3.73 \(\int \frac{\csc ^{10}(c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=109 \[ \frac{\cot ^{11}(c+d x)}{11 a d}+\frac{4 \cot ^9(c+d x)}{9 a d}+\frac{6 \cot ^7(c+d x)}{7 a d}+\frac{4 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^{11}(c+d x)}{11 a d} \]

[Out]

Cot[c + d*x]^3/(3*a*d) + (4*Cot[c + d*x]^5)/(5*a*d) + (6*Cot[c + d*x]^7)/(7*a*d) + (4*Cot[c + d*x]^9)/(9*a*d)
+ Cot[c + d*x]^11/(11*a*d) - Csc[c + d*x]^11/(11*a*d)

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Rubi [A]  time = 0.155081, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3872, 2839, 2606, 30, 2607, 270} \[ \frac{\cot ^{11}(c+d x)}{11 a d}+\frac{4 \cot ^9(c+d x)}{9 a d}+\frac{6 \cot ^7(c+d x)}{7 a d}+\frac{4 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^{11}(c+d x)}{11 a d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^10/(a + a*Sec[c + d*x]),x]

[Out]

Cot[c + d*x]^3/(3*a*d) + (4*Cot[c + d*x]^5)/(5*a*d) + (6*Cot[c + d*x]^7)/(7*a*d) + (4*Cot[c + d*x]^9)/(9*a*d)
+ Cot[c + d*x]^11/(11*a*d) - Csc[c + d*x]^11/(11*a*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 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -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 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{\csc ^{10}(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac{\cot (c+d x) \csc ^9(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac{\int \cot ^2(c+d x) \csc ^{10}(c+d x) \, dx}{a}+\frac{\int \cot (c+d x) \csc ^{11}(c+d x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int x^{10} \, dx,x,\csc (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int x^2 \left (1+x^2\right )^4 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac{\csc ^{11}(c+d x)}{11 a d}-\frac{\operatorname{Subst}\left (\int \left (x^2+4 x^4+6 x^6+4 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac{\cot ^3(c+d x)}{3 a d}+\frac{4 \cot ^5(c+d x)}{5 a d}+\frac{6 \cot ^7(c+d x)}{7 a d}+\frac{4 \cot ^9(c+d x)}{9 a d}+\frac{\cot ^{11}(c+d x)}{11 a d}-\frac{\csc ^{11}(c+d x)}{11 a d}\\ \end{align*}

Mathematica [B]  time = 1.4677, size = 242, normalized size = 2.22 \[ \frac{\csc (c) (5000940 \sin (c+d x)+833490 \sin (2 (c+d x))-3333960 \sin (3 (c+d x))-952560 \sin (4 (c+d x))+1428840 \sin (5 (c+d x))+535815 \sin (6 (c+d x))-357210 \sin (7 (c+d x))-158760 \sin (8 (c+d x))+39690 \sin (9 (c+d x))+19845 \sin (10 (c+d x))+1376256 \sin (c+2 d x)-5505024 \sin (2 c+3 d x)-1572864 \sin (3 c+4 d x)+2359296 \sin (4 c+5 d x)+884736 \sin (5 c+6 d x)-589824 \sin (6 c+7 d x)-262144 \sin (7 c+8 d x)+65536 \sin (8 c+9 d x)+32768 \sin (9 c+10 d x)-45416448 \sin (c)+8257536 \sin (d x)) \csc ^9(c+d x) \sec (c+d x)}{454164480 a d (\sec (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^10/(a + a*Sec[c + d*x]),x]

[Out]

(Csc[c]*Csc[c + d*x]^9*Sec[c + d*x]*(-45416448*Sin[c] + 8257536*Sin[d*x] + 5000940*Sin[c + d*x] + 833490*Sin[2
*(c + d*x)] - 3333960*Sin[3*(c + d*x)] - 952560*Sin[4*(c + d*x)] + 1428840*Sin[5*(c + d*x)] + 535815*Sin[6*(c
+ d*x)] - 357210*Sin[7*(c + d*x)] - 158760*Sin[8*(c + d*x)] + 39690*Sin[9*(c + d*x)] + 19845*Sin[10*(c + d*x)]
+ 1376256*Sin[c + 2*d*x] - 5505024*Sin[2*c + 3*d*x] - 1572864*Sin[3*c + 4*d*x] + 2359296*Sin[4*c + 5*d*x] + 8
84736*Sin[5*c + 6*d*x] - 589824*Sin[6*c + 7*d*x] - 262144*Sin[7*c + 8*d*x] + 65536*Sin[8*c + 9*d*x] + 32768*Si
n[9*c + 10*d*x]))/(454164480*a*d*(1 + Sec[c + d*x]))

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Maple [A]  time = 0.065, size = 140, normalized size = 1.3 \begin{align*}{\frac{1}{1024\,da} \left ( -{\frac{1}{11} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11}}-{\frac{8}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{27}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{48}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-14\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-3}-42\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}-{\frac{27}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{8}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}-{\frac{1}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-9}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^10/(a+a*sec(d*x+c)),x)

[Out]

1/1024/d/a*(-1/11*tan(1/2*d*x+1/2*c)^11-8/9*tan(1/2*d*x+1/2*c)^9-27/7*tan(1/2*d*x+1/2*c)^7-48/5*tan(1/2*d*x+1/
2*c)^5-14*tan(1/2*d*x+1/2*c)^3-16/tan(1/2*d*x+1/2*c)^3-42/tan(1/2*d*x+1/2*c)-27/5/tan(1/2*d*x+1/2*c)^5-8/7/tan
(1/2*d*x+1/2*c)^7-1/9/tan(1/2*d*x+1/2*c)^9)

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Maxima [B]  time = 1.03037, size = 292, normalized size = 2.68 \begin{align*} -\frac{\frac{\frac{48510 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{33264 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{13365 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{3080 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a} + \frac{11 \,{\left (\frac{360 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1701 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{5040 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{13230 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 35\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}{a \sin \left (d x + c\right )^{9}}}{3548160 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^10/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

-1/3548160*((48510*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 33264*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 13365*sin
(d*x + c)^7/(cos(d*x + c) + 1)^7 + 3080*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 315*sin(d*x + c)^11/(cos(d*x + c
) + 1)^11)/a + 11*(360*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1701*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 5040*s
in(d*x + c)^6/(cos(d*x + c) + 1)^6 + 13230*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 35)*(cos(d*x + c) + 1)^9/(a*s
in(d*x + c)^9))/d

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Fricas [B]  time = 1.71921, size = 602, normalized size = 5.52 \begin{align*} -\frac{128 \, \cos \left (d x + c\right )^{10} + 128 \, \cos \left (d x + c\right )^{9} - 576 \, \cos \left (d x + c\right )^{8} - 576 \, \cos \left (d x + c\right )^{7} + 1008 \, \cos \left (d x + c\right )^{6} + 1008 \, \cos \left (d x + c\right )^{5} - 840 \, \cos \left (d x + c\right )^{4} - 840 \, \cos \left (d x + c\right )^{3} + 315 \, \cos \left (d x + c\right )^{2} + 315 \, \cos \left (d x + c\right ) + 315}{3465 \,{\left (a d \cos \left (d x + c\right )^{9} + a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{7} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{5} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{3} - 4 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a 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)^10/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/3465*(128*cos(d*x + c)^10 + 128*cos(d*x + c)^9 - 576*cos(d*x + c)^8 - 576*cos(d*x + c)^7 + 1008*cos(d*x + c
)^6 + 1008*cos(d*x + c)^5 - 840*cos(d*x + c)^4 - 840*cos(d*x + c)^3 + 315*cos(d*x + c)^2 + 315*cos(d*x + c) +
315)/((a*d*cos(d*x + c)^9 + a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^7 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x +
c)^5 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^3 - 4*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*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)**10/(a+a*sec(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.33764, size = 217, normalized size = 1.99 \begin{align*} -\frac{\frac{11 \,{\left (13230 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 5040 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1701 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 360 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35\right )}}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}} + \frac{315 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 3080 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 13365 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 33264 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 48510 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}{a^{11}}}{3548160 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^10/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

-1/3548160*(11*(13230*tan(1/2*d*x + 1/2*c)^8 + 5040*tan(1/2*d*x + 1/2*c)^6 + 1701*tan(1/2*d*x + 1/2*c)^4 + 360
*tan(1/2*d*x + 1/2*c)^2 + 35)/(a*tan(1/2*d*x + 1/2*c)^9) + (315*a^10*tan(1/2*d*x + 1/2*c)^11 + 3080*a^10*tan(1
/2*d*x + 1/2*c)^9 + 13365*a^10*tan(1/2*d*x + 1/2*c)^7 + 33264*a^10*tan(1/2*d*x + 1/2*c)^5 + 48510*a^10*tan(1/2
*d*x + 1/2*c)^3)/a^11)/d