Integrand size = 21, antiderivative size = 267 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}-\frac {3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}+\frac {\left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}+\frac {b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}-\frac {3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}-\frac {b \cos ^6(c+d x)}{3 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^3}{a^9 d (b+a \cos (c+d x))}+\frac {2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))}{a^9 d} \]
-(a^2-7*b^2)*(a^2-b^2)^2*cos(d*x+c)/a^8/d-3*b*(a^2-b^2)^2*cos(d*x+c)^2/a^7 /d+1/3*(3*a^4-9*a^2*b^2+5*b^4)*cos(d*x+c)^3/a^6/d+1/2*b*(3*a^2-2*b^2)*cos( d*x+c)^4/a^5/d-3/5*(a^2-b^2)*cos(d*x+c)^5/a^4/d-1/3*b*cos(d*x+c)^6/a^3/d+1 /7*cos(d*x+c)^7/a^2/d+b^2*(a^2-b^2)^3/a^9/d/(b+a*cos(d*x+c))+2*b*(a^2-4*b^ 2)*(a^2-b^2)^2*ln(b+a*cos(d*x+c))/a^9/d
Time = 3.19 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.56 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {-3675 a^8+61320 a^6 b^2-132720 a^4 b^4+87360 a^2 b^6-13440 b^8-140 \left (21 a^8-228 a^6 b^2+400 a^4 b^4-192 a^2 b^6\right ) \cos (2 (c+d x))-3780 a^7 b \cos (3 (c+d x))+8400 a^5 b^3 \cos (3 (c+d x))-4480 a^3 b^5 \cos (3 (c+d x))+588 a^8 \cos (4 (c+d x))-1848 a^6 b^2 \cos (4 (c+d x))+1120 a^4 b^4 \cos (4 (c+d x))+476 a^7 b \cos (5 (c+d x))-336 a^5 b^3 \cos (5 (c+d x))-132 a^8 \cos (6 (c+d x))+112 a^6 b^2 \cos (6 (c+d x))-40 a^7 b \cos (7 (c+d x))+15 a^8 \cos (8 (c+d x))+26880 a^6 b^2 \log (b+a \cos (c+d x))-161280 a^4 b^4 \log (b+a \cos (c+d x))+241920 a^2 b^6 \log (b+a \cos (c+d x))-107520 b^8 \log (b+a \cos (c+d x))+1680 a b \cos (c+d x) \left (-8 a^6+67 a^4 b^2-116 a^2 b^4+56 b^6+16 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))\right )}{13440 a^9 d (b+a \cos (c+d x))} \]
(-3675*a^8 + 61320*a^6*b^2 - 132720*a^4*b^4 + 87360*a^2*b^6 - 13440*b^8 - 140*(21*a^8 - 228*a^6*b^2 + 400*a^4*b^4 - 192*a^2*b^6)*Cos[2*(c + d*x)] - 3780*a^7*b*Cos[3*(c + d*x)] + 8400*a^5*b^3*Cos[3*(c + d*x)] - 4480*a^3*b^5 *Cos[3*(c + d*x)] + 588*a^8*Cos[4*(c + d*x)] - 1848*a^6*b^2*Cos[4*(c + d*x )] + 1120*a^4*b^4*Cos[4*(c + d*x)] + 476*a^7*b*Cos[5*(c + d*x)] - 336*a^5* b^3*Cos[5*(c + d*x)] - 132*a^8*Cos[6*(c + d*x)] + 112*a^6*b^2*Cos[6*(c + d *x)] - 40*a^7*b*Cos[7*(c + d*x)] + 15*a^8*Cos[8*(c + d*x)] + 26880*a^6*b^2 *Log[b + a*Cos[c + d*x]] - 161280*a^4*b^4*Log[b + a*Cos[c + d*x]] + 241920 *a^2*b^6*Log[b + a*Cos[c + d*x]] - 107520*b^8*Log[b + a*Cos[c + d*x]] + 16 80*a*b*Cos[c + d*x]*(-8*a^6 + 67*a^4*b^2 - 116*a^2*b^4 + 56*b^6 + 16*(a^2 - 4*b^2)*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]]))/(13440*a^9*d*(b + a*Cos[c + d*x]))
Time = 0.63 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4360, 3042, 25, 3316, 27, 522, 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.
| \(\displaystyle \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^7}{\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {\sin ^7(c+d x) \cos ^2(c+d x)}{(-a \cos (c+d x)-b)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \cos \left (c+d x+\frac {\pi }{2}\right )^7}{\left (-a \sin \left (c+d x+\frac {\pi }{2}\right )-b\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2}{\left (b+a \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (a^2-a^2 \cos ^2(c+d x)\right )^3}{(b+a \cos (c+d x))^2}d(a \cos (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {a^2 \cos ^2(c+d x) \left (a^2-a^2 \cos ^2(c+d x)\right )^3}{(b+a \cos (c+d x))^2}d(a \cos (c+d x))}{a^9 d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (-a^6 \cos ^6(c+d x)+2 a^5 b \cos ^5(c+d x)+3 a^4 \left (a^2-b^2\right ) \cos ^4(c+d x)+2 a^3 b \left (2 b^2-3 a^2\right ) \cos ^3(c+d x)-a^2 \left (3 a^4-9 b^2 a^2+5 b^4\right ) \cos ^2(c+d x)+6 a b \left (b^2-a^2\right )^2 \cos (c+d x)+\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2+\frac {2 b \left (b^2-a^2\right )^2 \left (4 b^2-a^2\right )}{b+a \cos (c+d x)}-\frac {b^2 \left (b^2-a^2\right )^3}{(b+a \cos (c+d x))^2}\right )d(a \cos (c+d x))}{a^9 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)+\frac {1}{3} a^6 b \cos ^6(c+d x)+3 a^2 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)+a \left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)-\frac {b^2 \left (a^2-b^2\right )^3}{a \cos (c+d x)+b}-2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)+\frac {3}{5} a^5 \left (a^2-b^2\right ) \cos ^5(c+d x)-\frac {1}{2} a^4 b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)-\frac {1}{3} a^3 \left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^9 d}\) |
-((a*(a^2 - 7*b^2)*(a^2 - b^2)^2*Cos[c + d*x] + 3*a^2*b*(a^2 - b^2)^2*Cos[ c + d*x]^2 - (a^3*(3*a^4 - 9*a^2*b^2 + 5*b^4)*Cos[c + d*x]^3)/3 - (a^4*b*( 3*a^2 - 2*b^2)*Cos[c + d*x]^4)/2 + (3*a^5*(a^2 - b^2)*Cos[c + d*x]^5)/5 + (a^6*b*Cos[c + d*x]^6)/3 - (a^7*Cos[c + d*x]^7)/7 - (b^2*(a^2 - b^2)^3)/(b + a*Cos[c + d*x]) - 2*b*(a^2 - 4*b^2)*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x ]])/(a^9*d))
3.3.9.3.1 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[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[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 = 2.09 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{3}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4} a^{5} b}{2}-\cos \left (d x +c \right )^{4} a^{3} b^{3}+\cos \left (d x +c \right )^{3} a^{6}-3 \cos \left (d x +c \right )^{3} a^{4} b^{2}+\frac {5 \cos \left (d x +c \right )^{3} a^{2} b^{4}}{3}-3 \cos \left (d x +c \right )^{2} a^{5} b +6 \cos \left (d x +c \right )^{2} a^{3} b^{3}-3 \cos \left (d x +c \right )^{2} a \,b^{5}-\cos \left (d x +c \right ) a^{6}+9 \cos \left (d x +c \right ) a^{4} b^{2}-15 \cos \left (d x +c \right ) a^{2} b^{4}+7 b^{6} \cos \left (d x +c \right )}{a^{8}}+\frac {b^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{a^{9} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{9}}}{d}\) | \(321\) |
| default | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{3}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4} a^{5} b}{2}-\cos \left (d x +c \right )^{4} a^{3} b^{3}+\cos \left (d x +c \right )^{3} a^{6}-3 \cos \left (d x +c \right )^{3} a^{4} b^{2}+\frac {5 \cos \left (d x +c \right )^{3} a^{2} b^{4}}{3}-3 \cos \left (d x +c \right )^{2} a^{5} b +6 \cos \left (d x +c \right )^{2} a^{3} b^{3}-3 \cos \left (d x +c \right )^{2} a \,b^{5}-\cos \left (d x +c \right ) a^{6}+9 \cos \left (d x +c \right ) a^{4} b^{2}-15 \cos \left (d x +c \right ) a^{2} b^{4}+7 b^{6} \cos \left (d x +c \right )}{a^{8}}+\frac {b^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{a^{9} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{9}}}{d}\) | \(321\) |
| parallelrisch | \(\frac {2240 b \left (a -2 b \right ) \left (a +2 b \right ) \left (a -b \right )^{2} \left (a +b \right )^{2} \left (b +a \cos \left (d x +c \right )\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-2240 b \left (a -2 b \right ) \left (a +2 b \right ) \left (a -b \right )^{2} \left (a +b \right )^{2} \left (b +a \cos \left (d x +c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-11 \left (\frac {245 a \left (a^{6}-\frac {76}{7} a^{4} b^{2}+\frac {400}{21} a^{2} b^{4}-\frac {64}{7} b^{6}\right ) \cos \left (2 d x +2 c \right )}{11}+\frac {35 \left (9 b \,a^{6}-20 a^{4} b^{3}+\frac {32}{3} b^{5} a^{2}\right ) \cos \left (3 d x +3 c \right )}{11}+7 \left (2 a^{5} b^{2}-\frac {40}{33} b^{4} a^{3}-\frac {7}{11} a^{7}\right ) \cos \left (4 d x +4 c \right )+\frac {7 \left (-\frac {17}{3} b \,a^{6}+4 a^{4} b^{3}\right ) \cos \left (5 d x +5 c \right )}{11}+\left (-\frac {28}{33} a^{5} b^{2}+a^{7}\right ) \cos \left (6 d x +6 c \right )+\frac {10 a^{6} b \cos \left (7 d x +7 c \right )}{33}-\frac {5 a^{7} \cos \left (8 d x +8 c \right )}{44}+\frac {4 \left (-2128 a^{5} b^{2}+\frac {12880}{3} b^{4} a^{3}+\frac {175}{3} b \,a^{6}-1960 a^{4} b^{3}+4200 b^{5} a^{2}-2240 b^{7}+128 a^{7}-2240 a \,b^{6}\right ) \cos \left (d x +c \right )}{11}+\frac {1225 a^{7}}{44}-\frac {6720 a \,b^{6}}{11}+\frac {12600 b^{4} a^{3}}{11}+\frac {51520 b^{5} a^{2}}{33}-\frac {8960 b^{7}}{11}+\frac {512 b \,a^{6}}{11}-\frac {17990 a^{5} b^{2}}{33}-\frac {8512 a^{4} b^{3}}{11}\right ) a}{1120 \left (b +a \cos \left (d x +c \right )\right ) a^{9} d}\) | \(428\) |
| risch | \(\frac {7 \cos \left (3 d x +3 c \right )}{64 d \,a^{2}}-\frac {35 \,{\mathrm e}^{i \left (d x +c \right )}}{128 a^{2} d}-\frac {35 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 a^{2} d}+\frac {\cos \left (7 d x +7 c \right )}{448 a^{2} d}-\frac {2 b^{2} \left (-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right ) {\mathrm e}^{i \left (d x +c \right )}}{a^{9} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {24 i b^{3} c}{a^{5} d}-\frac {36 i b^{5} c}{a^{7} d}+\frac {16 i b^{7} c}{a^{9} d}-\frac {7 \cos \left (5 d x +5 c \right )}{320 d \,a^{2}}-\frac {b \cos \left (6 d x +6 c \right )}{96 d \,a^{3}}+\frac {3 \cos \left (5 d x +5 c \right ) b^{2}}{80 d \,a^{4}}+\frac {b \cos \left (4 d x +4 c \right )}{8 a^{3} d}-\frac {b^{3} \cos \left (4 d x +4 c \right )}{8 a^{5} d}-\frac {9 \cos \left (3 d x +3 c \right ) b^{2}}{16 a^{4} d}+\frac {5 \cos \left (3 d x +3 c \right ) b^{4}}{12 a^{6} d}-\frac {4 i b c}{a^{3} d}-\frac {29 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{64 a^{3} d}+\frac {5 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{5} d}-\frac {3 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{7} d}+\frac {57 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 a^{4} d}-\frac {55 \,{\mathrm e}^{i \left (d x +c \right )} b^{4}}{8 a^{6} d}+\frac {7 \,{\mathrm e}^{i \left (d x +c \right )} b^{6}}{2 a^{8} d}+\frac {57 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 a^{4} d}-\frac {55 \,{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{8 a^{6} d}+\frac {7 \,{\mathrm e}^{-i \left (d x +c \right )} b^{6}}{2 a^{8} d}-\frac {29 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{64 a^{3} d}+\frac {5 b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{5} d}-\frac {3 b^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{7} d}+\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{3} d}-\frac {12 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{5} d}+\frac {18 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{7} d}-\frac {8 b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{9} d}-\frac {2 i b x}{a^{3}}+\frac {12 i b^{3} x}{a^{5}}-\frac {18 i b^{5} x}{a^{7}}+\frac {8 i b^{7} x}{a^{9}}\) | \(749\) |
| norman | \(\frac {\frac {\left (96 a^{7}+96 b \,a^{6}+380 a^{5} b^{2}-1596 a^{4} b^{3}-1188 b^{4} a^{3}+3220 b^{5} a^{2}+720 a \,b^{6}-1680 b^{7}\right ) \left (a +b \right )}{210 a^{8} d b}-\frac {\left (32 a^{8}-160 a^{7} b -32 a^{6} b^{2}+172 a^{5} b^{3}-644 a^{4} b^{4}-196 a^{3} b^{5}+1180 a^{2} b^{6}+80 a \,b^{7}-560 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{5 a^{8} d b}+\frac {\left (32 a^{8}+160 a^{7} b -32 a^{6} b^{2}-172 a^{5} b^{3}-644 a^{4} b^{4}+196 a^{3} b^{5}+1180 a^{2} b^{6}-80 a \,b^{7}-560 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5 a^{8} d b}-\frac {\left (96 a^{8}-288 a^{7} b +212 a^{6} b^{2}+896 a^{5} b^{3}-2160 a^{4} b^{4}-1144 a^{3} b^{5}+3620 a^{2} b^{6}+480 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{15 a^{8} d b}-\frac {\left (96 a^{8}-192 a^{7} b +476 a^{6} b^{2}+1216 a^{5} b^{3}-2784 a^{4} b^{4}-2032 a^{3} b^{5}+3940 a^{2} b^{6}+960 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{210 d \,a^{8} b}+\frac {\left (96 a^{8}+288 a^{7} b +212 a^{6} b^{2}-896 a^{5} b^{3}-2160 a^{4} b^{4}+1144 a^{3} b^{5}+3620 a^{2} b^{6}-480 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{15 a^{8} d b}-\frac {\left (288 a^{8}-672 a^{7} b +1104 a^{6} b^{2}+3268 a^{5} b^{3}-7428 a^{4} b^{4}-4908 a^{3} b^{5}+11260 a^{2} b^{6}+2160 a \,b^{7}-5040 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{105 a^{8} d b}+\frac {\left (288 a^{8}+672 a^{7} b +1104 a^{6} b^{2}-3268 a^{5} b^{3}-7428 a^{4} b^{4}+4908 a^{3} b^{5}+11260 a^{2} b^{6}-2160 a \,b^{7}-5040 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{105 d \,a^{8} b}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{9} d}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{a^{9} d}\) | \(836\) |
1/d*(1/a^8*(1/7*cos(d*x+c)^7*a^6-1/3*b*cos(d*x+c)^6*a^5-3/5*a^6*cos(d*x+c) ^5+3/5*a^4*b^2*cos(d*x+c)^5+3/2*cos(d*x+c)^4*a^5*b-cos(d*x+c)^4*a^3*b^3+co s(d*x+c)^3*a^6-3*cos(d*x+c)^3*a^4*b^2+5/3*cos(d*x+c)^3*a^2*b^4-3*cos(d*x+c )^2*a^5*b+6*cos(d*x+c)^2*a^3*b^3-3*cos(d*x+c)^2*a*b^5-cos(d*x+c)*a^6+9*cos (d*x+c)*a^4*b^2-15*cos(d*x+c)*a^2*b^4+7*b^6*cos(d*x+c))+b^2*(a^6-3*a^4*b^2 +3*a^2*b^4-b^6)/a^9/(b+a*cos(d*x+c))+2/a^9*b*(a^6-6*a^4*b^2+9*a^2*b^4-4*b^ 6)*ln(b+a*cos(d*x+c)))
Time = 0.35 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {120 \, a^{8} \cos \left (d x + c\right )^{8} - 160 \, a^{7} b \cos \left (d x + c\right )^{7} + 1715 \, a^{6} b^{2} - 4725 \, a^{4} b^{4} + 3780 \, a^{2} b^{6} - 840 \, b^{8} - 56 \, {\left (9 \, a^{8} - 4 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{6} + 84 \, {\left (9 \, a^{7} b - 4 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{5} + 140 \, {\left (6 \, a^{8} - 9 \, a^{6} b^{2} + 4 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{4} - 280 \, {\left (6 \, a^{7} b - 9 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 840 \, {\left (a^{8} - 6 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 35 \, {\left (a^{7} b + 153 \, a^{5} b^{3} - 324 \, a^{3} b^{5} + 168 \, a b^{7}\right )} \cos \left (d x + c\right ) + 1680 \, {\left (a^{6} b^{2} - 6 \, a^{4} b^{4} + 9 \, a^{2} b^{6} - 4 \, b^{8} + {\left (a^{7} b - 6 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{840 \, {\left (a^{10} d \cos \left (d x + c\right ) + a^{9} b d\right )}} \]
1/840*(120*a^8*cos(d*x + c)^8 - 160*a^7*b*cos(d*x + c)^7 + 1715*a^6*b^2 - 4725*a^4*b^4 + 3780*a^2*b^6 - 840*b^8 - 56*(9*a^8 - 4*a^6*b^2)*cos(d*x + c )^6 + 84*(9*a^7*b - 4*a^5*b^3)*cos(d*x + c)^5 + 140*(6*a^8 - 9*a^6*b^2 + 4 *a^4*b^4)*cos(d*x + c)^4 - 280*(6*a^7*b - 9*a^5*b^3 + 4*a^3*b^5)*cos(d*x + c)^3 - 840*(a^8 - 6*a^6*b^2 + 9*a^4*b^4 - 4*a^2*b^6)*cos(d*x + c)^2 + 35* (a^7*b + 153*a^5*b^3 - 324*a^3*b^5 + 168*a*b^7)*cos(d*x + c) + 1680*(a^6*b ^2 - 6*a^4*b^4 + 9*a^2*b^6 - 4*b^8 + (a^7*b - 6*a^5*b^3 + 9*a^3*b^5 - 4*a* b^7)*cos(d*x + c))*log(a*cos(d*x + c) + b))/(a^10*d*cos(d*x + c) + a^9*b*d )
Timed out. \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.01 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {210 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )}}{a^{10} \cos \left (d x + c\right ) + a^{9} b} + \frac {30 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 126 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{5} b - 2 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (3 \, a^{6} - 9 \, a^{4} b^{2} + 5 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 630 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 210 \, {\left (a^{6} - 9 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )}{a^{8}} + \frac {420 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 4 \, b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{9}}}{210 \, d} \]
1/210*(210*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)/(a^10*cos(d*x + c) + a^ 9*b) + (30*a^6*cos(d*x + c)^7 - 70*a^5*b*cos(d*x + c)^6 - 126*(a^6 - a^4*b ^2)*cos(d*x + c)^5 + 105*(3*a^5*b - 2*a^3*b^3)*cos(d*x + c)^4 + 70*(3*a^6 - 9*a^4*b^2 + 5*a^2*b^4)*cos(d*x + c)^3 - 630*(a^5*b - 2*a^3*b^3 + a*b^5)* cos(d*x + c)^2 - 210*(a^6 - 9*a^4*b^2 + 15*a^2*b^4 - 7*b^6)*cos(d*x + c))/ a^8 + 420*(a^6*b - 6*a^4*b^3 + 9*a^2*b^5 - 4*b^7)*log(a*cos(d*x + c) + b)/ a^9)/d
Leaf count of result is larger than twice the leaf count of optimal. 1861 vs. \(2 (257) = 514\).
Time = 0.37 (sec) , antiderivative size = 1861, normalized size of antiderivative = 6.97 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
1/210*(420*(a^7*b - a^6*b^2 - 6*a^5*b^3 + 6*a^4*b^4 + 9*a^3*b^5 - 9*a^2*b^ 6 - 4*a*b^7 + 4*b^8)*log(abs(a + b + a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/(a^10 - a^9*b) - 420*(a^6*b - 6*a^4*b^3 + 9*a^2*b^5 - 4*b^7)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c ) + 1) + 1))/a^9 - 420*(a^7*b - 7*a^5*b^3 - 4*a^4*b^4 + 11*a^3*b^5 + 8*a^2 *b^6 - 5*a*b^7 - 4*b^8 + a^7*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a^6 *b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 6*a^5*b^3*(cos(d*x + c) - 1)/ (cos(d*x + c) + 1) + 6*a^4*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*a ^3*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 9*a^2*b^6*(cos(d*x + c) - 1 )/(cos(d*x + c) + 1) - 4*a*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 4*b ^8*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((a + b + a*(cos(d*x + c) - 1)/( cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))*a^9) + (192*a ^7 - 1089*a^6*b - 2772*a^5*b^2 + 6534*a^4*b^3 + 5600*a^3*b^4 - 9801*a^2*b^ 5 - 2940*a*b^6 + 4356*b^7 - 1344*a^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 8463*a^6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 18144*a^5*b^2*(cos(d *x + c) - 1)/(cos(d*x + c) + 1) - 49098*a^4*b^3*(cos(d*x + c) - 1)/(cos(d* x + c) + 1) - 35000*a^3*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 71127* a^2*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 17640*a*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 30492*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 4032*a^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 28749*a^6*b*(cos...
Time = 0.19 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.20 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {b^3}{2\,a^5}+\frac {b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{2\,a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b^2\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{2\,a^2}+\frac {b\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a^2}-\frac {3\,b^2}{5\,a^4}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2\,d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a^2}-\frac {2\,b\,\left (\frac {b^2\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a^2}+\frac {2\,b\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a}\right )}{a}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{3\,a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{3\,a}\right )}{d}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{3\,a^3\,d}-\frac {-a^6\,b^2+3\,a^4\,b^4-3\,a^2\,b^6+b^8}{a\,d\,\left (\cos \left (c+d\,x\right )\,a^9+b\,a^8\right )}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^6\,b-12\,a^4\,b^3+18\,a^2\,b^5-8\,b^7\right )}{a^9\,d} \]
(cos(c + d*x)^4*(b^3/(2*a^5) + (b*(3/a^2 - (3*b^2)/a^4))/(2*a)))/d - (cos( c + d*x)^2*((b^2*((2*b^3)/a^5 + (2*b*(3/a^2 - (3*b^2)/a^4))/a))/(2*a^2) + (b*(3/a^2 + (b^2*(3/a^2 - (3*b^2)/a^4))/a^2 - (2*b*((2*b^3)/a^5 + (2*b*(3/ a^2 - (3*b^2)/a^4))/a))/a))/a))/d - (cos(c + d*x)^5*(3/(5*a^2) - (3*b^2)/( 5*a^4)))/d + cos(c + d*x)^7/(7*a^2*d) - (cos(c + d*x)*(1/a^2 + (b^2*(3/a^2 + (b^2*(3/a^2 - (3*b^2)/a^4))/a^2 - (2*b*((2*b^3)/a^5 + (2*b*(3/a^2 - (3* b^2)/a^4))/a))/a))/a^2 - (2*b*((b^2*((2*b^3)/a^5 + (2*b*(3/a^2 - (3*b^2)/a ^4))/a))/a^2 + (2*b*(3/a^2 + (b^2*(3/a^2 - (3*b^2)/a^4))/a^2 - (2*b*((2*b^ 3)/a^5 + (2*b*(3/a^2 - (3*b^2)/a^4))/a))/a))/a))/a))/d + (cos(c + d*x)^3*( 1/a^2 + (b^2*(3/a^2 - (3*b^2)/a^4))/(3*a^2) - (2*b*((2*b^3)/a^5 + (2*b*(3/ a^2 - (3*b^2)/a^4))/a))/(3*a)))/d - (b*cos(c + d*x)^6)/(3*a^3*d) - (b^8 - 3*a^2*b^6 + 3*a^4*b^4 - a^6*b^2)/(a*d*(a^9*cos(c + d*x) + a^8*b)) + (log(b + a*cos(c + d*x))*(2*a^6*b - 8*b^7 + 18*a^2*b^5 - 12*a^4*b^3))/(a^9*d)