3.85 \(\int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3888, 3886, 3473, 8, 2606, 194, 2607, 30} \[ -\frac {2 \cot ^9(c+d x)}{9 a^2 d}+\frac {\cot ^7(c+d x)}{7 a^2 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}+\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {2 \csc ^9(c+d x)}{9 a^2 d}-\frac {8 \csc ^7(c+d x)}{7 a^2 d}+\frac {12 \csc ^5(c+d x)}{5 a^2 d}-\frac {8 \csc ^3(c+d x)}{3 a^2 d}+\frac {2 \csc (c+d x)}{a^2 d}-\frac {x}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]

[Out]

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

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rubi steps

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

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Mathematica [B]  time = 6.57, size = 802, normalized size = 4.48 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right )}{288 d (\sec (c+d x) a+a)^2}+\frac {\sec ^2(c+d x) \tan \left (\frac {c}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{288 d (\sec (c+d x) a+a)^2}-\frac {109 \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right )}{2016 d (\sec (c+d x) a+a)^2}-\frac {109 \sec ^2(c+d x) \tan \left (\frac {c}{2}\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2016 d (\sec (c+d x) a+a)^2}+\frac {313 \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right )}{840 d (\sec (c+d x) a+a)^2}-\frac {17 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \cot ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{160 d (\sec (c+d x) a+a)^2}+\frac {201 \cos ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \cot \left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{160 d (\sec (c+d x) a+a)^2}+\frac {\cot ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{160 d (\sec (c+d x) a+a)^2}+\frac {63881 \cos ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{10080 d (\sec (c+d x) a+a)^2}-\frac {7891 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{5040 d (\sec (c+d x) a+a)^2}-\frac {7891 \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \tan \left (\frac {c}{2}\right )}{5040 d (\sec (c+d x) a+a)^2}+\frac {313 \sec ^2(c+d x) \tan \left (\frac {c}{2}\right )}{840 d (\sec (c+d x) a+a)^2}-\frac {4 x \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{(\sec (c+d x) a+a)^2}-\frac {\cot \left (\frac {c}{2}\right ) \cot ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{160 d (\sec (c+d x) a+a)^2}+\frac {17 \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \cot \left (\frac {c}{2}\right ) \cot ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{160 d (\sec (c+d x) a+a)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]

[Out]

(-4*x*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2 + (17*Cos[c/2 + (d*x)/2]^2*Cot[c/2]*Cot[c/2
+ (d*x)/2]^2*Sec[c + d*x]^2)/(160*d*(a + a*Sec[c + d*x])^2) - (Cot[c/2]*Cot[c/2 + (d*x)/2]^4*Sec[c + d*x]^2)/(
160*d*(a + a*Sec[c + d*x])^2) + (201*Cos[c/2 + (d*x)/2]^3*Cot[c/2 + (d*x)/2]*Csc[c/2]*Sec[c + d*x]^2*Sin[(d*x)
/2])/(160*d*(a + a*Sec[c + d*x])^2) - (17*Cos[c/2 + (d*x)/2]*Cot[c/2 + (d*x)/2]^3*Csc[c/2]*Sec[c + d*x]^2*Sin[
(d*x)/2])/(160*d*(a + a*Sec[c + d*x])^2) + (Cot[c/2 + (d*x)/2]^4*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c + d*x]^2*Si
n[(d*x)/2])/(160*d*(a + a*Sec[c + d*x])^2) - (7891*Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(5
040*d*(a + a*Sec[c + d*x])^2) + (63881*Cos[c/2 + (d*x)/2]^3*Sec[c/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(10080*d*(a
+ a*Sec[c + d*x])^2) + (313*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(840*d*(a + a*Sec[c + d*x
])^2) - (109*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sec[c + d*x]^2*Sin[(d*x)/2])/(2016*d*(a + a*Sec[c + d*x])^2) + (Sec
[c/2]*Sec[c/2 + (d*x)/2]^5*Sec[c + d*x]^2*Sin[(d*x)/2])/(288*d*(a + a*Sec[c + d*x])^2) + (313*Sec[c + d*x]^2*T
an[c/2])/(840*d*(a + a*Sec[c + d*x])^2) - (7891*Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]^2*Tan[c/2])/(5040*d*(a + a*S
ec[c + d*x])^2) - (109*Sec[c/2 + (d*x)/2]^2*Sec[c + d*x]^2*Tan[c/2])/(2016*d*(a + a*Sec[c + d*x])^2) + (Sec[c/
2 + (d*x)/2]^4*Sec[c + d*x]^2*Tan[c/2])/(288*d*(a + a*Sec[c + d*x])^2)

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fricas [A]  time = 0.70, size = 250, normalized size = 1.40 \[ -\frac {598 \, \cos \left (d x + c\right )^{7} + 566 \, \cos \left (d x + c\right )^{6} - 1212 \, \cos \left (d x + c\right )^{5} - 1310 \, \cos \left (d x + c\right )^{4} + 860 \, \cos \left (d x + c\right )^{3} + 1014 \, \cos \left (d x + c\right )^{2} + 315 \, {\left (d x \cos \left (d x + c\right )^{6} + 2 \, d x \cos \left (d x + c\right )^{5} - d x \cos \left (d x + c\right )^{4} - 4 \, d x \cos \left (d x + c\right )^{3} - d x \cos \left (d x + c\right )^{2} + 2 \, d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) - 197 \, \cos \left (d x + c\right ) - 256}{315 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.45, size = 144, normalized size = 0.80 \[ -\frac {\frac {40320 \, {\left (d x + c\right )}}{a^{2}} + \frac {63 \, {\left (185 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {35 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 405 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2331 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9765 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 51345 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{18}}}{40320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

-1/40320*(40320*(d*x + c)/a^2 + 63*(185*tan(1/2*d*x + 1/2*c)^4 - 15*tan(1/2*d*x + 1/2*c)^2 + 1)/(a^2*tan(1/2*d
*x + 1/2*c)^5) - (35*a^16*tan(1/2*d*x + 1/2*c)^9 - 405*a^16*tan(1/2*d*x + 1/2*c)^7 + 2331*a^16*tan(1/2*d*x + 1
/2*c)^5 - 9765*a^16*tan(1/2*d*x + 1/2*c)^3 + 51345*a^16*tan(1/2*d*x + 1/2*c))/a^18)/d

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maple [A]  time = 0.85, size = 170, normalized size = 0.95 \[ \frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{1152 a^{2} d}-\frac {9 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{896 a^{2} d}+\frac {37 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 a^{2} d}-\frac {31 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{2} d}+\frac {163 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a^{2} d}-\frac {1}{640 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{128 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {37}{128 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6/(a+a*sec(d*x+c))^2,x)

[Out]

1/1152/a^2/d*tan(1/2*d*x+1/2*c)^9-9/896/a^2/d*tan(1/2*d*x+1/2*c)^7+37/640/a^2/d*tan(1/2*d*x+1/2*c)^5-31/128/a^
2/d*tan(1/2*d*x+1/2*c)^3+163/128/a^2/d*tan(1/2*d*x+1/2*c)-1/640/a^2/d/tan(1/2*d*x+1/2*c)^5+3/128/a^2/d/tan(1/2
*d*x+1/2*c)^3-37/128/a^2/d/tan(1/2*d*x+1/2*c)-2/a^2/d*arctan(tan(1/2*d*x+1/2*c))

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maxima [A]  time = 0.89, size = 197, normalized size = 1.10 \[ \frac {\frac {\frac {51345 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9765 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2331 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {405 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{2}} - \frac {80640 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {63 \, {\left (\frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {185 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{2} \sin \left (d x + c\right )^{5}}}{40320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

1/40320*((51345*sin(d*x + c)/(cos(d*x + c) + 1) - 9765*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2331*sin(d*x + c)
^5/(cos(d*x + c) + 1)^5 - 405*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^
2 - 80640*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + 63*(15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 185*sin(d
*x + c)^4/(cos(d*x + c) + 1)^4 - 1)*(cos(d*x + c) + 1)^5/(a^2*sin(d*x + c)^5))/d

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mupad [B]  time = 2.51, size = 230, normalized size = 1.28 \[ -\frac {63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+405\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-2331\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+9765\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-51345\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+11655\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+40320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{40320\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{6}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6/(a+a*sec(d*x+c))**2,x)

[Out]

Integral(cot(c + d*x)**6/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x)/a**2

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