3.71 \(\int \frac{\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

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

[Out]

-Csc[c + d*x]^2/(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)

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Rubi [A]  time = 0.102394, antiderivative size = 199, normalized size of antiderivative = 1., 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 = {2707, 88} \[ -\frac{\csc ^{12}(c+d x)}{12 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}+\frac{3 \csc ^{10}(c+d x)}{10 a^2 d}-\frac{8 \csc ^9(c+d x)}{9 a^2 d}-\frac{\csc ^8(c+d x)}{4 a^2 d}+\frac{12 \csc ^7(c+d x)}{7 a^2 d}-\frac{\csc ^6(c+d x)}{3 a^2 d}-\frac{8 \csc ^5(c+d x)}{5 a^2 d}+\frac{3 \csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^13/(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) + (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 2707

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]

Rule 88

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]))

Rubi steps

\begin{align*} \int \frac{\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^6 (a+x)^4}{x^{13}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{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*}

Mathematica [A]  time = 0.329218, size = 118, normalized size = 0.59 \[ -\frac{\csc ^2(c+d x) \left (1155 \csc ^{10}(c+d x)-2520 \csc ^9(c+d x)-4158 \csc ^8(c+d x)+12320 \csc ^7(c+d x)+3465 \csc ^6(c+d x)-23760 \csc ^5(c+d x)+4620 \csc ^4(c+d x)+22176 \csc ^3(c+d x)-10395 \csc ^2(c+d x)-9240 \csc (c+d x)+6930\right )}{13860 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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))/(13860*a^2*d)

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Maple [A]  time = 0.165, size = 119, normalized size = 0.6 \begin{align*}{\frac{1}{d{a}^{2}} \left ({\frac{3}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}+{\frac{12}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{2}{11\, \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{8}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{3}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{8}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{1}{12\, \left ( \sin \left ( dx+c \right ) \right ) ^{12}}}-{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{2}{3\, \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)^13/(a+a*sin(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 1.83541, size = 157, normalized size = 0.79 \begin{align*} -\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{align*}

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 = 1.65369, size = 541, normalized size = 2.72 \begin{align*} \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{align*}

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(-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)**13/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 2.1608, size = 157, normalized size = 0.79 \begin{align*} -\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{align*}

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)