

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Fricas [A]  time = 1.67582, size = 431, normalized size = 2.97 \begin{align*} -\frac{630 \, \cos \left (d x + c\right )^{8} - 1890 \, \cos \left (d x + c\right )^{6} + 1890 \, \cos \left (d x + c\right )^{4} - 945 \, \cos \left (d x + c\right )^{2} + 8 \,{\left (105 \, \cos \left (d x + c\right )^{6} - 126 \, \cos \left (d x + c\right )^{4} + 72 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 189}{1260 \,{\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/1260*(630*cos(d*x + c)^8 - 1890*cos(d*x + c)^6 + 1890*cos(d*x + c)^4 - 945*cos(d*x + c)^2 + 8*(105*cos(d*x
+ c)^6 - 126*cos(d*x + c)^4 + 72*cos(d*x + c)^2 - 16)*sin(d*x + c) + 189)/(a^2*d*cos(d*x + c)^10 - 5*a^2*d*cos
(d*x + c)^8 + 10*a^2*d*cos(d*x + c)^6 - 10*a^2*d*cos(d*x + c)^4 + 5*a^2*d*cos(d*x + c)^2 - a^2*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**11/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.93118, size = 116, normalized size = 0.8 \begin{align*} \frac{630 \, \sin \left (d x + c\right )^{8} - 840 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 1512 \, \sin \left (d x + c\right )^{5} - 1080 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 126}{1260 \, a^{2} d \sin \left (d x + c\right )^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1260*(630*sin(d*x + c)^8 - 840*sin(d*x + c)^7 - 630*sin(d*x + c)^6 + 1512*sin(d*x + c)^5 - 1080*sin(d*x + c)
^3 + 315*sin(d*x + c)^2 + 280*sin(d*x + c) - 126)/(a^2*d*sin(d*x + c)^10)