3.89 \(\int \frac{\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=109 \[ -\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{2 \csc ^7(c+d x)}{7 a^2 d} \]

[Out]

-Cot[c + d*x]^3/(3*a^2*d) - (4*Cot[c + d*x]^5)/(5*a^2*d) - (5*Cot[c + d*x]^7)/(7*a^2*d) - (2*Cot[c + d*x]^9)/(
9*a^2*d) - (2*Csc[c + d*x]^7)/(7*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d)

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Rubi [A]  time = 0.35068, antiderivative size = 109, 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} \[ -\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{2 \csc ^7(c+d x)}{7 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]

[Out]

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

Mathematica [A]  time = 0.97313, size = 191, normalized size = 1.75 \[ \frac{\csc (c) (25875 \sin (c+d x)+11500 \sin (2 (c+d x))-10925 \sin (3 (c+d x))-9200 \sin (4 (c+d x))+575 \sin (5 (c+d x))+2300 \sin (6 (c+d x))+575 \sin (7 (c+d x))-107520 \sin (2 c+d x)-10240 \sin (c+2 d x)+9728 \sin (2 c+3 d x)+8192 \sin (3 c+4 d x)-512 \sin (4 c+5 d x)-2048 \sin (5 c+6 d x)-512 \sin (6 c+7 d x)-61440 \sin (c)+84480 \sin (d x)) \csc ^5(c+d x) \sec ^2(c+d x)}{1290240 a^2 d (\sec (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]

[Out]

(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]^2*(-61440*Sin[c] + 84480*Sin[d*x] + 25875*Sin[c + d*x] + 11500*Sin[2*(c +
d*x)] - 10925*Sin[3*(c + d*x)] - 9200*Sin[4*(c + d*x)] + 575*Sin[5*(c + d*x)] + 2300*Sin[6*(c + d*x)] + 575*Si
n[7*(c + d*x)] - 107520*Sin[2*c + d*x] - 10240*Sin[c + 2*d*x] + 9728*Sin[2*c + 3*d*x] + 8192*Sin[3*c + 4*d*x]
- 512*Sin[4*c + 5*d*x] - 2048*Sin[5*c + 6*d*x] - 512*Sin[6*c + 7*d*x]))/(1290240*a^2*d*(1 + Sec[c + d*x])^2)

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Maple [A]  time = 0.07, size = 112, normalized size = 1. \begin{align*}{\frac{1}{128\,d{a}^{2}} \left ({\frac{1}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{3}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{1}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{5}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-5\,\tan \left ( 1/2\,dx+c/2 \right ) - \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}- \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}-{\frac{1}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^6/(a+a*sec(d*x+c))^2,x)

[Out]

1/128/d/a^2*(1/9*tan(1/2*d*x+1/2*c)^9+3/7*tan(1/2*d*x+1/2*c)^7+1/5*tan(1/2*d*x+1/2*c)^5-5/3*tan(1/2*d*x+1/2*c)
^3-5*tan(1/2*d*x+1/2*c)-1/tan(1/2*d*x+1/2*c)^3-1/tan(1/2*d*x+1/2*c)-1/5/tan(1/2*d*x+1/2*c)^5)

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Maxima [A]  time = 0.99652, size = 235, normalized size = 2.16 \begin{align*} -\frac{\frac{\frac{1575 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{525 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{135 \, \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^{2}} + \frac{63 \,{\left (\frac{5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{5 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{2} \sin \left (d x + c\right )^{5}}}{40320 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/40320*((1575*sin(d*x + c)/(cos(d*x + c) + 1) + 525*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 63*sin(d*x + c)^5/
(cos(d*x + c) + 1)^5 - 135*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^2 +
63*(5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1)*(cos(d*x + c) + 1)^5/(
a^2*sin(d*x + c)^5))/d

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Fricas [A]  time = 1.73779, size = 423, normalized size = 3.88 \begin{align*} \frac{8 \, \cos \left (d x + c\right )^{7} + 16 \, \cos \left (d x + c\right )^{6} - 12 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right )^{2} - 40 \, \cos \left (d x + c\right ) - 20}{315 \,{\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - 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)^6/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/315*(8*cos(d*x + c)^7 + 16*cos(d*x + c)^6 - 12*cos(d*x + c)^5 - 40*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + 30*co
s(d*x + c)^2 - 40*cos(d*x + c) - 20)/((a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 -
4*a^2*d*cos(d*x + c)^3 - 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)**6/(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.38236, size = 181, normalized size = 1.66 \begin{align*} -\frac{\frac{63 \,{\left (5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} - \frac{35 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 135 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 63 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 525 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1575 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{18}}}{40320 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

-1/40320*(63*(5*tan(1/2*d*x + 1/2*c)^4 + 5*tan(1/2*d*x + 1/2*c)^2 + 1)/(a^2*tan(1/2*d*x + 1/2*c)^5) - (35*a^16
*tan(1/2*d*x + 1/2*c)^9 + 135*a^16*tan(1/2*d*x + 1/2*c)^7 + 63*a^16*tan(1/2*d*x + 1/2*c)^5 - 525*a^16*tan(1/2*
d*x + 1/2*c)^3 - 1575*a^16*tan(1/2*d*x + 1/2*c))/a^18)/d