3.106 \(\int \frac{\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=127 \[ \frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{11 \cot ^9(c+d x)}{9 a^3 d}+\frac{10 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{7 \csc ^9(c+d x)}{9 a^3 d}-\frac{3 \csc ^7(c+d x)}{7 a^3 d} \]

[Out]

(3*Cot[c + d*x]^5)/(5*a^3*d) + (10*Cot[c + d*x]^7)/(7*a^3*d) + (11*Cot[c + d*x]^9)/(9*a^3*d) + (4*Cot[c + d*x]
^11)/(11*a^3*d) - (3*Csc[c + d*x]^7)/(7*a^3*d) + (7*Csc[c + d*x]^9)/(9*a^3*d) - (4*Csc[c + d*x]^11)/(11*a^3*d)

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Rubi [A]  time = 0.408049, antiderivative size = 127, 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} \[ \frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{11 \cot ^9(c+d x)}{9 a^3 d}+\frac{10 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{7 \csc ^9(c+d x)}{9 a^3 d}-\frac{3 \csc ^7(c+d x)}{7 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]

[Out]

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

Mathematica [A]  time = 1.21646, size = 223, normalized size = 1.76 \[ \frac{\csc (c) (524150 \sin (c+d x)+314490 \sin (2 (c+d x))-162010 \sin (3 (c+d x))-238250 \sin (4 (c+d x))-47650 \sin (5 (c+d x))+47650 \sin (6 (c+d x))+28590 \sin (7 (c+d x))+4765 \sin (8 (c+d x))-2027520 \sin (2 c+d x)+1486848 \sin (c+2 d x)-2365440 \sin (3 c+2 d x)+452608 \sin (2 c+3 d x)+665600 \sin (3 c+4 d x)+133120 \sin (4 c+5 d x)-133120 \sin (5 c+6 d x)-79872 \sin (6 c+7 d x)-13312 \sin (7 c+8 d x)-3886080 \sin (c)+563200 \sin (d x)) \csc ^5(c+d x) \sec ^3(c+d x)}{56770560 a^3 d (\sec (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]

[Out]

(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]^3*(-3886080*Sin[c] + 563200*Sin[d*x] + 524150*Sin[c + d*x] + 314490*Sin[2*
(c + d*x)] - 162010*Sin[3*(c + d*x)] - 238250*Sin[4*(c + d*x)] - 47650*Sin[5*(c + d*x)] + 47650*Sin[6*(c + d*x
)] + 28590*Sin[7*(c + d*x)] + 4765*Sin[8*(c + d*x)] - 2027520*Sin[2*c + d*x] + 1486848*Sin[c + 2*d*x] - 236544
0*Sin[3*c + 2*d*x] + 452608*Sin[2*c + 3*d*x] + 665600*Sin[3*c + 4*d*x] + 133120*Sin[4*c + 5*d*x] - 133120*Sin[
5*c + 6*d*x] - 79872*Sin[6*c + 7*d*x] - 13312*Sin[7*c + 8*d*x]))/(56770560*a^3*d*(1 + Sec[c + d*x])^3)

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Maple [A]  time = 0.079, size = 112, normalized size = 0.9 \begin{align*}{\frac{1}{256\,d{a}^{3}} \left ( -{\frac{1}{11} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11}}-{\frac{2}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{2}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{6}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-6\,\tan \left ( 1/2\,dx+c/2 \right ) -{\frac{2}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+2\, \left ( \tan \left ( 1/2\,dx+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))^3,x)

[Out]

1/256/d/a^3*(-1/11*tan(1/2*d*x+1/2*c)^11-2/9*tan(1/2*d*x+1/2*c)^9+2/7*tan(1/2*d*x+1/2*c)^7+6/5*tan(1/2*d*x+1/2
*c)^5-6*tan(1/2*d*x+1/2*c)-2/3/tan(1/2*d*x+1/2*c)^3+2/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 = 1.12955, size = 235, normalized size = 1.85 \begin{align*} -\frac{\frac{\frac{20790 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{4158 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{990 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{770 \, \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^{3}} + \frac{231 \,{\left (\frac{10 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{30 \, \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 )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{887040 \, 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))^3,x, algorithm="maxima")

[Out]

-1/887040*((20790*sin(d*x + c)/(cos(d*x + c) + 1) - 4158*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 990*sin(d*x + c
)^7/(cos(d*x + c) + 1)^7 + 770*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11
)/a^3 + 231*(10*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 30*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3)*(cos(d*x + c
) + 1)^5/(a^3*sin(d*x + c)^5))/d

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Fricas [A]  time = 1.98898, size = 495, normalized size = 3.9 \begin{align*} \frac{104 \, \cos \left (d x + c\right )^{8} + 312 \, \cos \left (d x + c\right )^{7} + 52 \, \cos \left (d x + c\right )^{6} - 676 \, \cos \left (d x + c\right )^{5} - 585 \, \cos \left (d x + c\right )^{4} + 325 \, \cos \left (d x + c\right )^{3} - 25 \, \cos \left (d x + c\right )^{2} - 150 \, \cos \left (d x + c\right ) - 50}{3465 \,{\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + 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(csc(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/3465*(104*cos(d*x + c)^8 + 312*cos(d*x + c)^7 + 52*cos(d*x + c)^6 - 676*cos(d*x + c)^5 - 585*cos(d*x + c)^4
+ 325*cos(d*x + c)^3 - 25*cos(d*x + c)^2 - 150*cos(d*x + c) - 50)/((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c
)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + 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(-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))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.31689, size = 181, normalized size = 1.43 \begin{align*} \frac{\frac{231 \,{\left (30 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3\right )}}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} - \frac{315 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 770 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 990 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 4158 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 20790 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{33}}}{887040 \, 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))^3,x, algorithm="giac")

[Out]

1/887040*(231*(30*tan(1/2*d*x + 1/2*c)^4 - 10*tan(1/2*d*x + 1/2*c)^2 - 3)/(a^3*tan(1/2*d*x + 1/2*c)^5) - (315*
a^30*tan(1/2*d*x + 1/2*c)^11 + 770*a^30*tan(1/2*d*x + 1/2*c)^9 - 990*a^30*tan(1/2*d*x + 1/2*c)^7 - 4158*a^30*t
an(1/2*d*x + 1/2*c)^5 + 20790*a^30*tan(1/2*d*x + 1/2*c))/a^33)/d