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[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)

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Rubi [A]  time = 0.253495, antiderivative size = 177, normalized size of antiderivative = 1., 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 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]

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]

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]

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*}

Mathematica [B]  time = 1.17305, 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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Maple [A]  time = 0.079, size = 151, normalized size = 0.9 \begin{align*} -{\frac{1}{576\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{1}{56\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{29}{320\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{1}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{99}{64\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-{\frac{1}{192\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \end{align*}

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/d/a^3*tan(1/2*d*x+1/2*c)^9+1/56/d/a^3*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)+2/d/a^3*arctan(tan(1/2*d*x+1/2*c))-1/192/d/a^3/tan(1/2*d*x
+1/2*c)^3+1/8/d/a^3/tan(1/2*d*x+1/2*c)

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Maxima [A]  time = 1.61626, size = 239, normalized size = 1.35 \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 1.48758, size = 566, normalized size = 3.2 \begin{align*} \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 )} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}

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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Giac [A]  time = 1.44733, size = 177, normalized size = 1. \begin{align*} \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} \end{align*}

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