3.81 \(\int \frac{\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=145 \[ -\frac{\csc ^{12}(c+d x)}{12 a^3 d}+\frac{3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{8 \csc ^9(c+d x)}{9 a^3 d}+\frac{3 \csc ^8(c+d x)}{4 a^3 d}+\frac{6 \csc ^7(c+d x)}{7 a^3 d}-\frac{4 \csc ^6(c+d x)}{3 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d} \]

[Out]

-Csc[c + d*x]^3/(3*a^3*d) + (3*Csc[c + d*x]^4)/(4*a^3*d) - (4*Csc[c + d*x]^6)/(3*a^3*d) + (6*Csc[c + d*x]^7)/(
7*a^3*d) + (3*Csc[c + d*x]^8)/(4*a^3*d) - (8*Csc[c + d*x]^9)/(9*a^3*d) + (3*Csc[c + d*x]^11)/(11*a^3*d) - Csc[
c + d*x]^12/(12*a^3*d)

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Rubi [A]  time = 0.0783676, 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} \[ -\frac{\csc ^{12}(c+d x)}{12 a^3 d}+\frac{3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{8 \csc ^9(c+d x)}{9 a^3 d}+\frac{3 \csc ^8(c+d x)}{4 a^3 d}+\frac{6 \csc ^7(c+d x)}{7 a^3 d}-\frac{4 \csc ^6(c+d x)}{3 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.117899, size = 88, normalized size = 0.61 \[ \frac{\csc ^3(c+d x) \left (-231 \csc ^9(c+d x)+756 \csc ^8(c+d x)-2464 \csc ^6(c+d x)+2079 \csc ^5(c+d x)+2376 \csc ^4(c+d x)-3696 \csc ^3(c+d x)+2079 \csc (c+d x)-924\right )}{2772 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^3*(-924 + 2079*Csc[c + d*x] - 3696*Csc[c + d*x]^3 + 2376*Csc[c + d*x]^4 + 2079*Csc[c + d*x]^5 -
2464*Csc[c + d*x]^6 + 756*Csc[c + d*x]^8 - 231*Csc[c + d*x]^9))/(2772*a^3*d)

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Maple [A]  time = 0.174, size = 89, normalized size = 0.6 \begin{align*}{\frac{1}{d{a}^{3}} \left ({\frac{6}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{3}{11\, \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}+{\frac{3}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}+{\frac{3}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{8}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{1}{12\, \left ( \sin \left ( dx+c \right ) \right ) ^{12}}}-{\frac{4}{3\, \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)^13/(a+a*sin(d*x+c))^3,x)

[Out]

1/d/a^3*(6/7/sin(d*x+c)^7+3/11/sin(d*x+c)^11+3/4/sin(d*x+c)^8+3/4/sin(d*x+c)^4-8/9/sin(d*x+c)^9-1/12/sin(d*x+c
)^12-4/3/sin(d*x+c)^6-1/3/sin(d*x+c)^3)

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Maxima [A]  time = 2.73298, size = 116, normalized size = 0.8 \begin{align*} -\frac{924 \, \sin \left (d x + c\right )^{9} - 2079 \, \sin \left (d x + c\right )^{8} + 3696 \, \sin \left (d x + c\right )^{6} - 2376 \, \sin \left (d x + c\right )^{5} - 2079 \, \sin \left (d x + c\right )^{4} + 2464 \, \sin \left (d x + c\right )^{3} - 756 \, \sin \left (d x + c\right ) + 231}{2772 \, a^{3} d \sin \left (d x + c\right )^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2772*(924*sin(d*x + c)^9 - 2079*sin(d*x + c)^8 + 3696*sin(d*x + c)^6 - 2376*sin(d*x + c)^5 - 2079*sin(d*x +
c)^4 + 2464*sin(d*x + c)^3 - 756*sin(d*x + c) + 231)/(a^3*d*sin(d*x + c)^12)

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Fricas [A]  time = 1.66886, size = 498, normalized size = 3.43 \begin{align*} \frac{2079 \, \cos \left (d x + c\right )^{8} - 4620 \, \cos \left (d x + c\right )^{6} + 3465 \, \cos \left (d x + c\right )^{4} - 1386 \, \cos \left (d x + c\right )^{2} - 4 \,{\left (231 \, \cos \left (d x + c\right )^{8} - 924 \, \cos \left (d x + c\right )^{6} + 792 \, \cos \left (d x + c\right )^{4} - 352 \, \cos \left (d x + c\right )^{2} + 64\right )} \sin \left (d x + c\right ) + 231}{2772 \,{\left (a^{3} d \cos \left (d x + c\right )^{12} - 6 \, a^{3} d \cos \left (d x + c\right )^{10} + 15 \, a^{3} d \cos \left (d x + c\right )^{8} - 20 \, a^{3} d \cos \left (d x + c\right )^{6} + 15 \, a^{3} d \cos \left (d x + c\right )^{4} - 6 \, 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)^13/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/2772*(2079*cos(d*x + c)^8 - 4620*cos(d*x + c)^6 + 3465*cos(d*x + c)^4 - 1386*cos(d*x + c)^2 - 4*(231*cos(d*x
+ c)^8 - 924*cos(d*x + c)^6 + 792*cos(d*x + c)^4 - 352*cos(d*x + c)^2 + 64)*sin(d*x + c) + 231)/(a^3*d*cos(d*
x + c)^12 - 6*a^3*d*cos(d*x + c)^10 + 15*a^3*d*cos(d*x + c)^8 - 20*a^3*d*cos(d*x + c)^6 + 15*a^3*d*cos(d*x + c
)^4 - 6*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)**13/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.99282, size = 116, normalized size = 0.8 \begin{align*} -\frac{924 \, \sin \left (d x + c\right )^{9} - 2079 \, \sin \left (d x + c\right )^{8} + 3696 \, \sin \left (d x + c\right )^{6} - 2376 \, \sin \left (d x + c\right )^{5} - 2079 \, \sin \left (d x + c\right )^{4} + 2464 \, \sin \left (d x + c\right )^{3} - 756 \, \sin \left (d x + c\right ) + 231}{2772 \, a^{3} d \sin \left (d x + c\right )^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2772*(924*sin(d*x + c)^9 - 2079*sin(d*x + c)^8 + 3696*sin(d*x + c)^6 - 2376*sin(d*x + c)^5 - 2079*sin(d*x +
c)^4 + 2464*sin(d*x + c)^3 - 756*sin(d*x + c) + 231)/(a^3*d*sin(d*x + c)^12)