\(\int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx\) [71]

Optimal result

Integrand size = 21, antiderivative size = 73 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cot ^3(c+d x)}{3 a d}+\frac {2 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^7(c+d x)}{7 a d}-\frac {\csc ^7(c+d x)}{7 a d} \]

[Out]

1/3*cot(d*x+c)^3/a/d+2/5*cot(d*x+c)^5/a/d+1/7*cot(d*x+c)^7/a/d-1/7*csc(d*x+c)^7/a/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2918, 2686, 30, 2687, 276} \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cot ^7(c+d x)}{7 a d}+\frac {2 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^7(c+d x)}{7 a d} \]

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 276

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 2686

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 2687

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 2918

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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot (c+d x) \csc ^5(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = -\frac {\int \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^7(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int x^6 \, dx,x,\csc (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = -\frac {\csc ^7(c+d x)}{7 a d}-\frac {\text {Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = \frac {\cot ^3(c+d x)}{3 a d}+\frac {2 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^7(c+d x)}{7 a d}-\frac {\csc ^7(c+d x)}{7 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(73)=146\).

Time = 0.83 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.16 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\csc (c) \csc ^5(c+d x) \sec (c+d x) (-8960 \sin (c)+2560 \sin (d x)+1500 \sin (c+d x)+375 \sin (2 (c+d x))-750 \sin (3 (c+d x))-300 \sin (4 (c+d x))+150 \sin (5 (c+d x))+75 \sin (6 (c+d x))+640 \sin (c+2 d x)-1280 \sin (2 c+3 d x)-512 \sin (3 c+4 d x)+256 \sin (4 c+5 d x)+128 \sin (5 c+6 d x))}{53760 a d (1+\sec (c+d x))} \]

[In]

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

[Out]

(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]*(-8960*Sin[c] + 2560*Sin[d*x] + 1500*Sin[c + d*x] + 375*Sin[2*(c + d*x)] -
750*Sin[3*(c + d*x)] - 300*Sin[4*(c + d*x)] + 150*Sin[5*(c + d*x)] + 75*Sin[6*(c + d*x)] + 640*Sin[c + 2*d*x]
- 1280*Sin[2*c + 3*d*x] - 512*Sin[3*c + 4*d*x] + 256*Sin[4*c + 5*d*x] + 128*Sin[5*c + 6*d*x]))/(53760*a*d*(1
+ Sec[c + d*x]))

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.18

method	result	size
parallelrisch	\(\frac {-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-84 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-21 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-175 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-140 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-525 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{6720 d a}\)	\(86\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d a}\)	\(88\)
default	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d a}\)	\(88\)
risch	\(-\frac {16 i \left (70 \,{\mathrm e}^{6 i \left (d x +c \right )}+20 \,{\mathrm e}^{5 i \left (d x +c \right )}+5 \,{\mathrm e}^{4 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}-4 \,{\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{105 a d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5}}\)	\(104\)
norman	\(\frac {-\frac {1}{320 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{448 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{48 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{64 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{192 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{80 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}\)	\(117\)

[In]

int(csc(d*x+c)^6/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/6720*(-15*tan(1/2*d*x+1/2*c)^7-84*tan(1/2*d*x+1/2*c)^5-21*cot(1/2*d*x+1/2*c)^5-175*tan(1/2*d*x+1/2*c)^3-140*
cot(1/2*d*x+1/2*c)^3-525*cot(1/2*d*x+1/2*c))/d/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (65) = 130\).

Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.79 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {8 \, \cos \left (d x + c\right )^{6} + 8 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{2} + 15 \, \cos \left (d x + c\right ) + 15}{105 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

-1/105*(8*cos(d*x + c)^6 + 8*cos(d*x + c)^5 - 20*cos(d*x + c)^4 - 20*cos(d*x + c)^3 + 15*cos(d*x + c)^2 + 15*c
os(d*x + c) + 15)/((a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*
d*cos(d*x + c) + a*d)*sin(d*x + c))

Sympy [F]

\[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\csc ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

integrate(csc(d*x+c)**6/(a+a*sec(d*x+c)),x)

[Out]

Integral(csc(c + d*x)**6/(sec(c + d*x) + 1), x)/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (65) = 130\).

Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.86 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {\frac {175 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {84 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} + \frac {7 \, {\left (\frac {20 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a \sin \left (d x + c\right )^{5}}}{6720 \, d} \]

[In]

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

[Out]

-1/6720*((175*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 84*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 15*sin(d*x + c)^7
/(cos(d*x + c) + 1)^7)/a + 7*(20*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 75*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
+ 3)*(cos(d*x + c) + 1)^5/(a*sin(d*x + c)^5))/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {7 \, {\left (75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 20 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 175 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a^{7}}}{6720 \, d} \]

[In]

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

[Out]

-1/6720*(7*(75*tan(1/2*d*x + 1/2*c)^4 + 20*tan(1/2*d*x + 1/2*c)^2 + 3)/(a*tan(1/2*d*x + 1/2*c)^5) + (15*a^6*ta
n(1/2*d*x + 1/2*c)^7 + 84*a^6*tan(1/2*d*x + 1/2*c)^5 + 175*a^6*tan(1/2*d*x + 1/2*c)^3)/a^7)/d

Mupad [B] (verification not implemented)

Time = 14.04 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.10 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+175\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{6720\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]

[In]

int(1/(sin(c + d*x)^6*(a + a/cos(c + d*x))),x)

[Out]

-(21*cos(c/2 + (d*x)/2)^12 + 15*sin(c/2 + (d*x)/2)^12 + 84*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 175*co
s(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 525*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 140*cos(c/2 + (d*x)/
2)^10*sin(c/2 + (d*x)/2)^2)/(6720*a*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5)