3.80 \(\int \frac{\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0805763, 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} \[ -\frac{\csc ^{10}(c+d x)}{10 a^3 d}+\frac{\csc ^9(c+d x)}{3 a^3 d}-\frac{\csc ^8(c+d x)}{8 a^3 d}-\frac{5 \csc ^7(c+d x)}{7 a^3 d}+\frac{5 \csc ^6(c+d x)}{6 a^3 d}+\frac{\csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^3,x]

[Out]

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

Mathematica [A]  time = 0.11307, size = 88, normalized size = 0.61 \[ \frac{\csc ^3(c+d x) \left (-84 \csc ^7(c+d x)+280 \csc ^6(c+d x)-105 \csc ^5(c+d x)-600 \csc ^4(c+d x)+700 \csc ^3(c+d x)+168 \csc ^2(c+d x)-630 \csc (c+d x)+280\right )}{840 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^3*(280 - 630*Csc[c + d*x] + 168*Csc[c + d*x]^2 + 700*Csc[c + d*x]^3 - 600*Csc[c + d*x]^4 - 105*C
sc[c + d*x]^5 + 280*Csc[c + d*x]^6 - 84*Csc[c + d*x]^7))/(840*a^3*d)

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Maple [A]  time = 0.153, size = 89, normalized size = 0.6 \begin{align*}{\frac{1}{d{a}^{3}} \left ( -{\frac{1}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{5}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}+{\frac{1}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}+{\frac{5}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \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))^3,x)

[Out]

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

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Maxima [A]  time = 1.96466, size = 116, normalized size = 0.8 \begin{align*} \frac{280 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 168 \, \sin \left (d x + c\right )^{5} + 700 \, \sin \left (d x + c\right )^{4} - 600 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 84}{840 \, a^{3} 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))^3,x, algorithm="maxima")

[Out]

1/840*(280*sin(d*x + c)^7 - 630*sin(d*x + c)^6 + 168*sin(d*x + c)^5 + 700*sin(d*x + c)^4 - 600*sin(d*x + c)^3
- 105*sin(d*x + c)^2 + 280*sin(d*x + c) - 84)/(a^3*d*sin(d*x + c)^10)

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Fricas [A]  time = 1.59062, size = 398, normalized size = 2.74 \begin{align*} -\frac{630 \, \cos \left (d x + c\right )^{6} - 1190 \, \cos \left (d x + c\right )^{4} + 595 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (35 \, \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 ) - 119}{840 \,{\left (a^{3} d \cos \left (d x + c\right )^{10} - 5 \, a^{3} d \cos \left (d x + c\right )^{8} + 10 \, a^{3} d \cos \left (d x + c\right )^{6} - 10 \, a^{3} d \cos \left (d x + c\right )^{4} + 5 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} 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))^3,x, algorithm="fricas")

[Out]

-1/840*(630*cos(d*x + c)^6 - 1190*cos(d*x + c)^4 + 595*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^6 - 126*cos(d*x + c
)^4 + 72*cos(d*x + c)^2 - 16)*sin(d*x + c) - 119)/(a^3*d*cos(d*x + c)^10 - 5*a^3*d*cos(d*x + c)^8 + 10*a^3*d*c
os(d*x + c)^6 - 10*a^3*d*cos(d*x + c)^4 + 5*a^3*d*cos(d*x + c)^2 - a^3*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))**3,x)

[Out]

Timed out

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Giac [A]  time = 2.01292, size = 116, normalized size = 0.8 \begin{align*} \frac{280 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 168 \, \sin \left (d x + c\right )^{5} + 700 \, \sin \left (d x + c\right )^{4} - 600 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 84}{840 \, a^{3} 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))^3,x, algorithm="giac")

[Out]

1/840*(280*sin(d*x + c)^7 - 630*sin(d*x + c)^6 + 168*sin(d*x + c)^5 + 700*sin(d*x + c)^4 - 600*sin(d*x + c)^3
- 105*sin(d*x + c)^2 + 280*sin(d*x + c) - 84)/(a^3*d*sin(d*x + c)^10)