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

Optimal. Leaf size=84 \[ -\frac{\cot ^8(c+d x)}{8 a d}+\frac{\csc ^7(c+d x)}{7 a d}-\frac{3 \csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{a d}-\frac{\csc (c+d x)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0997991, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2606, 194} \[ -\frac{\cot ^8(c+d x)}{8 a d}+\frac{\csc ^7(c+d x)}{7 a d}-\frac{3 \csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{a d}-\frac{\csc (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.208641, size = 77, normalized size = 0.92 \[ \frac{\csc ^8(c+d x) (-513 \sin (c+d x)+371 \sin (3 (c+d x))-105 \sin (5 (c+d x))+35 \sin (7 (c+d x))-245 \cos (2 (c+d x))-35 \cos (6 (c+d x)))}{2240 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*x]^8*(-245*Cos[2*(c + d*x)] - 35*Cos[6*(c + d*x)] - 513*Sin[c + d*x] + 371*Sin[3*(c + d*x)] - 105*S
in[5*(c + d*x)] + 35*Sin[7*(c + d*x)]))/(2240*a*d)

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Maple [A]  time = 0.115, size = 87, normalized size = 1. \begin{align*}{\frac{1}{da} \left ({\frac{1}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}- \left ( \sin \left ( dx+c \right ) \right ) ^{-1}-{\frac{1}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{3}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+ \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)),x)

[Out]

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

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Maxima [A]  time = 1.10832, size = 116, normalized size = 1.38 \begin{align*} -\frac{280 \, \sin \left (d x + c\right )^{7} - 140 \, \sin \left (d x + c\right )^{6} - 280 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} + 168 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) + 35}{280 \, a 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)),x, algorithm="maxima")

[Out]

-1/280*(280*sin(d*x + c)^7 - 140*sin(d*x + c)^6 - 280*sin(d*x + c)^5 + 210*sin(d*x + c)^4 + 168*sin(d*x + c)^3
- 140*sin(d*x + c)^2 - 40*sin(d*x + c) + 35)/(a*d*sin(d*x + c)^8)

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Fricas [A]  time = 1.51708, size = 343, normalized size = 4.08 \begin{align*} -\frac{140 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{4} + 140 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (35 \, \cos \left (d x + c\right )^{6} - 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) - 35}{280 \,{\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a 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)),x, algorithm="fricas")

[Out]

-1/280*(140*cos(d*x + c)^6 - 210*cos(d*x + c)^4 + 140*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^6 - 70*cos(d*x + c)^
4 + 56*cos(d*x + c)^2 - 16)*sin(d*x + c) - 35)/(a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x + c)
^4 - 4*a*d*cos(d*x + c)^2 + a*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)),x)

[Out]

Timed out

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Giac [A]  time = 1.43815, size = 116, normalized size = 1.38 \begin{align*} -\frac{280 \, \sin \left (d x + c\right )^{7} - 140 \, \sin \left (d x + c\right )^{6} - 280 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} + 168 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) + 35}{280 \, a 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)),x, algorithm="giac")

[Out]

-1/280*(280*sin(d*x + c)^7 - 140*sin(d*x + c)^6 - 280*sin(d*x + c)^5 + 210*sin(d*x + c)^4 + 168*sin(d*x + c)^3
- 140*sin(d*x + c)^2 - 40*sin(d*x + c) + 35)/(a*d*sin(d*x + c)^8)