3.107 \(\int \frac{\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=145 \[ \frac{4 \cot ^{13}(c+d x)}{13 a^3 d}+\frac{15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{7 \cot ^9(c+d x)}{3 a^3 d}+\frac{13 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{13}(c+d x)}{13 a^3 d}+\frac{7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{\csc ^9(c+d x)}{3 a^3 d} \]

[Out]

(3*Cot[c + d*x]^5)/(5*a^3*d) + (13*Cot[c + d*x]^7)/(7*a^3*d) + (7*Cot[c + d*x]^9)/(3*a^3*d) + (15*Cot[c + d*x]
^11)/(11*a^3*d) + (4*Cot[c + d*x]^13)/(13*a^3*d) - Csc[c + d*x]^9/(3*a^3*d) + (7*Csc[c + d*x]^11)/(11*a^3*d) -
(4*Csc[c + d*x]^13)/(13*a^3*d)

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Rubi [A]  time = 0.419321, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3872, 2875, 2873, 2607, 270, 2606, 14} \[ \frac{4 \cot ^{13}(c+d x)}{13 a^3 d}+\frac{15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{7 \cot ^9(c+d x)}{3 a^3 d}+\frac{13 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{13}(c+d x)}{13 a^3 d}+\frac{7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{\csc ^9(c+d x)}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]

[Out]

(3*Cot[c + d*x]^5)/(5*a^3*d) + (13*Cot[c + d*x]^7)/(7*a^3*d) + (7*Cot[c + d*x]^9)/(3*a^3*d) + (15*Cot[c + d*x]
^11)/(11*a^3*d) + (4*Cot[c + d*x]^13)/(13*a^3*d) - Csc[c + d*x]^9/(3*a^3*d) + (7*Csc[c + d*x]^11)/(11*a^3*d) -
(4*Csc[c + d*x]^13)/(13*a^3*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2875

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^n)/(a - b*Sin[e +
f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]

Rule 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 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 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac{\cot ^3(c+d x) \csc ^5(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=-\frac{\int (-a+a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^{11}(c+d x) \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \cot ^6(c+d x) \csc ^8(c+d x)+3 a^3 \cot ^5(c+d x) \csc ^9(c+d x)-3 a^3 \cot ^4(c+d x) \csc ^{10}(c+d x)+a^3 \cot ^3(c+d x) \csc ^{11}(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \cot ^6(c+d x) \csc ^8(c+d x) \, dx}{a^3}+\frac{\int \cot ^3(c+d x) \csc ^{11}(c+d x) \, dx}{a^3}+\frac{3 \int \cot ^5(c+d x) \csc ^9(c+d x) \, dx}{a^3}-\frac{3 \int \cot ^4(c+d x) \csc ^{10}(c+d x) \, dx}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int x^{10} \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac{\operatorname{Subst}\left (\int x^6 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^8 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^4 \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-x^{10}+x^{12}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac{\operatorname{Subst}\left (\int \left (x^6+3 x^8+3 x^{10}+x^{12}\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (x^8-2 x^{10}+x^{12}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (x^4+4 x^6+6 x^8+4 x^{10}+x^{12}\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac{3 \cot ^5(c+d x)}{5 a^3 d}+\frac{13 \cot ^7(c+d x)}{7 a^3 d}+\frac{7 \cot ^9(c+d x)}{3 a^3 d}+\frac{15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{4 \cot ^{13}(c+d x)}{13 a^3 d}-\frac{\csc ^9(c+d x)}{3 a^3 d}+\frac{7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{4 \csc ^{13}(c+d x)}{13 a^3 d}\\ \end{align*}

Mathematica [A]  time = 1.76125, size = 265, normalized size = 1.83 \[ -\frac{\csc (c) (-2764580 \sin (c+d x)-1382290 \sin (2 (c+d x))+1275960 \sin (3 (c+d x))+1336720 \sin (4 (c+d x))-60760 \sin (5 (c+d x))-524055 \sin (6 (c+d x))-167090 \sin (7 (c+d x))+60760 \sin (8 (c+d x))+45570 \sin (9 (c+d x))+7595 \sin (10 (c+d x))[phone] \sin (2 c+d x)-23668736 \sin (c+2 d x)[phone] \sin (3 c+2 d x)-6537216 \sin (2 c+3 d x)-6848512 \sin (3 c+4 d x)+311296 \sin (4 c+5 d x)+2684928 \sin (5 c+6 d x)+856064 \sin (6 c+7 d x)-311296 \sin (7 c+8 d x)-233472 \sin (8 c+9 d x)-38912 \sin (9 c+10 d x)[phone] \sin (c)-6336512 \sin (d x)) \csc ^7(c+d x) \sec ^3(c+d x)}{984023040 a^3 d (\sec (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]

[Out]

-(Csc[c]*Csc[c + d*x]^7*Sec[c + d*x]^3*(49201152*Sin[c] - 6336512*Sin[d*x] - 2764580*Sin[c + d*x] - 1382290*Si
n[2*(c + d*x)] + 1275960*Sin[3*(c + d*x)] + 1336720*Sin[4*(c + d*x)] - 60760*Sin[5*(c + d*x)] - 524055*Sin[6*(
c + d*x)] - 167090*Sin[7*(c + d*x)] + 60760*Sin[8*(c + d*x)] + 45570*Sin[9*(c + d*x)] + 7595*Sin[10*(c + d*x)]
+ 20500480*Sin[2*c + d*x] - 23668736*Sin[c + 2*d*x] + 30750720*Sin[3*c + 2*d*x] - 6537216*Sin[2*c + 3*d*x] -
6848512*Sin[3*c + 4*d*x] + 311296*Sin[4*c + 5*d*x] + 2684928*Sin[5*c + 6*d*x] + 856064*Sin[6*c + 7*d*x] - 3112
96*Sin[7*c + 8*d*x] - 233472*Sin[8*c + 9*d*x] - 38912*Sin[9*c + 10*d*x]))/(984023040*a^3*d*(1 + Sec[c + d*x])^
3)

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Maple [A]  time = 0.083, size = 138, normalized size = 1. \begin{align*}{\frac{1}{1024\,d{a}^{3}} \left ( -{\frac{1}{13} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{13}}-{\frac{4}{11} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11}}-{\frac{1}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{8}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{14}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-14\,\tan \left ( 1/2\,dx+c/2 \right ) - \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}-{\frac{4}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{1}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^8/(a+a*sec(d*x+c))^3,x)

[Out]

1/1024/d/a^3*(-1/13*tan(1/2*d*x+1/2*c)^13-4/11*tan(1/2*d*x+1/2*c)^11-1/3*tan(1/2*d*x+1/2*c)^9+8/7*tan(1/2*d*x+
1/2*c)^7+14/5*tan(1/2*d*x+1/2*c)^5-14*tan(1/2*d*x+1/2*c)-1/tan(1/2*d*x+1/2*c)^3+8/tan(1/2*d*x+1/2*c)-4/5/tan(1
/2*d*x+1/2*c)^5-1/7/tan(1/2*d*x+1/2*c)^7)

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Maxima [A]  time = 1.15545, size = 289, normalized size = 1.99 \begin{align*} -\frac{\frac{\frac{210210 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{42042 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{17160 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{5005 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{5460 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac{1155 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}}{a^{3}} + \frac{429 \,{\left (\frac{28 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{35 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{280 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 5\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{3} \sin \left (d x + c\right )^{7}}}{15375360 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^8/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/15375360*((210210*sin(d*x + c)/(cos(d*x + c) + 1) - 42042*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 17160*sin(d
*x + c)^7/(cos(d*x + c) + 1)^7 + 5005*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 5460*sin(d*x + c)^11/(cos(d*x + c)
+ 1)^11 + 1155*sin(d*x + c)^13/(cos(d*x + c) + 1)^13)/a^3 + 429*(28*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 35*
sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 280*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5)*(cos(d*x + c) + 1)^7/(a^3*s
in(d*x + c)^7))/d

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Fricas [A]  time = 2.05892, size = 567, normalized size = 3.91 \begin{align*} \frac{304 \, \cos \left (d x + c\right )^{10} + 912 \, \cos \left (d x + c\right )^{9} - 152 \, \cos \left (d x + c\right )^{8} - 2888 \, \cos \left (d x + c\right )^{7} - 1862 \, \cos \left (d x + c\right )^{6} + 2926 \, \cos \left (d x + c\right )^{5} + 3325 \, \cos \left (d x + c\right )^{4} - 665 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )^{2} + 210 \, \cos \left (d x + c\right ) + 70}{15015 \,{\left (a^{3} d \cos \left (d x + c\right )^{9} + 3 \, a^{3} d \cos \left (d x + c\right )^{8} - 8 \, a^{3} d \cos \left (d x + c\right )^{6} - 6 \, a^{3} d \cos \left (d x + c\right )^{5} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} + 8 \, a^{3} d \cos \left (d x + c\right )^{3} - 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(csc(d*x+c)^8/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/15015*(304*cos(d*x + c)^10 + 912*cos(d*x + c)^9 - 152*cos(d*x + c)^8 - 2888*cos(d*x + c)^7 - 1862*cos(d*x +
c)^6 + 2926*cos(d*x + c)^5 + 3325*cos(d*x + c)^4 - 665*cos(d*x + c)^3 - 35*cos(d*x + c)^2 + 210*cos(d*x + c) +
70)/((a^3*d*cos(d*x + c)^9 + 3*a^3*d*cos(d*x + c)^8 - 8*a^3*d*cos(d*x + c)^6 - 6*a^3*d*cos(d*x + c)^5 + 6*a^3
*d*cos(d*x + c)^4 + 8*a^3*d*cos(d*x + c)^3 - 3*a^3*d*cos(d*x + c) - a^3*d)*sin(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**8/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.35844, size = 220, normalized size = 1.52 \begin{align*} \frac{\frac{429 \,{\left (280 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 28 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 5\right )}}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}} - \frac{1155 \, a^{36} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 5460 \, a^{36} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 5005 \, a^{36} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 17160 \, a^{36} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 42042 \, a^{36} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 210210 \, a^{36} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{39}}}{15375360 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^8/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/15375360*(429*(280*tan(1/2*d*x + 1/2*c)^6 - 35*tan(1/2*d*x + 1/2*c)^4 - 28*tan(1/2*d*x + 1/2*c)^2 - 5)/(a^3*
tan(1/2*d*x + 1/2*c)^7) - (1155*a^36*tan(1/2*d*x + 1/2*c)^13 + 5460*a^36*tan(1/2*d*x + 1/2*c)^11 + 5005*a^36*t
an(1/2*d*x + 1/2*c)^9 - 17160*a^36*tan(1/2*d*x + 1/2*c)^7 - 42042*a^36*tan(1/2*d*x + 1/2*c)^5 + 210210*a^36*ta
n(1/2*d*x + 1/2*c))/a^39)/d