3.90 \(\int \frac{\csc ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=125 \[ -\frac{2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac{7 \cot ^9(c+d x)}{9 a^2 d}-\frac{9 \cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^5(c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac{2 \csc ^9(c+d x)}{9 a^2 d} \]

[Out]

-Cot[c + d*x]^3/(3*a^2*d) - Cot[c + d*x]^5/(a^2*d) - (9*Cot[c + d*x]^7)/(7*a^2*d) - (7*Cot[c + d*x]^9)/(9*a^2*
d) - (2*Cot[c + d*x]^11)/(11*a^2*d) - (2*Csc[c + d*x]^9)/(9*a^2*d) + (2*Csc[c + d*x]^11)/(11*a^2*d)

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Rubi [A]  time = 0.367212, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ -\frac{2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac{7 \cot ^9(c+d x)}{9 a^2 d}-\frac{9 \cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^5(c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac{2 \csc ^9(c+d x)}{9 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^2,x]

[Out]

-Cot[c + d*x]^3/(3*a^2*d) - Cot[c + d*x]^5/(a^2*d) - (9*Cot[c + d*x]^7)/(7*a^2*d) - (7*Cot[c + d*x]^9)/(9*a^2*
d) - (2*Cot[c + d*x]^11)/(11*a^2*d) - (2*Csc[c + d*x]^9)/(9*a^2*d) + (2*Csc[c + d*x]^11)/(11*a^2*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))^2} \, dx &=\int \frac{\cot ^2(c+d x) \csc ^6(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac{\int (-a+a \cos (c+d x))^2 \cot ^2(c+d x) \csc ^{10}(c+d x) \, dx}{a^4}\\ &=\frac{\int \left (a^2 \cot ^4(c+d x) \csc ^8(c+d x)-2 a^2 \cot ^3(c+d x) \csc ^9(c+d x)+a^2 \cot ^2(c+d x) \csc ^{10}(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \cot ^4(c+d x) \csc ^8(c+d x) \, dx}{a^2}+\frac{\int \cot ^2(c+d x) \csc ^{10}(c+d x) \, dx}{a^2}-\frac{2 \int \cot ^3(c+d x) \csc ^9(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int x^2 \left (1+x^2\right )^4 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac{2 \operatorname{Subst}\left (\int x^8 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 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^2 d}+\frac{2 \operatorname{Subst}\left (\int \left (-x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^5(c+d x)}{a^2 d}-\frac{9 \cot ^7(c+d x)}{7 a^2 d}-\frac{7 \cot ^9(c+d x)}{9 a^2 d}-\frac{2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac{2 \csc ^9(c+d x)}{9 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}\\ \end{align*}

Mathematica [A]  time = 1.40975, size = 233, normalized size = 1.86 \[ -\frac{\csc (c) (-218834 \sin (c+d x)-79576 \sin (2 (c+d x))+119364 \sin (3 (c+d x))+79576 \sin (4 (c+d x))-28420 \sin (5 (c+d x))-34104 \sin (6 (c+d x))-1421 \sin (7 (c+d x))+5684 \sin (8 (c+d x))+1421 \sin (9 (c+d x))+1419264 \sin (2 c+d x)+114688 \sin (c+2 d x)-172032 \sin (2 c+3 d x)-114688 \sin (3 c+4 d x)+40960 \sin (4 c+5 d x)+49152 \sin (5 c+6 d x)+2048 \sin (6 c+7 d x)-8192 \sin (7 c+8 d x)-2048 \sin (8 c+9 d x)+630784 \sin (c)-1103872 \sin (d x)) \csc ^7(c+d x) \sec ^2(c+d x)}{22708224 a^2 d (\sec (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^2,x]

[Out]

-(Csc[c]*Csc[c + d*x]^7*Sec[c + d*x]^2*(630784*Sin[c] - 1103872*Sin[d*x] - 218834*Sin[c + d*x] - 79576*Sin[2*(
c + d*x)] + 119364*Sin[3*(c + d*x)] + 79576*Sin[4*(c + d*x)] - 28420*Sin[5*(c + d*x)] - 34104*Sin[6*(c + d*x)]
- 1421*Sin[7*(c + d*x)] + 5684*Sin[8*(c + d*x)] + 1421*Sin[9*(c + d*x)] + 1419264*Sin[2*c + d*x] + 114688*Sin
[c + 2*d*x] - 172032*Sin[2*c + 3*d*x] - 114688*Sin[3*c + 4*d*x] + 40960*Sin[4*c + 5*d*x] + 49152*Sin[5*c + 6*d
*x] + 2048*Sin[6*c + 7*d*x] - 8192*Sin[7*c + 8*d*x] - 2048*Sin[8*c + 9*d*x]))/(22708224*a^2*d*(1 + Sec[c + d*x
])^2)

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Maple [A]  time = 0.078, size = 112, normalized size = 0.9 \begin{align*}{\frac{1}{512\,d{a}^{2}} \left ({\frac{1}{11} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11}}+{\frac{5}{9} \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}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-14\,\tan \left ( 1/2\,dx+c/2 \right ) -{\frac{8}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}- \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))^2,x)

[Out]

1/512/d/a^2*(1/11*tan(1/2*d*x+1/2*c)^11+5/9*tan(1/2*d*x+1/2*c)^9+8/7*tan(1/2*d*x+1/2*c)^7-14/3*tan(1/2*d*x+1/2
*c)^3-14*tan(1/2*d*x+1/2*c)-8/3/tan(1/2*d*x+1/2*c)^3-1/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 = 1.00744, size = 235, normalized size = 1.88 \begin{align*} -\frac{\frac{\frac{9702 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3234 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{792 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{385 \, \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}}}{a^{2}} + \frac{33 \,{\left (\frac{21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{56 \, \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 )}^{7}}{a^{2} \sin \left (d x + c\right )^{7}}}{354816 \, 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))^2,x, algorithm="maxima")

[Out]

-1/354816*((9702*sin(d*x + c)/(cos(d*x + c) + 1) + 3234*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 792*sin(d*x + c)
^7/(cos(d*x + c) + 1)^7 - 385*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 63*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/
a^2 + 33*(21*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 56*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3)*(cos(d*x + c) +
1)^7/(a^2*sin(d*x + c)^7))/d

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Fricas [A]  time = 1.85061, size = 521, normalized size = 4.17 \begin{align*} \frac{16 \, \cos \left (d x + c\right )^{9} + 32 \, \cos \left (d x + c\right )^{8} - 40 \, \cos \left (d x + c\right )^{7} - 112 \, \cos \left (d x + c\right )^{6} + 14 \, \cos \left (d x + c\right )^{5} + 140 \, \cos \left (d x + c\right )^{4} + 35 \, \cos \left (d x + c\right )^{3} - 70 \, \cos \left (d x + c\right )^{2} + 56 \, \cos \left (d x + c\right ) + 28}{693 \,{\left (a^{2} d \cos \left (d x + c\right )^{8} + 2 \, a^{2} d \cos \left (d x + c\right )^{7} - 2 \, a^{2} d \cos \left (d x + c\right )^{6} - 6 \, a^{2} d \cos \left (d x + c\right )^{5} + 6 \, a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} 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))^2,x, algorithm="fricas")

[Out]

1/693*(16*cos(d*x + c)^9 + 32*cos(d*x + c)^8 - 40*cos(d*x + c)^7 - 112*cos(d*x + c)^6 + 14*cos(d*x + c)^5 + 14
0*cos(d*x + c)^4 + 35*cos(d*x + c)^3 - 70*cos(d*x + c)^2 + 56*cos(d*x + c) + 28)/((a^2*d*cos(d*x + c)^8 + 2*a^
2*d*cos(d*x + c)^7 - 2*a^2*d*cos(d*x + c)^6 - 6*a^2*d*cos(d*x + c)^5 + 6*a^2*d*cos(d*x + c)^3 + 2*a^2*d*cos(d*
x + c)^2 - 2*a^2*d*cos(d*x + c) - a^2*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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.39539, size = 181, normalized size = 1.45 \begin{align*} -\frac{\frac{33 \,{\left (56 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3\right )}}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}} - \frac{63 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 385 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 792 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 3234 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9702 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{22}}}{354816 \, 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))^2,x, algorithm="giac")

[Out]

-1/354816*(33*(56*tan(1/2*d*x + 1/2*c)^4 + 21*tan(1/2*d*x + 1/2*c)^2 + 3)/(a^2*tan(1/2*d*x + 1/2*c)^7) - (63*a
^20*tan(1/2*d*x + 1/2*c)^11 + 385*a^20*tan(1/2*d*x + 1/2*c)^9 + 792*a^20*tan(1/2*d*x + 1/2*c)^7 - 3234*a^20*ta
n(1/2*d*x + 1/2*c)^3 - 9702*a^20*tan(1/2*d*x + 1/2*c))/a^22)/d