3.69 \(\int \frac{\cot ^9(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\csc ^8(c+d x)}{8 a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}+\frac{\csc ^6(c+d x)}{6 a^2 d}-\frac{4 \csc ^5(c+d x)}{5 a^2 d}+\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d} \]

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

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Rubi [A]  time = 0.0738176, antiderivative size = 127, 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} \[ -\frac{\csc ^8(c+d x)}{8 a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}+\frac{\csc ^6(c+d x)}{6 a^2 d}-\frac{4 \csc ^5(c+d x)}{5 a^2 d}+\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.142755, size = 78, normalized size = 0.61 \[ \frac{\csc ^2(c+d x) \left (-105 \csc ^6(c+d x)+240 \csc ^5(c+d x)+140 \csc ^4(c+d x)-672 \csc ^3(c+d x)+210 \csc ^2(c+d x)+560 \csc (c+d x)-420\right )}{840 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^9/(a + a*Sin[c + d*x])^2,x]

[Out]

(Csc[c + d*x]^2*(-420 + 560*Csc[c + d*x] + 210*Csc[c + d*x]^2 - 672*Csc[c + d*x]^3 + 140*Csc[c + d*x]^4 + 240*
Csc[c + d*x]^5 - 105*Csc[c + d*x]^6))/(840*a^2*d)

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Maple [A]  time = 0.125, size = 79, normalized size = 0.6 \begin{align*}{\frac{1}{d{a}^{2}} \left ({\frac{2}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{4}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\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)^9/(a+a*sin(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 1.76225, size = 103, normalized size = 0.81 \begin{align*} -\frac{420 \, \sin \left (d x + c\right )^{6} - 560 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} + 672 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 240 \, \sin \left (d x + c\right ) + 105}{840 \, a^{2} d \sin \left (d x + c\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^9/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/840*(420*sin(d*x + c)^6 - 560*sin(d*x + c)^5 - 210*sin(d*x + c)^4 + 672*sin(d*x + c)^3 - 140*sin(d*x + c)^2
- 240*sin(d*x + c) + 105)/(a^2*d*sin(d*x + c)^8)

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Fricas [A]  time = 1.44614, size = 331, normalized size = 2.61 \begin{align*} \frac{420 \, \cos \left (d x + c\right )^{6} - 1050 \, \cos \left (d x + c\right )^{4} + 700 \, \cos \left (d x + c\right )^{2} + 16 \,{\left (35 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 175}{840 \,{\left (a^{2} d \cos \left (d x + c\right )^{8} - 4 \, a^{2} d \cos \left (d x + c\right )^{6} + 6 \, a^{2} d \cos \left (d x + c\right )^{4} - 4 \, 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)^9/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/840*(420*cos(d*x + c)^6 - 1050*cos(d*x + c)^4 + 700*cos(d*x + c)^2 + 16*(35*cos(d*x + c)^4 - 28*cos(d*x + c)
^2 + 8)*sin(d*x + c) - 175)/(a^2*d*cos(d*x + c)^8 - 4*a^2*d*cos(d*x + c)^6 + 6*a^2*d*cos(d*x + c)^4 - 4*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)**9/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 2.22548, size = 103, normalized size = 0.81 \begin{align*} -\frac{420 \, \sin \left (d x + c\right )^{6} - 560 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} + 672 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 240 \, \sin \left (d x + c\right ) + 105}{840 \, a^{2} d \sin \left (d x + c\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^9/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/840*(420*sin(d*x + c)^6 - 560*sin(d*x + c)^5 - 210*sin(d*x + c)^4 + 672*sin(d*x + c)^3 - 140*sin(d*x + c)^2
- 240*sin(d*x + c) + 105)/(a^2*d*sin(d*x + c)^8)