3.102 \(\int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=177 \[ \frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}+\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {\cot (c+d x)}{a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d}+\frac {15 \csc ^7(c+d x)}{7 a^3 d}-\frac {21 \csc ^5(c+d x)}{5 a^3 d}+\frac {13 \csc ^3(c+d x)}{3 a^3 d}-\frac {3 \csc (c+d x)}{a^3 d}+\frac {x}{a^3} \]

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

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Rubi [A]  time = 0.25, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ \frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}+\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {\cot (c+d x)}{a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d}+\frac {15 \csc ^7(c+d x)}{7 a^3 d}-\frac {21 \csc ^5(c+d x)}{5 a^3 d}+\frac {13 \csc ^3(c+d x)}{3 a^3 d}-\frac {3 \csc (c+d x)}{a^3 d}+\frac {x}{a^3} \]

Antiderivative was successfully verified.

[In]

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

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Mathematica [B]  time = 1.18, size = 366, normalized size = 2.07 \[ \frac {\csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \csc ^3(2 (c+d x)) (675036 \sin (c+d x)+506277 \sin (2 (c+d x))-37502 \sin (3 (c+d x))-225012 \sin (4 (c+d x))-112506 \sin (5 (c+d x))-18751 \sin (6 (c+d x))-431424 \sin (2 c+d x)-375552 \sin (c+2 d x)-201600 \sin (3 c+2 d x)-41248 \sin (2 c+3 d x)+84000 \sin (4 c+3 d x)+155712 \sin (3 c+4 d x)+100800 \sin (5 c+4 d x)+98016 \sin (4 c+5 d x)+30240 \sin (6 c+5 d x)+21376 \sin (5 c+6 d x)-181440 d x \cos (2 c+d x)+136080 d x \cos (c+2 d x)-136080 d x \cos (3 c+2 d x)-10080 d x \cos (2 c+3 d x)+10080 d x \cos (4 c+3 d x)-60480 d x \cos (3 c+4 d x)+60480 d x \cos (5 c+4 d x)-30240 d x \cos (4 c+5 d x)+30240 d x \cos (6 c+5 d x)-5040 d x \cos (5 c+6 d x)+5040 d x \cos (7 c+6 d x)-169344 \sin (c)-338112 \sin (d x)+181440 d x \cos (d x))}{80640 a^3 d (\sec (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]

[Out]

(Csc[c/2]*Csc[2*(c + d*x)]^3*Sec[c/2]*(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] + 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*Si
n[6*c + 5*d*x] + 21376*Sin[5*c + 6*d*x]))/(80640*a^3*d*(1 + Sec[c + d*x])^3)

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fricas [A]  time = 1.21, size = 216, normalized size = 1.22 \[ \frac {668 \, \cos \left (d x + c\right )^{6} + 1059 \, \cos \left (d x + c\right )^{5} - 573 \, \cos \left (d x + c\right )^{4} - 1813 \, \cos \left (d x + c\right )^{3} - 393 \, \cos \left (d x + c\right )^{2} + 315 \, {\left (d x \cos \left (d x + c\right )^{5} + 3 \, d x \cos \left (d x + c\right )^{4} + 2 \, d x \cos \left (d x + c\right )^{3} - 2 \, d x \cos \left (d x + c\right )^{2} - 3 \, d x \cos \left (d x + c\right ) - d x\right )} \sin \left (d x + c\right ) + 789 \, \cos \left (d x + c\right ) + 368}{315 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/315*(668*cos(d*x + c)^6 + 1059*cos(d*x + c)^5 - 573*cos(d*x + c)^4 - 1813*cos(d*x + c)^3 - 393*cos(d*x + c)^
2 + 315*(d*x*cos(d*x + c)^5 + 3*d*x*cos(d*x + c)^4 + 2*d*x*cos(d*x + c)^3 - 2*d*x*cos(d*x + c)^2 - 3*d*x*cos(d
*x + c) - d*x)*sin(d*x + c) + 789*cos(d*x + c) + 368)/((a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*
d*cos(d*x + c)^3 - 2*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 = 0.47, size = 131, normalized size = 0.74 \[ \frac {\frac {20160 \, {\left (d x + c\right )}}{a^{3}} + \frac {105 \, {\left (24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {35 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1827 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6720 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 31185 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{27}}}{20160 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/20160*(20160*(d*x + c)/a^3 + 105*(24*tan(1/2*d*x + 1/2*c)^2 - 1)/(a^3*tan(1/2*d*x + 1/2*c)^3) - (35*a^24*tan
(1/2*d*x + 1/2*c)^9 - 360*a^24*tan(1/2*d*x + 1/2*c)^7 + 1827*a^24*tan(1/2*d*x + 1/2*c)^5 - 6720*a^24*tan(1/2*d
*x + 1/2*c)^3 + 31185*a^24*tan(1/2*d*x + 1/2*c))/a^27)/d

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maple [A]  time = 0.88, size = 151, normalized size = 0.85 \[ -\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{576 a^{3} d}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 a^{3} d}-\frac {29 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d \,a^{3}}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d \,a^{3}}-\frac {99 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d \,a^{3}}-\frac {1}{192 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{8 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4/(a+a*sec(d*x+c))^3,x)

[Out]

-1/576/a^3/d*tan(1/2*d*x+1/2*c)^9+1/56/a^3/d*tan(1/2*d*x+1/2*c)^7-29/320/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)^3-99/64/d/a^3*tan(1/2*d*x+1/2*c)-1/192/a^3/d/tan(1/2*d*x+1/2*c)^3+1/8/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 = 0.43, size = 177, normalized size = 1.00 \[ -\frac {\frac {\frac {31185 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6720 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1827 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {360 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{3}} - \frac {40320 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {105 \, {\left (\frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a^{3} \sin \left (d x + c\right )^{3}}}{20160 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/20160*((31185*sin(d*x + c)/(cos(d*x + c) + 1) - 6720*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1827*sin(d*x + c
)^5/(cos(d*x + c) + 1)^5 - 360*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a
^3 - 40320*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - 105*(24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)*(cos
(d*x + c) + 1)^3/(a^3*sin(d*x + c)^3))/d

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mupad [B]  time = 2.08, size = 205, normalized size = 1.16 \[ \frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (c+d\,x\right )}{a^3\,d\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\right )}-\frac {\frac {668\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{315}-\frac {983\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{210}+\frac {346\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{105}-\frac {2291\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2520}+\frac {173\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{840}-\frac {19\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{672}+\frac {1}{576}}{a^3\,d\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4/(a + a/cos(c + d*x))^3,x)

[Out]

(cos(c/2 + (d*x)/2)^9*(c + d*x) - cos(c/2 + (d*x)/2)^11*(c + d*x))/(a^3*d*(cos(c/2 + (d*x)/2)^9 - cos(c/2 + (d
*x)/2)^11)) - ((173*cos(c/2 + (d*x)/2)^4)/840 - (19*cos(c/2 + (d*x)/2)^2)/672 - (2291*cos(c/2 + (d*x)/2)^6)/25
20 + (346*cos(c/2 + (d*x)/2)^8)/105 - (983*cos(c/2 + (d*x)/2)^10)/210 + (668*cos(c/2 + (d*x)/2)^12)/315 + 1/57
6)/(a^3*d*(cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2) - cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{4}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(cot(c + d*x)**4/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

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