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

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Rubi [A]  time = 0.15, antiderivative size = 179, normalized size of antiderivative = 1.00, 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 8

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

Rule 30

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

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*}

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Mathematica [B]  time = 1.99, 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]

-1/330301440*(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*Co
s[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] - 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*S
in[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]))
/d

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fricas [A]  time = 0.61, size = 274, normalized size = 1.53 \[ -\frac {598 \, a^{2} \cos \left (d x + c\right )^{7} - 566 \, a^{2} \cos \left (d x + c\right )^{6} - 1212 \, a^{2} \cos \left (d x + c\right )^{5} + 1310 \, a^{2} \cos \left (d x + c\right )^{4} + 860 \, a^{2} \cos \left (d x + c\right )^{3} - 1014 \, a^{2} \cos \left (d x + c\right )^{2} - 197 \, a^{2} \cos \left (d x + c\right ) + 256 \, a^{2} + 315 \, {\left (a^{2} d x \cos \left (d x + c\right )^{6} - 2 \, a^{2} d x \cos \left (d x + c\right )^{5} - a^{2} d x \cos \left (d x + c\right )^{4} + 4 \, a^{2} d x \cos \left (d x + c\right )^{3} - a^{2} d x \cos \left (d x + c\right )^{2} - 2 \, a^{2} d x \cos \left (d x + c\right ) + a^{2} d x\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} + 4 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \]

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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giac [A]  time = 0.43, size = 145, normalized size = 0.81 \[ \frac {63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 945 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40320 \, {\left (d x + c\right )} a^{2} + 11655 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {51345 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 9765 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2331 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 405 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{40320 \, d} \]

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

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maple [A]  time = 1.03, size = 231, normalized size = 1.29 \[ \frac {a^{2} \left (-\frac {\left (\cot ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\cot ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a^{2} \left (-\frac {\cos ^{10}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos ^{10}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{10}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{10}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{10}\left (d x +c \right )}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}\right )-\frac {a^{2} \left (\cos ^{9}\left (d x +c \right )\right )}{9 \sin \left (d x +c \right )^{9}}}{d} \]

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 = 0.76, size = 137, normalized size = 0.77 \[ -\frac {{\left (315 \, d x + 315 \, c + \frac {315 \, \tan \left (d x + c\right )^{8} - 105 \, \tan \left (d x + c\right )^{6} + 63 \, \tan \left (d x + c\right )^{4} - 45 \, \tan \left (d x + c\right )^{2} + 35}{\tan \left (d x + c\right )^{9}}\right )} a^{2} + \frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{8} - 420 \, \sin \left (d x + c\right )^{6} + 378 \, \sin \left (d x + c\right )^{4} - 180 \, \sin \left (d x + c\right )^{2} + 35\right )} a^{2}}{\sin \left (d x + c\right )^{9}} + \frac {35 \, a^{2}}{\tan \left (d x + c\right )^{9}}}{315 \, d} \]

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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mupad [B]  time = 2.78, size = 230, normalized size = 1.28 \[ -\frac {a^2\,\left (35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-63\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-11655\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+51345\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-9765\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2331\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-405\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+40320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )\right )}{40320\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^10*(a + a/cos(c + d*x))^2,x)

[Out]

-(a^2*(35*cos(c/2 + (d*x)/2)^14 - 63*sin(c/2 + (d*x)/2)^14 + 945*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 -
11655*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 + 51345*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 - 9765*cos(
c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 2331*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 - 405*cos(c/2 + (d*x)/
2)^12*sin(c/2 + (d*x)/2)^2 + 40320*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9*(c + d*x)))/(40320*d*cos(c/2 + (d
*x)/2)^5*sin(c/2 + (d*x)/2)^9)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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