3.1.71 \(\int \frac {\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [71]

Optimal. Leaf size=199 \[ -\frac {\csc ^2(c+d x)}{2 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {3 \csc ^4(c+d x)}{4 a^2 d}-\frac {8 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{3 a^2 d}+\frac {12 \csc ^7(c+d x)}{7 a^2 d}-\frac {\csc ^8(c+d x)}{4 a^2 d}-\frac {8 \csc ^9(c+d x)}{9 a^2 d}+\frac {3 \csc ^{10}(c+d x)}{10 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac {\csc ^{12}(c+d x)}{12 a^2 d} \]

[Out]

-1/2*csc(d*x+c)^2/a^2/d+2/3*csc(d*x+c)^3/a^2/d+3/4*csc(d*x+c)^4/a^2/d-8/5*csc(d*x+c)^5/a^2/d-1/3*csc(d*x+c)^6/
a^2/d+12/7*csc(d*x+c)^7/a^2/d-1/4*csc(d*x+c)^8/a^2/d-8/9*csc(d*x+c)^9/a^2/d+3/10*csc(d*x+c)^10/a^2/d+2/11*csc(
d*x+c)^11/a^2/d-1/12*csc(d*x+c)^12/a^2/d

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Rubi [A]
time = 0.07, antiderivative size = 199, 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^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}+\frac {3 \csc ^{10}(c+d x)}{10 a^2 d}-\frac {8 \csc ^9(c+d x)}{9 a^2 d}-\frac {\csc ^8(c+d x)}{4 a^2 d}+\frac {12 \csc ^7(c+d x)}{7 a^2 d}-\frac {\csc ^6(c+d x)}{3 a^2 d}-\frac {8 \csc ^5(c+d x)}{5 a^2 d}+\frac {3 \csc ^4(c+d x)}{4 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]^13/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/2*Csc[c + d*x]^2/(a^2*d) + (2*Csc[c + d*x]^3)/(3*a^2*d) + (3*Csc[c + d*x]^4)/(4*a^2*d) - (8*Csc[c + d*x]^5)
/(5*a^2*d) - Csc[c + d*x]^6/(3*a^2*d) + (12*Csc[c + d*x]^7)/(7*a^2*d) - Csc[c + d*x]^8/(4*a^2*d) - (8*Csc[c +
d*x]^9)/(9*a^2*d) + (3*Csc[c + d*x]^10)/(10*a^2*d) + (2*Csc[c + d*x]^11)/(11*a^2*d) - Csc[c + d*x]^12/(12*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 ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^6 (a+x)^4}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^{10}}{x^{13}}-\frac {2 a^9}{x^{12}}-\frac {3 a^8}{x^{11}}+\frac {8 a^7}{x^{10}}+\frac {2 a^6}{x^9}-\frac {12 a^5}{x^8}+\frac {2 a^4}{x^7}+\frac {8 a^3}{x^6}-\frac {3 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 {3 \csc ^4(c+d x)}{4 a^2 d}-\frac {8 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{3 a^2 d}+\frac {12 \csc ^7(c+d x)}{7 a^2 d}-\frac {\csc ^8(c+d x)}{4 a^2 d}-\frac {8 \csc ^9(c+d x)}{9 a^2 d}+\frac {3 \csc ^{10}(c+d x)}{10 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac {\csc ^{12}(c+d x)}{12 a^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 118, normalized size = 0.59 \begin {gather*} -\frac {\csc ^2(c+d x) \left (6930-9240 \csc (c+d x)-10395 \csc ^2(c+d x)+22176 \csc ^3(c+d x)+4620 \csc ^4(c+d x)-23760 \csc ^5(c+d x)+3465 \csc ^6(c+d x)+12320 \csc ^7(c+d x)-4158 \csc ^8(c+d x)-2520 \csc ^9(c+d x)+1155 \csc ^{10}(c+d x)\right )}{13860 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/13860*(Csc[c + d*x]^2*(6930 - 9240*Csc[c + d*x] - 10395*Csc[c + d*x]^2 + 22176*Csc[c + d*x]^3 + 4620*Csc[c
+ d*x]^4 - 23760*Csc[c + d*x]^5 + 3465*Csc[c + d*x]^6 + 12320*Csc[c + d*x]^7 - 4158*Csc[c + d*x]^8 - 2520*Csc[
c + d*x]^9 + 1155*Csc[c + d*x]^10))/(a^2*d)

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Maple [A]
time = 0.58, size = 119, normalized size = 0.60

method	result	size
derivativedivides	\(\frac {-\frac {8}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{4 \sin \left (d x +c \right )^{8}}+\frac {2}{3 \sin \left (d x +c \right )^{3}}+\frac {2}{11 \sin \left (d x +c \right )^{11}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {8}{5 \sin \left (d x +c \right )^{5}}+\frac {12}{7 \sin \left (d x +c \right )^{7}}+\frac {3}{10 \sin \left (d x +c \right )^{10}}}{d \,a^{2}}\)	\(119\)
default	\(\frac {-\frac {8}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{12 \sin \left (d x +c \right )^{12}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{4 \sin \left (d x +c \right )^{8}}+\frac {2}{3 \sin \left (d x +c \right )^{3}}+\frac {2}{11 \sin \left (d x +c \right )^{11}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {8}{5 \sin \left (d x +c \right )^{5}}+\frac {12}{7 \sin \left (d x +c \right )^{7}}+\frac {3}{10 \sin \left (d x +c \right )^{10}}}{d \,a^{2}}\)	\(119\)
risch	\(\frac {2 \,{\mathrm e}^{22 i \left (d x +c \right )}-8 \,{\mathrm e}^{20 i \left (d x +c \right )}+\frac {16 i {\mathrm e}^{3 i \left (d x +c \right )}}{3}+\frac {46 \,{\mathrm e}^{18 i \left (d x +c \right )}}{3}-\frac {16 i {\mathrm e}^{19 i \left (d x +c \right )}}{5}-96 \,{\mathrm e}^{16 i \left (d x +c \right )}-\frac {16 i {\mathrm e}^{21 i \left (d x +c \right )}}{3}+\frac {84 \,{\mathrm e}^{14 i \left (d x +c \right )}}{5}-\frac {1856 i {\mathrm e}^{17 i \left (d x +c \right )}}{35}-\frac {1008 \,{\mathrm e}^{12 i \left (d x +c \right )}}{5}+\frac {16 i {\mathrm e}^{5 i \left (d x +c \right )}}{5}+\frac {84 \,{\mathrm e}^{10 i \left (d x +c \right )}}{5}+\frac {4672 i {\mathrm e}^{15 i \left (d x +c \right )}}{315}-96 \,{\mathrm e}^{8 i \left (d x +c \right )}+\frac {1856 i {\mathrm e}^{7 i \left (d x +c \right )}}{35}+\frac {46 \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}-\frac {18784 i {\mathrm e}^{13 i \left (d x +c \right )}}{231}-8 \,{\mathrm e}^{4 i \left (d x +c \right )}-\frac {4672 i {\mathrm e}^{9 i \left (d x +c \right )}}{315}+2 \,{\mathrm e}^{2 i \left (d x +c \right )}+\frac {18784 i {\mathrm e}^{11 i \left (d x +c \right )}}{231}}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}\)	\(264\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^13/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d/a^2*(-8/9/sin(d*x+c)^9-1/12/sin(d*x+c)^12-1/2/sin(d*x+c)^2-1/4/sin(d*x+c)^8+2/3/sin(d*x+c)^3+2/11/sin(d*x+
c)^11+3/4/sin(d*x+c)^4-1/3/sin(d*x+c)^6-8/5/sin(d*x+c)^5+12/7/sin(d*x+c)^7+3/10/sin(d*x+c)^10)

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Maxima [A]
time = 0.28, size = 116, normalized size = 0.58 \begin {gather*} -\frac {6930 \, \sin \left (d x + c\right )^{10} - 9240 \, \sin \left (d x + c\right )^{9} - 10395 \, \sin \left (d x + c\right )^{8} + 22176 \, \sin \left (d x + c\right )^{7} + 4620 \, \sin \left (d x + c\right )^{6} - 23760 \, \sin \left (d x + c\right )^{5} + 3465 \, \sin \left (d x + c\right )^{4} + 12320 \, \sin \left (d x + c\right )^{3} - 4158 \, \sin \left (d x + c\right )^{2} - 2520 \, \sin \left (d x + c\right ) + 1155}{13860 \, a^{2} 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))^2,x, algorithm="maxima")

[Out]

-1/13860*(6930*sin(d*x + c)^10 - 9240*sin(d*x + c)^9 - 10395*sin(d*x + c)^8 + 22176*sin(d*x + c)^7 + 4620*sin(
d*x + c)^6 - 23760*sin(d*x + c)^5 + 3465*sin(d*x + c)^4 + 12320*sin(d*x + c)^3 - 4158*sin(d*x + c)^2 - 2520*si
n(d*x + c) + 1155)/(a^2*d*sin(d*x + c)^12)

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Fricas [A]
time = 0.38, size = 195, normalized size = 0.98 \begin {gather*} \frac {6930 \, \cos \left (d x + c\right )^{10} - 24255 \, \cos \left (d x + c\right )^{8} + 32340 \, \cos \left (d x + c\right )^{6} - 24255 \, \cos \left (d x + c\right )^{4} + 9702 \, \cos \left (d x + c\right )^{2} + 8 \, {\left (1155 \, \cos \left (d x + c\right )^{8} - 1848 \, \cos \left (d x + c\right )^{6} + 1584 \, \cos \left (d x + c\right )^{4} - 704 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) - 1617}{13860 \, {\left (a^{2} d \cos \left (d x + c\right )^{12} - 6 \, a^{2} d \cos \left (d x + c\right )^{10} + 15 \, a^{2} d \cos \left (d x + c\right )^{8} - 20 \, a^{2} d \cos \left (d x + c\right )^{6} + 15 \, a^{2} d \cos \left (d x + c\right )^{4} - 6 \, 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)^13/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/13860*(6930*cos(d*x + c)^10 - 24255*cos(d*x + c)^8 + 32340*cos(d*x + c)^6 - 24255*cos(d*x + c)^4 + 9702*cos(
d*x + c)^2 + 8*(1155*cos(d*x + c)^8 - 1848*cos(d*x + c)^6 + 1584*cos(d*x + c)^4 - 704*cos(d*x + c)^2 + 128)*si
n(d*x + c) - 1617)/(a^2*d*cos(d*x + c)^12 - 6*a^2*d*cos(d*x + c)^10 + 15*a^2*d*cos(d*x + c)^8 - 20*a^2*d*cos(d
*x + c)^6 + 15*a^2*d*cos(d*x + c)^4 - 6*a^2*d*cos(d*x + c)^2 + a^2*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))**2,x)

[Out]

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

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Giac [A]
time = 10.22, size = 116, normalized size = 0.58 \begin {gather*} -\frac {6930 \, \sin \left (d x + c\right )^{10} - 9240 \, \sin \left (d x + c\right )^{9} - 10395 \, \sin \left (d x + c\right )^{8} + 22176 \, \sin \left (d x + c\right )^{7} + 4620 \, \sin \left (d x + c\right )^{6} - 23760 \, \sin \left (d x + c\right )^{5} + 3465 \, \sin \left (d x + c\right )^{4} + 12320 \, \sin \left (d x + c\right )^{3} - 4158 \, \sin \left (d x + c\right )^{2} - 2520 \, \sin \left (d x + c\right ) + 1155}{13860 \, a^{2} 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))^2,x, algorithm="giac")

[Out]

-1/13860*(6930*sin(d*x + c)^10 - 9240*sin(d*x + c)^9 - 10395*sin(d*x + c)^8 + 22176*sin(d*x + c)^7 + 4620*sin(
d*x + c)^6 - 23760*sin(d*x + c)^5 + 3465*sin(d*x + c)^4 + 12320*sin(d*x + c)^3 - 4158*sin(d*x + c)^2 - 2520*si
n(d*x + c) + 1155)/(a^2*d*sin(d*x + c)^12)

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Mupad [B]
time = 6.85, size = 116, normalized size = 0.58 \begin {gather*} -\frac {\frac {{\sin \left (c+d\,x\right )}^{10}}{2}-\frac {2\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,{\sin \left (c+d\,x\right )}^8}{4}+\frac {8\,{\sin \left (c+d\,x\right )}^7}{5}+\frac {{\sin \left (c+d\,x\right )}^6}{3}-\frac {12\,{\sin \left (c+d\,x\right )}^5}{7}+\frac {{\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 )}^2}{10}-\frac {2\,\sin \left (c+d\,x\right )}{11}+\frac {1}{12}}{a^2\,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))^2,x)

[Out]

-((8*sin(c + d*x)^3)/9 - (3*sin(c + d*x)^2)/10 - (2*sin(c + d*x))/11 + sin(c + d*x)^4/4 - (12*sin(c + d*x)^5)/
7 + sin(c + d*x)^6/3 + (8*sin(c + d*x)^7)/5 - (3*sin(c + d*x)^8)/4 - (2*sin(c + d*x)^9)/3 + sin(c + d*x)^10/2
+ 1/12)/(a^2*d*sin(c + d*x)^12)

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