3.1.70 \(\int \frac {\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [70]

Optimal. Leaf size=145 \[ \frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^4(c+d x)}{2 a^2 d}+\frac {6 \csc ^5(c+d x)}{5 a^2 d}-\frac {6 \csc ^7(c+d x)}{7 a^2 d}+\frac {\csc ^8(c+d x)}{4 a^2 d}+\frac {2 \csc ^9(c+d x)}{9 a^2 d}-\frac {\csc ^{10}(c+d x)}{10 a^2 d} \]

[Out]

1/2*csc(d*x+c)^2/a^2/d-2/3*csc(d*x+c)^3/a^2/d-1/2*csc(d*x+c)^4/a^2/d+6/5*csc(d*x+c)^5/a^2/d-6/7*csc(d*x+c)^7/a
^2/d+1/4*csc(d*x+c)^8/a^2/d+2/9*csc(d*x+c)^9/a^2/d-1/10*csc(d*x+c)^10/a^2/d

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Rubi [A]
time = 0.06, antiderivative size = 145, 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 = {2786, 90} \begin {gather*} -\frac {\csc ^{10}(c+d x)}{10 a^2 d}+\frac {2 \csc ^9(c+d x)}{9 a^2 d}+\frac {\csc ^8(c+d x)}{4 a^2 d}-\frac {6 \csc ^7(c+d x)}{7 a^2 d}+\frac {6 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^4(c+d x)}{2 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {\csc ^2(c+d x)}{2 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^11/(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/(2*a^2*d) + (6*Csc[c + d*x]^5)/(5*a^2
*d) - (6*Csc[c + d*x]^7)/(7*a^2*d) + Csc[c + d*x]^8/(4*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d) - Csc[c + d*x]^10
/(10*a^2*d)

Rule 90

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]))

Rule 2786

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 ^{11}(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^5 (a+x)^3}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^8}{x^{11}}-\frac {2 a^7}{x^{10}}-\frac {2 a^6}{x^9}+\frac {6 a^5}{x^8}-\frac {6 a^3}{x^6}+\frac {2 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)}{2 a^2 d}+\frac {6 \csc ^5(c+d x)}{5 a^2 d}-\frac {6 \csc ^7(c+d x)}{7 a^2 d}+\frac {\csc ^8(c+d x)}{4 a^2 d}+\frac {2 \csc ^9(c+d x)}{9 a^2 d}-\frac {\csc ^{10}(c+d x)}{10 a^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 88, normalized size = 0.61 \begin {gather*} \frac {\csc ^2(c+d x) \left (630-840 \csc (c+d x)-630 \csc ^2(c+d x)+1512 \csc ^3(c+d x)-1080 \csc ^5(c+d x)+315 \csc ^6(c+d x)+280 \csc ^7(c+d x)-126 \csc ^8(c+d x)\right )}{1260 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^2,x]

[Out]

(Csc[c + d*x]^2*(630 - 840*Csc[c + d*x] - 630*Csc[c + d*x]^2 + 1512*Csc[c + d*x]^3 - 1080*Csc[c + d*x]^5 + 315
*Csc[c + d*x]^6 + 280*Csc[c + d*x]^7 - 126*Csc[c + d*x]^8))/(1260*a^2*d)

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Maple [A]
time = 0.46, size = 89, normalized size = 0.61

method	result	size
derivativedivides	\(\frac {\frac {2}{9 \sin \left (d x +c \right )^{9}}-\frac {2}{3 \sin \left (d x +c \right )^{3}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{4 \sin \left (d x +c \right )^{8}}-\frac {1}{2 \sin \left (d x +c \right )^{4}}-\frac {1}{10 \sin \left (d x +c \right )^{10}}+\frac {6}{5 \sin \left (d x +c \right )^{5}}-\frac {6}{7 \sin \left (d x +c \right )^{7}}}{d \,a^{2}}\)	\(89\)
default	\(\frac {\frac {2}{9 \sin \left (d x +c \right )^{9}}-\frac {2}{3 \sin \left (d x +c \right )^{3}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{4 \sin \left (d x +c \right )^{8}}-\frac {1}{2 \sin \left (d x +c \right )^{4}}-\frac {1}{10 \sin \left (d x +c \right )^{10}}+\frac {6}{5 \sin \left (d x +c \right )^{5}}-\frac {6}{7 \sin \left (d x +c \right )^{7}}}{d \,a^{2}}\)	\(89\)
risch	\(-\frac {2 \left (315 \,{\mathrm e}^{18 i \left (d x +c \right )}-1260 \,{\mathrm e}^{16 i \left (d x +c \right )}+840 i {\mathrm e}^{3 i \left (d x +c \right )}+1260 \,{\mathrm e}^{14 i \left (d x +c \right )}-840 i {\mathrm e}^{17 i \left (d x +c \right )}-8820 \,{\mathrm e}^{12 i \left (d x +c \right )}+168 i {\mathrm e}^{5 i \left (d x +c \right )}+882 \,{\mathrm e}^{10 i \left (d x +c \right )}-168 i {\mathrm e}^{15 i \left (d x +c \right )}-8820 \,{\mathrm e}^{8 i \left (d x +c \right )}+4680 i {\mathrm e}^{7 i \left (d x +c \right )}+1260 \,{\mathrm e}^{6 i \left (d x +c \right )}-4680 i {\mathrm e}^{13 i \left (d x +c \right )}-1260 \,{\mathrm e}^{4 i \left (d x +c \right )}-2840 i {\mathrm e}^{9 i \left (d x +c \right )}+315 \,{\mathrm e}^{2 i \left (d x +c \right )}+2840 i {\mathrm e}^{11 i \left (d x +c \right )}\right )}{315 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}\)	\(218\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d/a^2*(2/9/sin(d*x+c)^9-2/3/sin(d*x+c)^3+1/2/sin(d*x+c)^2+1/4/sin(d*x+c)^8-1/2/sin(d*x+c)^4-1/10/sin(d*x+c)^
10+6/5/sin(d*x+c)^5-6/7/sin(d*x+c)^7)

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Maxima [A]
time = 0.28, size = 86, normalized size = 0.59 \begin {gather*} \frac {630 \, \sin \left (d x + c\right )^{8} - 840 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 1512 \, \sin \left (d x + c\right )^{5} - 1080 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 126}{1260 \, a^{2} d \sin \left (d x + c\right )^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/1260*(630*sin(d*x + c)^8 - 840*sin(d*x + c)^7 - 630*sin(d*x + c)^6 + 1512*sin(d*x + c)^5 - 1080*sin(d*x + c)
^3 + 315*sin(d*x + c)^2 + 280*sin(d*x + c) - 126)/(a^2*d*sin(d*x + c)^10)

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Fricas [A]
time = 0.36, size = 162, normalized size = 1.12 \begin {gather*} -\frac {630 \, \cos \left (d x + c\right )^{8} - 1890 \, \cos \left (d x + c\right )^{6} + 1890 \, \cos \left (d x + c\right )^{4} - 945 \, \cos \left (d x + c\right )^{2} + 8 \, {\left (105 \, \cos \left (d x + c\right )^{6} - 126 \, \cos \left (d x + c\right )^{4} + 72 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 189}{1260 \, {\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/1260*(630*cos(d*x + c)^8 - 1890*cos(d*x + c)^6 + 1890*cos(d*x + c)^4 - 945*cos(d*x + c)^2 + 8*(105*cos(d*x
+ c)^6 - 126*cos(d*x + c)^4 + 72*cos(d*x + c)^2 - 16)*sin(d*x + c) + 189)/(a^2*d*cos(d*x + c)^10 - 5*a^2*d*cos
(d*x + c)^8 + 10*a^2*d*cos(d*x + c)^6 - 10*a^2*d*cos(d*x + c)^4 + 5*a^2*d*cos(d*x + c)^2 - a^2*d)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**11/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]
time = 14.52, size = 86, normalized size = 0.59 \begin {gather*} \frac {630 \, \sin \left (d x + c\right )^{8} - 840 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 1512 \, \sin \left (d x + c\right )^{5} - 1080 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 126}{1260 \, a^{2} d \sin \left (d x + c\right )^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1260*(630*sin(d*x + c)^8 - 840*sin(d*x + c)^7 - 630*sin(d*x + c)^6 + 1512*sin(d*x + c)^5 - 1080*sin(d*x + c)
^3 + 315*sin(d*x + c)^2 + 280*sin(d*x + c) - 126)/(a^2*d*sin(d*x + c)^10)

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Mupad [B]
time = 6.63, size = 85, normalized size = 0.59 \begin {gather*} \frac {\frac {{\sin \left (c+d\,x\right )}^8}{2}-\frac {2\,{\sin \left (c+d\,x\right )}^7}{3}-\frac {{\sin \left (c+d\,x\right )}^6}{2}+\frac {6\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {6\,{\sin \left (c+d\,x\right )}^3}{7}+\frac {{\sin \left (c+d\,x\right )}^2}{4}+\frac {2\,\sin \left (c+d\,x\right )}{9}-\frac {1}{10}}{a^2\,d\,{\sin \left (c+d\,x\right )}^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^11/(a + a*sin(c + d*x))^2,x)

[Out]

((2*sin(c + d*x))/9 + sin(c + d*x)^2/4 - (6*sin(c + d*x)^3)/7 + (6*sin(c + d*x)^5)/5 - sin(c + d*x)^6/2 - (2*s
in(c + d*x)^7)/3 + sin(c + d*x)^8/2 - 1/10)/(a^2*d*sin(c + d*x)^10)

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