3.79 \(\int \frac{\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=109 \[ -\frac{\csc ^8(c+d x)}{8 a^3 d}+\frac{3 \csc ^7(c+d x)}{7 a^3 d}-\frac{\csc ^6(c+d x)}{3 a^3 d}-\frac{2 \csc ^5(c+d x)}{5 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d} \]

[Out]

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

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Rubi [A]  time = 0.0689768, antiderivative size = 109, 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, 75} \[ -\frac{\csc ^8(c+d x)}{8 a^3 d}+\frac{3 \csc ^7(c+d x)}{7 a^3 d}-\frac{\csc ^6(c+d x)}{3 a^3 d}-\frac{2 \csc ^5(c+d x)}{5 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^9/(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) - (2*Csc[c + d*x]^5)/(5*a^3*d) - Csc[c + d*x]^6/(3*a^
3*d) + (3*Csc[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^8/(8*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 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

Mathematica [A]  time = 0.0721869, size = 68, normalized size = 0.62 \[ -\frac{\csc ^3(c+d x) \left (105 \csc ^5(c+d x)-360 \csc ^4(c+d x)+280 \csc ^3(c+d x)+336 \csc ^2(c+d x)-630 \csc (c+d x)+280\right )}{840 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^9/(a + a*Sin[c + d*x])^3,x]

[Out]

-(Csc[c + d*x]^3*(280 - 630*Csc[c + d*x] + 336*Csc[c + d*x]^2 + 280*Csc[c + d*x]^3 - 360*Csc[c + d*x]^4 + 105*
Csc[c + d*x]^5))/(840*a^3*d)

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Maple [A]  time = 0.136, size = 69, normalized size = 0.6 \begin{align*}{\frac{1}{d{a}^{3}} \left ({\frac{3}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{2}{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 ) ^{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)^9/(a+a*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.21198, size = 89, normalized size = 0.82 \begin{align*} -\frac{280 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4} + 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} - 360 \, \sin \left (d x + c\right ) + 105}{840 \, a^{3} 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))^3,x, algorithm="maxima")

[Out]

-1/840*(280*sin(d*x + c)^5 - 630*sin(d*x + c)^4 + 336*sin(d*x + c)^3 + 280*sin(d*x + c)^2 - 360*sin(d*x + c) +
105)/(a^3*d*sin(d*x + c)^8)

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Fricas [A]  time = 1.47629, size = 302, normalized size = 2.77 \begin{align*} \frac{630 \, \cos \left (d x + c\right )^{4} - 980 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (35 \, \cos \left (d x + c\right )^{4} - 112 \, \cos \left (d x + c\right )^{2} + 32\right )} \sin \left (d x + c\right ) + 245}{840 \,{\left (a^{3} d \cos \left (d x + c\right )^{8} - 4 \, a^{3} d \cos \left (d x + c\right )^{6} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, 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)^9/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/840*(630*cos(d*x + c)^4 - 980*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^4 - 112*cos(d*x + c)^2 + 32)*sin(d*x + c)
+ 245)/(a^3*d*cos(d*x + c)^8 - 4*a^3*d*cos(d*x + c)^6 + 6*a^3*d*cos(d*x + c)^4 - 4*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)**9/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.61935, size = 89, normalized size = 0.82 \begin{align*} -\frac{280 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4} + 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} - 360 \, \sin \left (d x + c\right ) + 105}{840 \, a^{3} 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))^3,x, algorithm="giac")

[Out]

-1/840*(280*sin(d*x + c)^5 - 630*sin(d*x + c)^4 + 336*sin(d*x + c)^3 + 280*sin(d*x + c)^2 - 360*sin(d*x + c) +
105)/(a^3*d*sin(d*x + c)^8)