3.1.80 \(\int \frac {\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [80]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, 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^3 d}+\frac {\csc ^9(c+d x)}{3 a^3 d}-\frac {\csc ^8(c+d x)}{8 a^3 d}-\frac {5 \csc ^7(c+d x)}{7 a^3 d}+\frac {5 \csc ^6(c+d x)}{6 a^3 d}+\frac {\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 88, normalized size = 0.61 \begin {gather*} \frac {\csc ^3(c+d x) \left (280-630 \csc (c+d x)+168 \csc ^2(c+d x)+700 \csc ^3(c+d x)-600 \csc ^4(c+d x)-105 \csc ^5(c+d x)+280 \csc ^6(c+d x)-84 \csc ^7(c+d x)\right )}{840 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^3*(280 - 630*Csc[c + d*x] + 168*Csc[c + d*x]^2 + 700*Csc[c + d*x]^3 - 600*Csc[c + d*x]^4 - 105*C
sc[c + d*x]^5 + 280*Csc[c + d*x]^6 - 84*Csc[c + d*x]^7))/(840*a^3*d)

________________________________________________________________________________________

Maple [A]
time = 0.45, size = 89, normalized size = 0.61

method	result	size
derivativedivides	\(\frac {-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {5}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{3 \sin \left (d x +c \right )^{9}}-\frac {5}{7 \sin \left (d x +c \right )^{7}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{10 \sin \left (d x +c \right )^{10}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{3}}\)	\(89\)
default	\(\frac {-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {5}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{3 \sin \left (d x +c \right )^{9}}-\frac {5}{7 \sin \left (d x +c \right )^{7}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{10 \sin \left (d x +c \right )^{10}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{3}}\)	\(89\)
risch	\(-\frac {4 i \left (-315 i {\mathrm e}^{16 i \left (d x +c \right )}+70 \,{\mathrm e}^{17 i \left (d x +c \right )}+490 i {\mathrm e}^{14 i \left (d x +c \right )}-658 \,{\mathrm e}^{15 i \left (d x +c \right )}+35 i {\mathrm e}^{12 i \left (d x +c \right )}-90 \,{\mathrm e}^{13 i \left (d x +c \right )}+2268 i {\mathrm e}^{10 i \left (d x +c \right )}-1410 \,{\mathrm e}^{11 i \left (d x +c \right )}+35 i {\mathrm e}^{8 i \left (d x +c \right )}+1410 \,{\mathrm e}^{9 i \left (d x +c \right )}+490 i {\mathrm e}^{6 i \left (d x +c \right )}+90 \,{\mathrm e}^{7 i \left (d x +c \right )}-315 i {\mathrm e}^{4 i \left (d x +c \right )}+658 \,{\mathrm e}^{5 i \left (d x +c \right )}-70 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{105 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}\)	\(196\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^11/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 86, normalized size = 0.59 \begin {gather*} \frac {280 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 168 \, \sin \left (d x + c\right )^{5} + 700 \, \sin \left (d x + c\right )^{4} - 600 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 84}{840 \, a^{3} 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))^3,x, algorithm="maxima")

[Out]

1/840*(280*sin(d*x + c)^7 - 630*sin(d*x + c)^6 + 168*sin(d*x + c)^5 + 700*sin(d*x + c)^4 - 600*sin(d*x + c)^3
- 105*sin(d*x + c)^2 + 280*sin(d*x + c) - 84)/(a^3*d*sin(d*x + c)^10)

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 152, normalized size = 1.05 \begin {gather*} -\frac {630 \, \cos \left (d x + c\right )^{6} - 1190 \, \cos \left (d x + c\right )^{4} + 595 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \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 ) - 119}{840 \, {\left (a^{3} d \cos \left (d x + c\right )^{10} - 5 \, a^{3} d \cos \left (d x + c\right )^{8} + 10 \, a^{3} d \cos \left (d x + c\right )^{6} - 10 \, a^{3} d \cos \left (d x + c\right )^{4} + 5 \, 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)^11/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/840*(630*cos(d*x + c)^6 - 1190*cos(d*x + c)^4 + 595*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^6 - 126*cos(d*x + c
)^4 + 72*cos(d*x + c)^2 - 16)*sin(d*x + c) - 119)/(a^3*d*cos(d*x + c)^10 - 5*a^3*d*cos(d*x + c)^8 + 10*a^3*d*c
os(d*x + c)^6 - 10*a^3*d*cos(d*x + c)^4 + 5*a^3*d*cos(d*x + c)^2 - a^3*d)

________________________________________________________________________________________

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))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 6.05, size = 86, normalized size = 0.59 \begin {gather*} \frac {280 \, \sin \left (d x + c\right )^{7} - 630 \, \sin \left (d x + c\right )^{6} + 168 \, \sin \left (d x + c\right )^{5} + 700 \, \sin \left (d x + c\right )^{4} - 600 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 280 \, \sin \left (d x + c\right ) - 84}{840 \, a^{3} 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))^3,x, algorithm="giac")

[Out]

1/840*(280*sin(d*x + c)^7 - 630*sin(d*x + c)^6 + 168*sin(d*x + c)^5 + 700*sin(d*x + c)^4 - 600*sin(d*x + c)^3
- 105*sin(d*x + c)^2 + 280*sin(d*x + c) - 84)/(a^3*d*sin(d*x + c)^10)

________________________________________________________________________________________

Mupad [B]
time = 6.81, size = 86, normalized size = 0.59 \begin {gather*} \frac {280\,{\sin \left (c+d\,x\right )}^7-630\,{\sin \left (c+d\,x\right )}^6+168\,{\sin \left (c+d\,x\right )}^5+700\,{\sin \left (c+d\,x\right )}^4-600\,{\sin \left (c+d\,x\right )}^3-105\,{\sin \left (c+d\,x\right )}^2+280\,\sin \left (c+d\,x\right )-84}{840\,a^3\,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))^3,x)

[Out]

(280*sin(c + d*x) - 105*sin(c + d*x)^2 - 600*sin(c + d*x)^3 + 700*sin(c + d*x)^4 + 168*sin(c + d*x)^5 - 630*si
n(c + d*x)^6 + 280*sin(c + d*x)^7 - 84)/(840*a^3*d*sin(c + d*x)^10)

________________________________________________________________________________________