3.36 \(\int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=179 \[ -\frac{2 a^2 \cot ^9(c+d x)}{9 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x \]

[Out]

-(a^2*x) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (a^2*Cot[c + d*x]^
7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Csc[c + d*x])/d + (8*a^2*Csc[c + d*x]^3)/(3*d) - (12*a^2*Csc[
c + d*x]^5)/(5*d) + (8*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x]^9)/(9*d)

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Rubi [A]  time = 0.154976, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30} \[ -\frac{2 a^2 \cot ^9(c+d x)}{9 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]

[Out]

-(a^2*x) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (a^2*Cot[c + d*x]^
7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Csc[c + d*x])/d + (8*a^2*Csc[c + d*x]^3)/(3*d) - (12*a^2*Csc[
c + d*x]^5)/(5*d) + (8*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*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]

Rubi steps

\begin{align*} \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^{10}(c+d x)+2 a^2 \cot ^9(c+d x) \csc (c+d x)+a^2 \cot ^8(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^{10}(c+d x) \, dx+a^2 \int \cot ^8(c+d x) \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^9(c+d x) \csc (c+d x) \, dx\\ &=-\frac{a^2 \cot ^9(c+d x)}{9 d}-a^2 \int \cot ^8(c+d x) \, dx+\frac{a^2 \operatorname{Subst}\left (\int x^8 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}+a^2 \int \cot ^6(c+d x) \, dx-\frac{\left (2 a^2\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^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}-a^2 \int \cot ^4(c+d x) \, dx\\ &=\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}+a^2 \int \cot ^2(c+d x) \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}-a^2 \int 1 \, dx\\ &=-a^2 x-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [B]  time = 1.87963, size = 428, normalized size = 2.39 \[ -\frac{a^2 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^9\left (\frac{1}{2} (c+d x)\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) (-1152405 \sin (c+d x)+512180 \sin (2 (c+d x))+486571 \sin (3 (c+d x))-409744 \sin (4 (c+d x))-25609 \sin (5 (c+d x))+102436 \sin (6 (c+d x))-25609 \sin (7 (c+d x))-825216 \sin (2 c+d x)+622976 \sin (c+2 d x)+142464 \sin (3 c+2 d x)+297088 \sin (2 c+3 d x)+430080 \sin (4 c+3 d x)-424192 \sin (3 c+4 d x)-188160 \sin (5 c+4 d x)+2048 \sin (4 c+5 d x)-40320 \sin (6 c+5 d x)+112768 \sin (5 c+6 d x)+40320 \sin (7 c+6 d x)-38272 \sin (6 c+7 d x)-453600 d x \cos (2 c+d x)-201600 d x \cos (c+2 d x)+201600 d x \cos (3 c+2 d x)-191520 d x \cos (2 c+3 d x)+191520 d x \cos (4 c+3 d x)+161280 d x \cos (3 c+4 d x)-161280 d x \cos (5 c+4 d x)+10080 d x \cos (4 c+5 d x)-10080 d x \cos (6 c+5 d x)-40320 d x \cos (5 c+6 d x)+40320 d x \cos (7 c+6 d x)+10080 d x \cos (6 c+7 d x)-10080 d x \cos (8 c+7 d x)+259584 \sin (c)-897024 \sin (d x)+453600 d x \cos (d x))}{330301440 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]

[Out]

-(a^2*Csc[c/2]*Csc[(c + d*x)/2]^9*Sec[c/2]*Sec[(c + d*x)/2]^5*(453600*d*x*Cos[d*x] - 453600*d*x*Cos[2*c + d*x]
- 201600*d*x*Cos[c + 2*d*x] + 201600*d*x*Cos[3*c + 2*d*x] - 191520*d*x*Cos[2*c + 3*d*x] + 191520*d*x*Cos[4*c
+ 3*d*x] + 161280*d*x*Cos[3*c + 4*d*x] - 161280*d*x*Cos[5*c + 4*d*x] + 10080*d*x*Cos[4*c + 5*d*x] - 10080*d*x*
Cos[6*c + 5*d*x] - 40320*d*x*Cos[5*c + 6*d*x] + 40320*d*x*Cos[7*c + 6*d*x] + 10080*d*x*Cos[6*c + 7*d*x] - 1008
0*d*x*Cos[8*c + 7*d*x] + 259584*Sin[c] - 897024*Sin[d*x] - 1152405*Sin[c + d*x] + 512180*Sin[2*(c + d*x)] + 48
6571*Sin[3*(c + d*x)] - 409744*Sin[4*(c + d*x)] - 25609*Sin[5*(c + d*x)] + 102436*Sin[6*(c + d*x)] - 25609*Sin
[7*(c + d*x)] - 825216*Sin[2*c + d*x] + 622976*Sin[c + 2*d*x] + 142464*Sin[3*c + 2*d*x] + 297088*Sin[2*c + 3*d
*x] + 430080*Sin[4*c + 3*d*x] - 424192*Sin[3*c + 4*d*x] - 188160*Sin[5*c + 4*d*x] + 2048*Sin[4*c + 5*d*x] - 40
320*Sin[6*c + 5*d*x] + 112768*Sin[5*c + 6*d*x] + 40320*Sin[7*c + 6*d*x] - 38272*Sin[6*c + 7*d*x]))/(330301440*
d)

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Maple [A]  time = 0.087, size = 231, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}-\cot \left ( dx+c \right ) -dx-c \right ) +2\,{a}^{2} \left ( -1/9\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{63\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{63\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-1/9\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{\sin \left ( dx+c \right ) }}-1/9\, \left ({\frac{128}{35}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{9}}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}} \right ) } \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))^2,x)

[Out]

1/d*(a^2*(-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)+2*a^2*(-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))-1/9*a^2/sin(d*x+c)^9*cos(d*x+c)^9)

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Maxima [A]  time = 2.10935, size = 185, normalized size = 1.03 \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^{2} + \frac{2 \,{\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^{2}}{\sin \left (d x + c\right )^{9}} + \frac{35 \, a^{2}}{\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))^2,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^2 + 2*(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^2/sin(d*x + c)^9 + 35*a^2/tan(d*x + c)^9)/d

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Fricas [A]  time = 0.947558, size = 694, normalized size = 3.88 \begin{align*} -\frac{598 \, a^{2} \cos \left (d x + c\right )^{7} - 566 \, a^{2} \cos \left (d x + c\right )^{6} - 1212 \, a^{2} \cos \left (d x + c\right )^{5} + 1310 \, a^{2} \cos \left (d x + c\right )^{4} + 860 \, a^{2} \cos \left (d x + c\right )^{3} - 1014 \, a^{2} \cos \left (d x + c\right )^{2} - 197 \, a^{2} \cos \left (d x + c\right ) + 256 \, a^{2} + 315 \,{\left (a^{2} d x \cos \left (d x + c\right )^{6} - 2 \, a^{2} d x \cos \left (d x + c\right )^{5} - a^{2} d x \cos \left (d x + c\right )^{4} + 4 \, a^{2} d x \cos \left (d x + c\right )^{3} - a^{2} d x \cos \left (d x + c\right )^{2} - 2 \, a^{2} d x \cos \left (d x + c\right ) + a^{2} d x\right )} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} + 4 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - 2 \, 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))^2,x, algorithm="fricas")

[Out]

-1/315*(598*a^2*cos(d*x + c)^7 - 566*a^2*cos(d*x + c)^6 - 1212*a^2*cos(d*x + c)^5 + 1310*a^2*cos(d*x + c)^4 +
860*a^2*cos(d*x + c)^3 - 1014*a^2*cos(d*x + c)^2 - 197*a^2*cos(d*x + c) + 256*a^2 + 315*(a^2*d*x*cos(d*x + c)^
6 - 2*a^2*d*x*cos(d*x + c)^5 - a^2*d*x*cos(d*x + c)^4 + 4*a^2*d*x*cos(d*x + c)^3 - a^2*d*x*cos(d*x + c)^2 - 2*
a^2*d*x*cos(d*x + c) + a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^6 - 2*d*cos(d*x + c)^5 - d*cos(d*x + c)^4 + 4*d
*cos(d*x + c)^3 - d*cos(d*x + c)^2 - 2*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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.54551, size = 196, normalized size = 1.09 \begin{align*} \frac{63 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 945 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40320 \,{\left (d x + c\right )} a^{2} + 11655 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{51345 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 9765 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2331 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 405 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}}}{40320 \, 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))^2,x, algorithm="giac")

[Out]

1/40320*(63*a^2*tan(1/2*d*x + 1/2*c)^5 - 945*a^2*tan(1/2*d*x + 1/2*c)^3 - 40320*(d*x + c)*a^2 + 11655*a^2*tan(
1/2*d*x + 1/2*c) - (51345*a^2*tan(1/2*d*x + 1/2*c)^8 - 9765*a^2*tan(1/2*d*x + 1/2*c)^6 + 2331*a^2*tan(1/2*d*x
+ 1/2*c)^4 - 405*a^2*tan(1/2*d*x + 1/2*c)^2 + 35*a^2)/tan(1/2*d*x + 1/2*c)^9)/d