3.54 \(\int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=179 \[ -\frac{4 a^3 \cot ^9(c+d x)}{9 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x \]

[Out]

-(a^3*x) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (a^3*Cot[c + d*x]^
7)/(7*d) - (4*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3*Csc[c + d*x])/d + (13*a^3*Csc[c + d*x]^3)/(3*d) - (21*a^3*Csc
[c + d*x]^5)/(5*d) + (15*a^3*Csc[c + d*x]^7)/(7*d) - (4*a^3*Csc[c + d*x]^9)/(9*d)

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Rubi [A]  time = 0.199362, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ -\frac{4 a^3 \cot ^9(c+d x)}{9 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]

[Out]

-(a^3*x) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (a^3*Cot[c + d*x]^
7)/(7*d) - (4*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3*Csc[c + d*x])/d + (13*a^3*Csc[c + d*x]^3)/(3*d) - (21*a^3*Csc
[c + d*x]^5)/(5*d) + (15*a^3*Csc[c + d*x]^7)/(7*d) - (4*a^3*Csc[c + d*x]^9)/(9*d)

Rule 3886

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^{10}(c+d x)+3 a^3 \cot ^9(c+d x) \csc (c+d x)+3 a^3 \cot ^8(c+d x) \csc ^2(c+d x)+a^3 \cot ^7(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^{10}(c+d x) \, dx+a^3 \int \cot ^7(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^9(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^8(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^9(c+d x)}{9 d}-a^3 \int \cot ^8(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^8 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}+a^3 \int \cot ^6(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}-a^3 \int \cot ^4(c+d x) \, dx\\ &=\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}+a^3 \int \cot ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}-a^3 \int 1 \, dx\\ &=-a^3 x-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [B]  time = 1.36962, size = 370, normalized size = 2.07 \[ \frac{a^3 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^9\left (\frac{1}{2} (c+d x)\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) (675036 \sin (c+d x)-506277 \sin (2 (c+d x))-37502 \sin (3 (c+d x))+225012 \sin (4 (c+d x))-112506 \sin (5 (c+d x))+18751 \sin (6 (c+d x))+431424 \sin (2 c+d x)-375552 \sin (c+2 d x)-201600 \sin (3 c+2 d x)+41248 \sin (2 c+3 d x)-84000 \sin (4 c+3 d x)+155712 \sin (3 c+4 d x)+100800 \sin (5 c+4 d x)-98016 \sin (4 c+5 d x)-30240 \sin (6 c+5 d x)+21376 \sin (5 c+6 d x)+181440 d x \cos (2 c+d x)+136080 d x \cos (c+2 d x)-136080 d x \cos (3 c+2 d x)+10080 d x \cos (2 c+3 d x)-10080 d x \cos (4 c+3 d x)-60480 d x \cos (3 c+4 d x)+60480 d x \cos (5 c+4 d x)+30240 d x \cos (4 c+5 d x)-30240 d x \cos (6 c+5 d x)-5040 d x \cos (5 c+6 d x)+5040 d x \cos (7 c+6 d x)-169344 \sin (c)+338112 \sin (d x)-181440 d x \cos (d x))}{41287680 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]

[Out]

(a^3*Csc[c/2]*Csc[(c + d*x)/2]^9*Sec[c/2]*Sec[(c + d*x)/2]^3*(-181440*d*x*Cos[d*x] + 181440*d*x*Cos[2*c + d*x]
+ 136080*d*x*Cos[c + 2*d*x] - 136080*d*x*Cos[3*c + 2*d*x] + 10080*d*x*Cos[2*c + 3*d*x] - 10080*d*x*Cos[4*c +
3*d*x] - 60480*d*x*Cos[3*c + 4*d*x] + 60480*d*x*Cos[5*c + 4*d*x] + 30240*d*x*Cos[4*c + 5*d*x] - 30240*d*x*Cos[
6*c + 5*d*x] - 5040*d*x*Cos[5*c + 6*d*x] + 5040*d*x*Cos[7*c + 6*d*x] - 169344*Sin[c] + 338112*Sin[d*x] + 67503
6*Sin[c + d*x] - 506277*Sin[2*(c + d*x)] - 37502*Sin[3*(c + d*x)] + 225012*Sin[4*(c + d*x)] - 112506*Sin[5*(c
+ d*x)] + 18751*Sin[6*(c + d*x)] + 431424*Sin[2*c + d*x] - 375552*Sin[c + 2*d*x] - 201600*Sin[3*c + 2*d*x] + 4
1248*Sin[2*c + 3*d*x] - 84000*Sin[4*c + 3*d*x] + 155712*Sin[3*c + 4*d*x] + 100800*Sin[5*c + 4*d*x] - 98016*Sin
[4*c + 5*d*x] - 30240*Sin[6*c + 5*d*x] + 21376*Sin[5*c + 6*d*x]))/(41287680*d)

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Maple [B]  time = 0.086, size = 364, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x)

[Out]

1/d*(a^3*(-1/9*cot(d*x+c)^9+1/7*cot(d*x+c)^7-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+3*a^3*(-1/9/s
in(d*x+c)^9*cos(d*x+c)^10+1/63/sin(d*x+c)^7*cos(d*x+c)^10-1/105/sin(d*x+c)^5*cos(d*x+c)^10+1/63/sin(d*x+c)^3*c
os(d*x+c)^10-1/9/sin(d*x+c)*cos(d*x+c)^10-1/9*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*c
os(d*x+c)^2)*sin(d*x+c))-1/3*a^3/sin(d*x+c)^9*cos(d*x+c)^9+a^3*(-1/9/sin(d*x+c)^9*cos(d*x+c)^8-1/63/sin(d*x+c)
^7*cos(d*x+c)^8+1/315/sin(d*x+c)^5*cos(d*x+c)^8-1/315/sin(d*x+c)^3*cos(d*x+c)^8+1/63/sin(d*x+c)*cos(d*x+c)^8+1
/63*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c)))

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Maxima [A]  time = 1.69417, size = 246, normalized size = 1.37 \begin{align*} -\frac{{\left (315 \, d x + 315 \, c + \frac{315 \, \tan \left (d x + c\right )^{8} - 105 \, \tan \left (d x + c\right )^{6} + 63 \, \tan \left (d x + c\right )^{4} - 45 \, \tan \left (d x + c\right )^{2} + 35}{\tan \left (d x + c\right )^{9}}\right )} a^{3} + \frac{3 \,{\left (315 \, \sin \left (d x + c\right )^{8} - 420 \, \sin \left (d x + c\right )^{6} + 378 \, \sin \left (d x + c\right )^{4} - 180 \, \sin \left (d x + c\right )^{2} + 35\right )} a^{3}}{\sin \left (d x + c\right )^{9}} - \frac{{\left (105 \, \sin \left (d x + c\right )^{6} - 189 \, \sin \left (d x + c\right )^{4} + 135 \, \sin \left (d x + c\right )^{2} - 35\right )} a^{3}}{\sin \left (d x + c\right )^{9}} + \frac{105 \, a^{3}}{\tan \left (d x + c\right )^{9}}}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/315*((315*d*x + 315*c + (315*tan(d*x + c)^8 - 105*tan(d*x + c)^6 + 63*tan(d*x + c)^4 - 45*tan(d*x + c)^2 +
35)/tan(d*x + c)^9)*a^3 + 3*(315*sin(d*x + c)^8 - 420*sin(d*x + c)^6 + 378*sin(d*x + c)^4 - 180*sin(d*x + c)^2
+ 35)*a^3/sin(d*x + c)^9 - (105*sin(d*x + c)^6 - 189*sin(d*x + c)^4 + 135*sin(d*x + c)^2 - 35)*a^3/sin(d*x +
c)^9 + 105*a^3/tan(d*x + c)^9)/d

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Fricas [A]  time = 1.17243, size = 605, normalized size = 3.38 \begin{align*} -\frac{668 \, a^{3} \cos \left (d x + c\right )^{6} - 1059 \, a^{3} \cos \left (d x + c\right )^{5} - 573 \, a^{3} \cos \left (d x + c\right )^{4} + 1813 \, a^{3} \cos \left (d x + c\right )^{3} - 393 \, a^{3} \cos \left (d x + c\right )^{2} - 789 \, a^{3} \cos \left (d x + c\right ) + 368 \, a^{3} + 315 \,{\left (a^{3} d x \cos \left (d x + c\right )^{5} - 3 \, a^{3} d x \cos \left (d x + c\right )^{4} + 2 \, a^{3} d x \cos \left (d x + c\right )^{3} + 2 \, a^{3} d x \cos \left (d x + c\right )^{2} - 3 \, a^{3} d x \cos \left (d x + c\right ) + a^{3} d x\right )} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - 3 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/315*(668*a^3*cos(d*x + c)^6 - 1059*a^3*cos(d*x + c)^5 - 573*a^3*cos(d*x + c)^4 + 1813*a^3*cos(d*x + c)^3 -
393*a^3*cos(d*x + c)^2 - 789*a^3*cos(d*x + c) + 368*a^3 + 315*(a^3*d*x*cos(d*x + c)^5 - 3*a^3*d*x*cos(d*x + c)
^4 + 2*a^3*d*x*cos(d*x + c)^3 + 2*a^3*d*x*cos(d*x + c)^2 - 3*a^3*d*x*cos(d*x + c) + a^3*d*x)*sin(d*x + c))/((d
*cos(d*x + c)^5 - 3*d*cos(d*x + c)^4 + 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 - 3*d*cos(d*x + c) + d)*sin(d*x
+ c))

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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)**10*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.55588, size = 173, normalized size = 0.97 \begin{align*} -\frac{105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 20160 \,{\left (d x + c\right )} a^{3} - 2520 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{31185 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 6720 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1827 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 360 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}}}{20160 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/20160*(105*a^3*tan(1/2*d*x + 1/2*c)^3 + 20160*(d*x + c)*a^3 - 2520*a^3*tan(1/2*d*x + 1/2*c) + (31185*a^3*ta
n(1/2*d*x + 1/2*c)^8 - 6720*a^3*tan(1/2*d*x + 1/2*c)^6 + 1827*a^3*tan(1/2*d*x + 1/2*c)^4 - 360*a^3*tan(1/2*d*x
+ 1/2*c)^2 + 35*a^3)/tan(1/2*d*x + 1/2*c)^9)/d