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]

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

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Rubi [A]  time = 0.07, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.08, 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]

-1/840*(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))/(a^3*d)

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fricas [A]  time = 0.42, size = 117, normalized size = 1.07 \[ \frac {630 \, \cos \left (d x + c\right )^{4} - 980 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 112 \, \cos \left (d x + c\right )^{2} + 32\right )} \sin \left (d x + c\right ) + 245}{840 \, {\left (a^{3} d \cos \left (d x + c\right )^{8} - 4 \, a^{3} d \cos \left (d x + c\right )^{6} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \]

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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giac [A]  time = 2.04, size = 66, normalized size = 0.61 \[ -\frac {280 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4} + 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} - 360 \, \sin \left (d x + c\right ) + 105}{840 \, a^{3} d \sin \left (d x + c\right )^{8}} \]

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)

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maple [A]  time = 0.36, size = 69, normalized size = 0.63 \[ \frac {-\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {2}{5 \sin \left (d x +c \right )^{5}}+\frac {3}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{3}} \]

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

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maxima [A]  time = 0.33, size = 66, normalized size = 0.61 \[ -\frac {280 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4} + 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} - 360 \, \sin \left (d x + c\right ) + 105}{840 \, a^{3} d \sin \left (d x + c\right )^{8}} \]

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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mupad [B]  time = 6.63, size = 66, normalized size = 0.61 \[ -\frac {280\,{\sin \left (c+d\,x\right )}^5-630\,{\sin \left (c+d\,x\right )}^4+336\,{\sin \left (c+d\,x\right )}^3+280\,{\sin \left (c+d\,x\right )}^2-360\,\sin \left (c+d\,x\right )+105}{840\,a^3\,d\,{\sin \left (c+d\,x\right )}^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^9/(a + a*sin(c + d*x))^3,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{9}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

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]

Integral(cot(c + d*x)**9/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3

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