Integrand size = 21, antiderivative size = 73 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^4(c+d x)}{4 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^6(c+d x)}{2 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^3 d} \]
-1/4*cos(d*x+c)^4/a^3/d+3/5*cos(d*x+c)^5/a^3/d-1/2*cos(d*x+c)^6/a^3/d+1/7* cos(d*x+c)^7/a^3/d
Time = 0.85 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {-2421+4060 \cos (c+d x)-3220 \cos (2 (c+d x))+2100 \cos (3 (c+d x))-1120 \cos (4 (c+d x))+476 \cos (5 (c+d x))-140 \cos (6 (c+d x))+20 \cos (7 (c+d x))}{8960 a^3 d} \]
(-2421 + 4060*Cos[c + d*x] - 3220*Cos[2*(c + d*x)] + 2100*Cos[3*(c + d*x)] - 1120*Cos[4*(c + d*x)] + 476*Cos[5*(c + d*x)] - 140*Cos[6*(c + d*x)] + 2 0*Cos[7*(c + d*x)])/(8960*a^3*d)
Time = 0.42 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 27, 49, 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.
\(\displaystyle \int \frac {\sin ^7(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^7}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\sin ^7(c+d x) \cos ^3(c+d x)}{(a (-\cos (c+d x))-a)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cos ^3(c+d x) \sin ^7(c+d x)}{(\cos (c+d x) a+a)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sin ^7(c+d x) \cos ^3(c+d x)}{(a \cos (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \cos \left (c+d x+\frac {\pi }{2}\right )^7}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}{\left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle -\frac {\int \cos ^3(c+d x) (a-a \cos (c+d x))^3d(a \cos (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int a^3 \cos ^3(c+d x) (a-a \cos (c+d x))^3d(a \cos (c+d x))}{a^{10} d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {\int \left (-\cos ^6(c+d x) a^6+3 \cos ^5(c+d x) a^6-3 \cos ^4(c+d x) a^6+\cos ^3(c+d x) a^6\right )d(a \cos (c+d x))}{a^{10} d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)+\frac {1}{2} a^7 \cos ^6(c+d x)-\frac {3}{5} a^7 \cos ^5(c+d x)+\frac {1}{4} a^7 \cos ^4(c+d x)}{a^{10} d}\) |
-(((a^7*Cos[c + d*x]^4)/4 - (3*a^7*Cos[c + d*x]^5)/5 + (a^7*Cos[c + d*x]^6 )/2 - (a^7*Cos[c + d*x]^7)/7)/(a^10*d))
3.1.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && 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 = 0.97 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{2}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\) | \(49\) |
default | \(\frac {\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{2}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\) | \(49\) |
parallelrisch | \(\frac {-805 \cos \left (2 d x +2 c \right )+2784-280 \cos \left (4 d x +4 c \right )-35 \cos \left (6 d x +6 c \right )+1015 \cos \left (d x +c \right )+525 \cos \left (3 d x +3 c \right )+119 \cos \left (5 d x +5 c \right )+5 \cos \left (7 d x +7 c \right )}{2240 a^{3} d}\) | \(85\) |
risch | \(\frac {29 \cos \left (d x +c \right )}{64 a^{3} d}+\frac {\cos \left (7 d x +7 c \right )}{448 d \,a^{3}}-\frac {\cos \left (6 d x +6 c \right )}{64 d \,a^{3}}+\frac {17 \cos \left (5 d x +5 c \right )}{320 d \,a^{3}}-\frac {\cos \left (4 d x +4 c \right )}{8 d \,a^{3}}+\frac {15 \cos \left (3 d x +3 c \right )}{64 d \,a^{3}}-\frac {23 \cos \left (2 d x +2 c \right )}{64 d \,a^{3}}\) | \(118\) |
norman | \(\frac {\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{a d}+\frac {52 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d a}+\frac {52}{35 a d}+\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d a}+\frac {24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}+\frac {52 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d a}+\frac {156 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7} a^{2}}\) | \(143\) |
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {20 \, \cos \left (d x + c\right )^{7} - 70 \, \cos \left (d x + c\right )^{6} + 84 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4}}{140 \, a^{3} d} \]
1/140*(20*cos(d*x + c)^7 - 70*cos(d*x + c)^6 + 84*cos(d*x + c)^5 - 35*cos( d*x + c)^4)/(a^3*d)
Timed out. \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {20 \, \cos \left (d x + c\right )^{7} - 70 \, \cos \left (d x + c\right )^{6} + 84 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4}}{140 \, a^{3} d} \]
1/140*(20*cos(d*x + c)^7 - 70*cos(d*x + c)^6 + 84*cos(d*x + c)^5 - 35*cos( d*x + c)^4)/(a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (65) = 130\).
Time = 0.39 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.23 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {4 \, {\left (\frac {91 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {273 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {455 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {490 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {210 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {140 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 13\right )}}{35 \, a^{3} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \]
4/35*(91*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 273*(cos(d*x + c) - 1)^2/ (cos(d*x + c) + 1)^2 + 455*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 490 *(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 210*(cos(d*x + c) - 1)^5/(cos (d*x + c) + 1)^5 - 140*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 13)/(a^ 3*d*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^7)
Time = 13.42 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a^3}+\frac {{\cos \left (c+d\,x\right )}^6}{2\,a^3}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^3}}{d} \]