3.1.81 \(\int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [81]

Optimal. Leaf size=145 \[ -\frac {\csc ^3(c+d x)}{3 a^3 d}+\frac {3 \csc ^4(c+d x)}{4 a^3 d}-\frac {4 \csc ^6(c+d x)}{3 a^3 d}+\frac {6 \csc ^7(c+d x)}{7 a^3 d}+\frac {3 \csc ^8(c+d x)}{4 a^3 d}-\frac {8 \csc ^9(c+d x)}{9 a^3 d}+\frac {3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {\csc ^{12}(c+d x)}{12 a^3 d} \]

[Out]

-1/3*csc(d*x+c)^3/a^3/d+3/4*csc(d*x+c)^4/a^3/d-4/3*csc(d*x+c)^6/a^3/d+6/7*csc(d*x+c)^7/a^3/d+3/4*csc(d*x+c)^8/
a^3/d-8/9*csc(d*x+c)^9/a^3/d+3/11*csc(d*x+c)^11/a^3/d-1/12*csc(d*x+c)^12/a^3/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 ^{12}(c+d x)}{12 a^3 d}+\frac {3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {8 \csc ^9(c+d x)}{9 a^3 d}+\frac {3 \csc ^8(c+d x)}{4 a^3 d}+\frac {6 \csc ^7(c+d x)}{7 a^3 d}-\frac {4 \csc ^6(c+d x)}{3 a^3 d}+\frac {3 \csc ^4(c+d x)}{4 a^3 d}-\frac {\csc ^3(c+d x)}{3 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^3,x]

[Out]

-1/3*Csc[c + d*x]^3/(a^3*d) + (3*Csc[c + d*x]^4)/(4*a^3*d) - (4*Csc[c + d*x]^6)/(3*a^3*d) + (6*Csc[c + d*x]^7)
/(7*a^3*d) + (3*Csc[c + d*x]^8)/(4*a^3*d) - (8*Csc[c + d*x]^9)/(9*a^3*d) + (3*Csc[c + d*x]^11)/(11*a^3*d) - Cs
c[c + d*x]^12/(12*a^3*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 ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^6 (a+x)^3}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^9}{x^{13}}-\frac {3 a^8}{x^{12}}+\frac {8 a^6}{x^{10}}-\frac {6 a^5}{x^9}-\frac {6 a^4}{x^8}+\frac {8 a^3}{x^7}-\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 {4 \csc ^6(c+d x)}{3 a^3 d}+\frac {6 \csc ^7(c+d x)}{7 a^3 d}+\frac {3 \csc ^8(c+d x)}{4 a^3 d}-\frac {8 \csc ^9(c+d x)}{9 a^3 d}+\frac {3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {\csc ^{12}(c+d x)}{12 a^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 88, normalized size = 0.61 \begin {gather*} \frac {\csc ^3(c+d x) \left (-924+2079 \csc (c+d x)-3696 \csc ^3(c+d x)+2376 \csc ^4(c+d x)+2079 \csc ^5(c+d x)-2464 \csc ^6(c+d x)+756 \csc ^8(c+d x)-231 \csc ^9(c+d x)\right )}{2772 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^3*(-924 + 2079*Csc[c + d*x] - 3696*Csc[c + d*x]^3 + 2376*Csc[c + d*x]^4 + 2079*Csc[c + d*x]^5 -
2464*Csc[c + d*x]^6 + 756*Csc[c + d*x]^8 - 231*Csc[c + d*x]^9))/(2772*a^3*d)

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Maple [A]
time = 0.57, size = 89, normalized size = 0.61

method	result	size
derivativedivides	\(\frac {\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {8}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}+\frac {3}{4 \sin \left (d x +c \right )^{8}}-\frac {4}{3 \sin \left (d x +c \right )^{6}}+\frac {6}{7 \sin \left (d x +c \right )^{7}}+\frac {3}{11 \sin \left (d x +c \right )^{11}}}{d \,a^{3}}\)	\(89\)
default	\(\frac {\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {8}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}+\frac {3}{4 \sin \left (d x +c \right )^{8}}-\frac {4}{3 \sin \left (d x +c \right )^{6}}+\frac {6}{7 \sin \left (d x +c \right )^{7}}+\frac {3}{11 \sin \left (d x +c \right )^{11}}}{d \,a^{3}}\)	\(89\)
risch	\(\frac {4 i \left (-2079 i {\mathrm e}^{20 i \left (d x +c \right )}+462 \,{\mathrm e}^{21 i \left (d x +c \right )}-2079 i {\mathrm e}^{4 i \left (d x +c \right )}-4158 \,{\mathrm e}^{19 i \left (d x +c \right )}+9702 i {\mathrm e}^{12 i \left (d x +c \right )}-2376 \,{\mathrm e}^{17 i \left (d x +c \right )}+27720 i {\mathrm e}^{10 i \left (d x +c \right )}-22616 \,{\mathrm e}^{15 i \left (d x +c \right )}-2772 i {\mathrm e}^{8 i \left (d x +c \right )}+7908 \,{\mathrm e}^{13 i \left (d x +c \right )}+1848 i {\mathrm e}^{6 i \left (d x +c \right )}-7908 \,{\mathrm e}^{11 i \left (d x +c \right )}-2772 i {\mathrm e}^{16 i \left (d x +c \right )}+22616 \,{\mathrm e}^{9 i \left (d x +c \right )}+27720 i {\mathrm e}^{14 i \left (d x +c \right )}+2376 \,{\mathrm e}^{7 i \left (d x +c \right )}+1848 i {\mathrm e}^{18 i \left (d x +c \right )}+4158 \,{\mathrm e}^{5 i \left (d x +c \right )}-462 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{693 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}\)	\(242\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d/a^3*(3/4/sin(d*x+c)^4-8/9/sin(d*x+c)^9-1/3/sin(d*x+c)^3-1/12/sin(d*x+c)^12+3/4/sin(d*x+c)^8-4/3/sin(d*x+c)
^6+6/7/sin(d*x+c)^7+3/11/sin(d*x+c)^11)

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Maxima [A]
time = 0.28, size = 86, normalized size = 0.59 \begin {gather*} -\frac {924 \, \sin \left (d x + c\right )^{9} - 2079 \, \sin \left (d x + c\right )^{8} + 3696 \, \sin \left (d x + c\right )^{6} - 2376 \, \sin \left (d x + c\right )^{5} - 2079 \, \sin \left (d x + c\right )^{4} + 2464 \, \sin \left (d x + c\right )^{3} - 756 \, \sin \left (d x + c\right ) + 231}{2772 \, a^{3} d \sin \left (d x + c\right )^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2772*(924*sin(d*x + c)^9 - 2079*sin(d*x + c)^8 + 3696*sin(d*x + c)^6 - 2376*sin(d*x + c)^5 - 2079*sin(d*x +
c)^4 + 2464*sin(d*x + c)^3 - 756*sin(d*x + c) + 231)/(a^3*d*sin(d*x + c)^12)

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Fricas [A]
time = 0.38, size = 185, normalized size = 1.28 \begin {gather*} \frac {2079 \, \cos \left (d x + c\right )^{8} - 4620 \, \cos \left (d x + c\right )^{6} + 3465 \, \cos \left (d x + c\right )^{4} - 1386 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (231 \, \cos \left (d x + c\right )^{8} - 924 \, \cos \left (d x + c\right )^{6} + 792 \, \cos \left (d x + c\right )^{4} - 352 \, \cos \left (d x + c\right )^{2} + 64\right )} \sin \left (d x + c\right ) + 231}{2772 \, {\left (a^{3} d \cos \left (d x + c\right )^{12} - 6 \, a^{3} d \cos \left (d x + c\right )^{10} + 15 \, a^{3} d \cos \left (d x + c\right )^{8} - 20 \, a^{3} d \cos \left (d x + c\right )^{6} + 15 \, a^{3} d \cos \left (d x + c\right )^{4} - 6 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/2772*(2079*cos(d*x + c)^8 - 4620*cos(d*x + c)^6 + 3465*cos(d*x + c)^4 - 1386*cos(d*x + c)^2 - 4*(231*cos(d*x
+ c)^8 - 924*cos(d*x + c)^6 + 792*cos(d*x + c)^4 - 352*cos(d*x + c)^2 + 64)*sin(d*x + c) + 231)/(a^3*d*cos(d*
x + c)^12 - 6*a^3*d*cos(d*x + c)^10 + 15*a^3*d*cos(d*x + c)^8 - 20*a^3*d*cos(d*x + c)^6 + 15*a^3*d*cos(d*x + c
)^4 - 6*a^3*d*cos(d*x + c)^2 + a^3*d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**13/(a+a*sin(d*x+c))**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3061 deep

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Giac [A]
time = 6.73, size = 86, normalized size = 0.59 \begin {gather*} -\frac {924 \, \sin \left (d x + c\right )^{9} - 2079 \, \sin \left (d x + c\right )^{8} + 3696 \, \sin \left (d x + c\right )^{6} - 2376 \, \sin \left (d x + c\right )^{5} - 2079 \, \sin \left (d x + c\right )^{4} + 2464 \, \sin \left (d x + c\right )^{3} - 756 \, \sin \left (d x + c\right ) + 231}{2772 \, a^{3} d \sin \left (d x + c\right )^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2772*(924*sin(d*x + c)^9 - 2079*sin(d*x + c)^8 + 3696*sin(d*x + c)^6 - 2376*sin(d*x + c)^5 - 2079*sin(d*x +
c)^4 + 2464*sin(d*x + c)^3 - 756*sin(d*x + c) + 231)/(a^3*d*sin(d*x + c)^12)

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Mupad [B]
time = 6.82, size = 85, normalized size = 0.59 \begin {gather*} \frac {-\frac {{\sin \left (c+d\,x\right )}^9}{3}+\frac {3\,{\sin \left (c+d\,x\right )}^8}{4}-\frac {4\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {6\,{\sin \left (c+d\,x\right )}^5}{7}+\frac {3\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {8\,{\sin \left (c+d\,x\right )}^3}{9}+\frac {3\,\sin \left (c+d\,x\right )}{11}-\frac {1}{12}}{a^3\,d\,{\sin \left (c+d\,x\right )}^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^13/(a + a*sin(c + d*x))^3,x)

[Out]

((3*sin(c + d*x))/11 - (8*sin(c + d*x)^3)/9 + (3*sin(c + d*x)^4)/4 + (6*sin(c + d*x)^5)/7 - (4*sin(c + d*x)^6)
/3 + (3*sin(c + d*x)^8)/4 - sin(c + d*x)^9/3 - 1/12)/(a^3*d*sin(c + d*x)^12)

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