3.103 \(\int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=215 \[ \frac {4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac {19 \csc ^9(c+d x)}{9 a^3 d}-\frac {36 \csc ^7(c+d x)}{7 a^3 d}+\frac {34 \csc ^5(c+d x)}{5 a^3 d}-\frac {16 \csc ^3(c+d x)}{3 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {x}{a^3} \]

[Out]

-x/a^3-cot(d*x+c)/a^3/d+1/3*cot(d*x+c)^3/a^3/d-1/5*cot(d*x+c)^5/a^3/d+1/7*cot(d*x+c)^7/a^3/d-1/9*cot(d*x+c)^9/
a^3/d+4/11*cot(d*x+c)^11/a^3/d+3*csc(d*x+c)/a^3/d-16/3*csc(d*x+c)^3/a^3/d+34/5*csc(d*x+c)^5/a^3/d-36/7*csc(d*x
+c)^7/a^3/d+19/9*csc(d*x+c)^9/a^3/d-4/11*csc(d*x+c)^11/a^3/d

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Rubi [A]  time = 0.28, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3888, 3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ \frac {4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac {19 \csc ^9(c+d x)}{9 a^3 d}-\frac {36 \csc ^7(c+d x)}{7 a^3 d}+\frac {34 \csc ^5(c+d x)}{5 a^3 d}-\frac {16 \csc ^3(c+d x)}{3 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {x}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]

[Out]

-(x/a^3) - Cot[c + d*x]/(a^3*d) + Cot[c + d*x]^3/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + Cot[c + d*x]^7/(7*a^3*
d) - Cot[c + d*x]^9/(9*a^3*d) + (4*Cot[c + d*x]^11)/(11*a^3*d) + (3*Csc[c + d*x])/(a^3*d) - (16*Csc[c + d*x]^3
)/(3*a^3*d) + (34*Csc[c + d*x]^5)/(5*a^3*d) - (36*Csc[c + d*x]^7)/(7*a^3*d) + (19*Csc[c + d*x]^9)/(9*a^3*d) -
(4*Csc[c + d*x]^11)/(11*a^3*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 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]

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))^3} \, dx &=\frac {\int \cot ^{12}(c+d x) (-a+a \sec (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \cot ^{12}(c+d x)+3 a^3 \cot ^{11}(c+d x) \csc (c+d x)-3 a^3 \cot ^{10}(c+d x) \csc ^2(c+d x)+a^3 \cot ^9(c+d x) \csc ^3(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \cot ^{12}(c+d x) \, dx}{a^3}+\frac {\int \cot ^9(c+d x) \csc ^3(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^{11}(c+d x) \csc (c+d x) \, dx}{a^3}-\frac {3 \int \cot ^{10}(c+d x) \csc ^2(c+d x) \, dx}{a^3}\\ &=\frac {\cot ^{11}(c+d x)}{11 a^3 d}+\frac {\int \cot ^{10}(c+d x) \, dx}{a^3}-\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^{10} \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (-1+x^2\right )^5 \, dx,x,\csc (c+d x)\right )}{a^3 d}\\ &=-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac {\int \cot ^8(c+d x) \, dx}{a^3}-\frac {\operatorname {Subst}\left (\int \left (x^2-4 x^4+6 x^6-4 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (-1+5 x^2-10 x^4+10 x^6-5 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}\\ &=\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {16 \csc ^3(c+d x)}{3 a^3 d}+\frac {34 \csc ^5(c+d x)}{5 a^3 d}-\frac {36 \csc ^7(c+d x)}{7 a^3 d}+\frac {19 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac {\int \cot ^6(c+d x) \, dx}{a^3}\\ &=-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {16 \csc ^3(c+d x)}{3 a^3 d}+\frac {34 \csc ^5(c+d x)}{5 a^3 d}-\frac {36 \csc ^7(c+d x)}{7 a^3 d}+\frac {19 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {\int \cot ^4(c+d x) \, dx}{a^3}\\ &=\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {16 \csc ^3(c+d x)}{3 a^3 d}+\frac {34 \csc ^5(c+d x)}{5 a^3 d}-\frac {36 \csc ^7(c+d x)}{7 a^3 d}+\frac {19 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac {\int \cot ^2(c+d x) \, dx}{a^3}\\ &=-\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {16 \csc ^3(c+d x)}{3 a^3 d}+\frac {34 \csc ^5(c+d x)}{5 a^3 d}-\frac {36 \csc ^7(c+d x)}{7 a^3 d}+\frac {19 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {\int 1 \, dx}{a^3}\\ &=-\frac {x}{a^3}-\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {16 \csc ^3(c+d x)}{3 a^3 d}+\frac {34 \csc ^5(c+d x)}{5 a^3 d}-\frac {36 \csc ^7(c+d x)}{7 a^3 d}+\frac {19 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}\\ \end {align*}

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Mathematica [A]  time = 3.73, size = 394, normalized size = 1.83 \[ -\frac {\tan \left (\frac {c}{2}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (315 \sec ^{10}\left (\frac {1}{2} (c+d x)\right )-5425 \sec ^8\left (\frac {1}{2} (c+d x)\right )+41320 \sec ^6\left (\frac {1}{2} (c+d x)\right )-184650 \sec ^4\left (\frac {1}{2} (c+d x)\right )+561145 \sec ^2\left (\frac {1}{2} (c+d x)\right )+6468 \sin (c) \csc ^3\left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \csc ^3\left (\frac {1}{2} (c+d x)\right )+231 \cot ^2\left (\frac {c}{2}\right ) (28 \cos (c+d x)-25) \csc ^4\left (\frac {1}{2} (c+d x)\right )+231 \cot \left (\frac {c}{2}\right ) \left (3840 d x-\csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \csc \left (\frac {1}{2} (c+d x)\right ) \left (3 \csc ^4\left (\frac {1}{2} (c+d x)\right )+743\right )\right )+315 \csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^{11}\left (\frac {1}{2} (c+d x)\right )-5425 \csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right )+41320 \csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right )-184650 \csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )+561145 \csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right )-1736335 \csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right )\right )}{110880 a^3 d (\sec (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]

[Out]

-1/110880*(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(231*(-25 + 28*Cos[c + d*x])*Cot[c/2]^2*Csc[(c + d*x)/2]^4 + 5611
45*Sec[(c + d*x)/2]^2 - 184650*Sec[(c + d*x)/2]^4 + 41320*Sec[(c + d*x)/2]^6 - 5425*Sec[(c + d*x)/2]^8 + 315*S
ec[(c + d*x)/2]^10 - 1736335*Csc[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] + 561145*Csc[c/2]*Sec[(c + d*x)/2]^3*Sin[(
d*x)/2] - 184650*Csc[c/2]*Sec[(c + d*x)/2]^5*Sin[(d*x)/2] + 41320*Csc[c/2]*Sec[(c + d*x)/2]^7*Sin[(d*x)/2] - 5
425*Csc[c/2]*Sec[(c + d*x)/2]^9*Sin[(d*x)/2] + 315*Csc[c/2]*Sec[(c + d*x)/2]^11*Sin[(d*x)/2] + 6468*Csc[c/2]^3
*Csc[(c + d*x)/2]^3*Sin[c]*Sin[(d*x)/2] + 231*Cot[c/2]*(3840*d*x - Csc[c/2]*Csc[(c + d*x)/2]*(743 + 3*Csc[(c +
d*x)/2]^4)*Sin[(d*x)/2]))*Tan[c/2])/(a^3*d*(1 + Sec[c + d*x])^3)

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fricas [A]  time = 1.16, size = 282, normalized size = 1.31 \[ -\frac {7453 \, \cos \left (d x + c\right )^{8} + 11964 \, \cos \left (d x + c\right )^{7} - 11866 \, \cos \left (d x + c\right )^{6} - 30542 \, \cos \left (d x + c\right )^{5} + 90 \, \cos \left (d x + c\right )^{4} + 26438 \, \cos \left (d x + c\right )^{3} + 8539 \, \cos \left (d x + c\right )^{2} + 3465 \, {\left (d x \cos \left (d x + c\right )^{7} + 3 \, d x \cos \left (d x + c\right )^{6} + d x \cos \left (d x + c\right )^{5} - 5 \, d x \cos \left (d x + c\right )^{4} - 5 \, d x \cos \left (d x + c\right )^{3} + d x \cos \left (d x + c\right )^{2} + 3 \, d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) - 7671 \, \cos \left (d x + c\right ) - 3712}{3465 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/3465*(7453*cos(d*x + c)^8 + 11964*cos(d*x + c)^7 - 11866*cos(d*x + c)^6 - 30542*cos(d*x + c)^5 + 90*cos(d*x
+ c)^4 + 26438*cos(d*x + c)^3 + 8539*cos(d*x + c)^2 + 3465*(d*x*cos(d*x + c)^7 + 3*d*x*cos(d*x + c)^6 + d*x*c
os(d*x + c)^5 - 5*d*x*cos(d*x + c)^4 - 5*d*x*cos(d*x + c)^3 + d*x*cos(d*x + c)^2 + 3*d*x*cos(d*x + c) + d*x)*s
in(d*x + c) - 7671*cos(d*x + c) - 3712)/((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5
- 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(
d*x + c))

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giac [A]  time = 1.47, size = 160, normalized size = 0.74 \[ -\frac {\frac {887040 \, {\left (d x + c\right )}}{a^{3}} + \frac {231 \, {\left (690 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 50 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {5 \, {\left (63 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 770 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 18018 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 59136 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 264726 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{33}}}{887040 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/887040*(887040*(d*x + c)/a^3 + 231*(690*tan(1/2*d*x + 1/2*c)^4 - 50*tan(1/2*d*x + 1/2*c)^2 + 3)/(a^3*tan(1/
2*d*x + 1/2*c)^5) + 5*(63*a^30*tan(1/2*d*x + 1/2*c)^11 - 770*a^30*tan(1/2*d*x + 1/2*c)^9 + 4554*a^30*tan(1/2*d
*x + 1/2*c)^7 - 18018*a^30*tan(1/2*d*x + 1/2*c)^5 + 59136*a^30*tan(1/2*d*x + 1/2*c)^3 - 264726*a^30*tan(1/2*d*
x + 1/2*c))/a^33)/d

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maple [A]  time = 1.02, size = 189, normalized size = 0.88 \[ -\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2816 a^{3} d}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1152 a^{3} d}-\frac {23 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{896 a^{3} d}+\frac {13 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{3}}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d \,a^{3}}+\frac {191 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{3}}-\frac {1}{1280 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {5}{384 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {23}{128 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x)

[Out]

-1/2816/a^3/d*tan(1/2*d*x+1/2*c)^11+5/1152/a^3/d*tan(1/2*d*x+1/2*c)^9-23/896/a^3/d*tan(1/2*d*x+1/2*c)^7+13/128
/d/a^3*tan(1/2*d*x+1/2*c)^5-1/3/d/a^3*tan(1/2*d*x+1/2*c)^3+191/128/d/a^3*tan(1/2*d*x+1/2*c)-1/1280/a^3/d/tan(1
/2*d*x+1/2*c)^5+5/384/a^3/d/tan(1/2*d*x+1/2*c)^3-23/128/a^3/d/tan(1/2*d*x+1/2*c)-2/d/a^3*arctan(tan(1/2*d*x+1/
2*c))

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maxima [A]  time = 0.56, size = 218, normalized size = 1.01 \[ \frac {\frac {5 \, {\left (\frac {264726 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {59136 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {18018 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {4554 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )}}{a^{3}} - \frac {1774080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {231 \, {\left (\frac {50 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {690 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{887040 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/887040*(5*(264726*sin(d*x + c)/(cos(d*x + c) + 1) - 59136*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 18018*sin(d*
x + c)^5/(cos(d*x + c) + 1)^5 - 4554*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 770*sin(d*x + c)^9/(cos(d*x + c) +
1)^9 - 63*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/a^3 - 1774080*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 + 2
31*(50*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 690*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3)*(cos(d*x + c) + 1)^5
/(a^3*sin(d*x + c)^5))/d

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mupad [B]  time = 3.24, size = 254, normalized size = 1.18 \[ -\frac {693\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+315\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-3850\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+22770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-90090\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+295680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-1323630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+159390\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-11550\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+887040\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{887040\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^3,x)

[Out]

-(693*cos(c/2 + (d*x)/2)^16 + 315*sin(c/2 + (d*x)/2)^16 - 3850*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^14 + 22
770*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^12 - 90090*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 295680*cos
(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8 - 1323630*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 + 159390*cos(c/2 +
(d*x)/2)^12*sin(c/2 + (d*x)/2)^4 - 11550*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2 + 887040*cos(c/2 + (d*x)/
2)^11*sin(c/2 + (d*x)/2)^5*(c + d*x))/(887040*a^3*d*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(cot(c + d*x)**6/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

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