Integrand size = 21, antiderivative size = 73 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cot ^3(c+d x)}{3 a d}+\frac {2 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^7(c+d x)}{7 a d}-\frac {\csc ^7(c+d x)}{7 a d} \]
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 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\csc (c) \csc ^5(c+d x) \sec (c+d x) (-8960 \sin (c)+2560 \sin (d x)+1500 \sin (c+d x)+375 \sin (2 (c+d x))-750 \sin (3 (c+d x))-300 \sin (4 (c+d x))+150 \sin (5 (c+d x))+75 \sin (6 (c+d x))+640 \sin (c+2 d x)-1280 \sin (2 c+3 d x)-512 \sin (3 c+4 d x)+256 \sin (4 c+5 d x)+128 \sin (5 c+6 d x))}{53760 a d (1+\sec (c+d x))} \]
(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]*(-8960*Sin[c] + 2560*Sin[d*x] + 1500*S in[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]))
Time = 0.51 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.88, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4360, 25, 25, 3042, 25, 3318, 25, 3042, 25, 3086, 15, 3087, 244, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\csc ^6(c+d x)}{a \sec (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^6 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\cot (c+d x) \csc ^5(c+d x)}{a (-\cos (c+d x))-a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cot (c+d x) \csc ^5(c+d x)}{\cos (c+d x) a+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\cot (c+d x) \csc ^5(c+d x)}{a \cos (c+d x)+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^6 \left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )}dx\) |
\(\Big \downarrow \) 3318 |
\(\displaystyle -\frac {\int \cot ^2(c+d x) \csc ^6(c+d x)dx}{a}-\frac {\int -\cot (c+d x) \csc ^7(c+d x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cot (c+d x) \csc ^7(c+d x)dx}{a}-\frac {\int \cot ^2(c+d x) \csc ^6(c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\sec \left (c+d x-\frac {\pi }{2}\right )^7 \tan \left (c+d x-\frac {\pi }{2}\right )dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^6 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^7 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^6 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {\int \csc ^6(c+d x)d\csc (c+d x)}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^6 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^6 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}-\frac {\csc ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle -\frac {\int \cot ^2(c+d x) \left (\cot ^2(c+d x)+1\right )^2d(-\cot (c+d x))}{a d}-\frac {\csc ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {\int \left (\cot ^6(c+d x)+2 \cot ^4(c+d x)+\cot ^2(c+d x)\right )d(-\cot (c+d x))}{a d}-\frac {\csc ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{7} \cot ^7(c+d x)-\frac {2}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}-\frac {\csc ^7(c+d x)}{7 a d}\) |
-((-1/3*Cot[c + d*x]^3 - (2*Cot[c + d*x]^5)/5 - Cot[c + d*x]^7/7)/(a*d)) - Csc[c + d*x]^7/(7*a*d)
3.1.71.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[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])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[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])
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
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 | \(\frac {-\frac {1}{320 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{448 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{48 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{64 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{192 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{80 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}\) | \(117\) |
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
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 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {8 \, \cos \left (d x + c\right )^{6} + 8 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{2} + 15 \, \cos \left (d x + c\right ) + 15}{105 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \]
-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*cos(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))
\[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\csc ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {\frac {175 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {84 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} + \frac {7 \, {\left (\frac {20 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a \sin \left (d x + c\right )^{5}}}{6720 \, d} \]
-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
Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {7 \, {\left (75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 20 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 175 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a^{7}}}{6720 \, d} \]
-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*tan(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
Time = 14.04 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.10 \[ \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+175\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{6720\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
-(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*cos(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)