Integrand size = 21, antiderivative size = 167 \[ \int \frac {\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {11 x}{128 a^2}+\frac {11 \cos (c+d x) \sin (c+d x)}{128 a^2 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a^2 d}-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {2 \sin ^5(c+d x)}{5 a^2 d}+\frac {2 \sin ^7(c+d x)}{7 a^2 d} \]
11/128*x/a^2+11/128*cos(d*x+c)*sin(d*x+c)/a^2/d-7/64*cos(d*x+c)^3*sin(d*x+ c)/a^2/d-1/16*cos(d*x+c)^5*sin(d*x+c)/a^2/d-1/6*cos(d*x+c)^3*sin(d*x+c)^3/ a^2/d-1/8*cos(d*x+c)^5*sin(d*x+c)^3/a^2/d-2/5*sin(d*x+c)^5/a^2/d+2/7*sin(d *x+c)^7/a^2/d
Time = 3.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.78 \[ \int \frac {\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (9240 d x-10080 \sin (c+d x)-1680 \sin (2 (c+d x))+3360 \sin (3 (c+d x))-2520 \sin (4 (c+d x))+672 \sin (5 (c+d x))+560 \sin (6 (c+d x))-480 \sin (7 (c+d x))+105 \sin (8 (c+d x))+980 \tan \left (\frac {c}{2}\right )\right )}{26880 a^2 d (1+\sec (c+d x))^2} \]
(Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(9240*d*x - 10080*Sin[c + d*x] - 1680*S in[2*(c + d*x)] + 3360*Sin[3*(c + d*x)] - 2520*Sin[4*(c + d*x)] + 672*Sin[ 5*(c + d*x)] + 560*Sin[6*(c + d*x)] - 480*Sin[7*(c + d*x)] + 105*Sin[8*(c + d*x)] + 980*Tan[c/2]))/(26880*a^2*d*(1 + Sec[c + d*x])^2)
Time = 0.75 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4360, 3042, 3354, 3042, 3352, 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.
| \(\displaystyle \int \frac {\sin ^8(c+d x)}{(a \sec (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^8}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {\sin ^8(c+d x) \cos ^2(c+d x)}{(a (-\cos (c+d x))-a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \cos \left (c+d x+\frac {\pi }{2}\right )^8}{\left (a \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )-a\right )^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) (a-a \cos (c+d x))^2 \sin ^4(c+d x)dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos \left (c+d x+\frac {\pi }{2}\right )^4 \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a-a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \cos ^4(c+d x) \sin ^4(c+d x)-2 a^2 \cos ^3(c+d x) \sin ^4(c+d x)+a^2 \cos ^2(c+d x) \sin ^4(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 a^2 \sin ^7(c+d x)}{7 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}-\frac {a^2 \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac {7 a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {11 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {11 a^2 x}{128}}{a^4}\) |
((11*a^2*x)/128 + (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (7*a^2*Cos[ c + d*x]^3*Sin[c + d*x])/(64*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d) - (2*a^2*Sin[c + d*x]^5)/(5*d) + (2*a^2*Sin[c + d*x]^7)/(7* d))/a^4
3.1.83.3.1 Defintions of rubi rules used
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] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(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]
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 = 1.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {9240 d x -10080 \sin \left (d x +c \right )+105 \sin \left (8 d x +8 c \right )+560 \sin \left (6 d x +6 c \right )-2520 \sin \left (4 d x +4 c \right )-1680 \sin \left (2 d x +2 c \right )-480 \sin \left (7 d x +7 c \right )+672 \sin \left (5 d x +5 c \right )+3360 \sin \left (3 d x +3 c \right )}{107520 a^{2} d}\) | \(99\) |
| derivativedivides | \(\frac {\frac {128 \left (-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}-\frac {253 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24576}-\frac {4213 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{122880}-\frac {55583 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{860160}+\frac {31007 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{860160}-\frac {20363 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{122880}+\frac {253 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{24576}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{8192}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}+\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{a^{2} d}\) | \(141\) |
| default | \(\frac {\frac {128 \left (-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}-\frac {253 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24576}-\frac {4213 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{122880}-\frac {55583 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{860160}+\frac {31007 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{860160}-\frac {20363 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{122880}+\frac {253 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{24576}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{8192}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}+\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{a^{2} d}\) | \(141\) |
| risch | \(\frac {11 x}{128 a^{2}}-\frac {3 \sin \left (d x +c \right )}{32 a^{2} d}+\frac {\sin \left (8 d x +8 c \right )}{1024 a^{2} d}-\frac {\sin \left (7 d x +7 c \right )}{224 a^{2} d}+\frac {\sin \left (6 d x +6 c \right )}{192 a^{2} d}+\frac {\sin \left (5 d x +5 c \right )}{160 a^{2} d}-\frac {3 \sin \left (4 d x +4 c \right )}{128 a^{2} d}+\frac {\sin \left (3 d x +3 c \right )}{32 a^{2} d}-\frac {\sin \left (2 d x +2 c \right )}{64 a^{2} d}\) | \(141\) |
| norman | \(\frac {\frac {11 x}{128 a}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}-\frac {253 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 a d}-\frac {4213 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{960 a d}-\frac {55583 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6720 a d}+\frac {31007 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6720 a d}-\frac {20363 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{960 a d}+\frac {253 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 a d}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 a d}+\frac {11 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 a}+\frac {77 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 a}+\frac {77 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 a}+\frac {385 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 a}+\frac {77 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16 a}+\frac {77 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{32 a}+\frac {11 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16 a}+\frac {11 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{128 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8} a}\) | \(313\) |
1/107520*(9240*d*x-10080*sin(d*x+c)+105*sin(8*d*x+8*c)+560*sin(6*d*x+6*c)- 2520*sin(4*d*x+4*c)-1680*sin(2*d*x+2*c)-480*sin(7*d*x+7*c)+672*sin(5*d*x+5 *c)+3360*sin(3*d*x+3*c))/a^2/d
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.54 \[ \int \frac {\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {1155 \, d x + {\left (1680 \, \cos \left (d x + c\right )^{7} - 3840 \, \cos \left (d x + c\right )^{6} - 280 \, \cos \left (d x + c\right )^{5} + 6144 \, \cos \left (d x + c\right )^{4} - 3710 \, \cos \left (d x + c\right )^{3} - 768 \, \cos \left (d x + c\right )^{2} + 1155 \, \cos \left (d x + c\right ) - 1536\right )} \sin \left (d x + c\right )}{13440 \, a^{2} d} \]
1/13440*(1155*d*x + (1680*cos(d*x + c)^7 - 3840*cos(d*x + c)^6 - 280*cos(d *x + c)^5 + 6144*cos(d*x + c)^4 - 3710*cos(d*x + c)^3 - 768*cos(d*x + c)^2 + 1155*cos(d*x + c) - 1536)*sin(d*x + c))/(a^2*d)
Timed out. \[ \int \frac {\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (151) = 302\).
Time = 0.29 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.26 \[ \int \frac {\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {\frac {1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {8855 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {29491 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {55583 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {31007 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {142541 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {8855 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {1155 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}}{a^{2} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {1155 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6720 \, d} \]
-1/6720*((1155*sin(d*x + c)/(cos(d*x + c) + 1) + 8855*sin(d*x + c)^3/(cos( d*x + c) + 1)^3 + 29491*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 55583*sin(d* x + c)^7/(cos(d*x + c) + 1)^7 - 31007*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 142541*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 8855*sin(d*x + c)^13/(cos (d*x + c) + 1)^13 - 1155*sin(d*x + c)^15/(cos(d*x + c) + 1)^15)/(a^2 + 8*a ^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 28*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 56*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 70*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 56*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 28*a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 8*a^2*sin(d*x + c)^14/(cos (d*x + c) + 1)^14 + a^2*sin(d*x + c)^16/(cos(d*x + c) + 1)^16) - 1155*arct an(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.83 \[ \int \frac {\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {1155 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 8855 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 142541 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 31007 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 55583 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 29491 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 8855 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a^{2}}}{13440 \, d} \]
1/13440*(1155*(d*x + c)/a^2 + 2*(1155*tan(1/2*d*x + 1/2*c)^15 + 8855*tan(1 /2*d*x + 1/2*c)^13 - 142541*tan(1/2*d*x + 1/2*c)^11 + 31007*tan(1/2*d*x + 1/2*c)^9 - 55583*tan(1/2*d*x + 1/2*c)^7 - 29491*tan(1/2*d*x + 1/2*c)^5 - 8 855*tan(1/2*d*x + 1/2*c)^3 - 1155*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/ 2*c)^2 + 1)^8*a^2))/d
Time = 16.85 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {11\,x}{128\,a^2}-\frac {-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {253\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+\frac {20363\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{960}-\frac {31007\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6720}+\frac {55583\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{6720}+\frac {4213\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960}+\frac {253\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]
(11*x)/(128*a^2) - ((11*tan(c/2 + (d*x)/2))/64 + (253*tan(c/2 + (d*x)/2)^3 )/192 + (4213*tan(c/2 + (d*x)/2)^5)/960 + (55583*tan(c/2 + (d*x)/2)^7)/672 0 - (31007*tan(c/2 + (d*x)/2)^9)/6720 + (20363*tan(c/2 + (d*x)/2)^11)/960 - (253*tan(c/2 + (d*x)/2)^13)/192 - (11*tan(c/2 + (d*x)/2)^15)/64)/(a^2*d* (tan(c/2 + (d*x)/2)^2 + 1)^8)