Integrand size = 21, antiderivative size = 329 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right ) \cos (c+d x)}{a^9 d}-\frac {3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) \cos ^2(c+d x)}{2 a^8 d}+\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^7 d}+\frac {b \left (9 a^2-10 b^2\right ) \cos ^4(c+d x)}{4 a^6 d}-\frac {3 \left (a^2-2 b^2\right ) \cos ^5(c+d x)}{5 a^5 d}-\frac {b \cos ^6(c+d x)}{2 a^4 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {b^3 \left (a^2-b^2\right )^3}{2 a^{10} d (b+a \cos (c+d x))^2}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{a^{10} d (b+a \cos (c+d x))}+\frac {3 b \left (a^2-b^2\right ) \left (a^4-9 a^2 b^2+12 b^4\right ) \log (b+a \cos (c+d x))}{a^{10} d} \] Output:
-(a^6-18*a^4*b^2+45*a^2*b^4-28*b^6)*cos(d*x+c)/a^9/d-3/2*b*(3*a^4-10*a^2*b ^2+7*b^4)*cos(d*x+c)^2/a^8/d+(a^4-6*a^2*b^2+5*b^4)*cos(d*x+c)^3/a^7/d+1/4* b*(9*a^2-10*b^2)*cos(d*x+c)^4/a^6/d-3/5*(a^2-2*b^2)*cos(d*x+c)^5/a^5/d-1/2 *b*cos(d*x+c)^6/a^4/d+1/7*cos(d*x+c)^7/a^3/d-1/2*b^3*(a^2-b^2)^3/a^10/d/(b +a*cos(d*x+c))^2+3*b^2*(a^2-3*b^2)*(a^2-b^2)^2/a^10/d/(b+a*cos(d*x+c))+3*b *(a^2-b^2)*(a^4-9*a^2*b^2+12*b^4)*ln(b+a*cos(d*x+c))/a^10/d
Time = 4.26 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.67 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {-7945 a^8 b+164080 a^6 b^3-502320 a^4 b^5+425600 a^2 b^7-76160 b^9-784 a^9 \cos (3 (c+d x))+17528 a^7 b^2 \cos (3 (c+d x))-43680 a^5 b^4 \cos (3 (c+d x))+26880 a^3 b^6 \cos (3 (c+d x))-1456 a^8 b \cos (4 (c+d x))+4872 a^6 b^3 \cos (4 (c+d x))-3360 a^4 b^5 \cos (4 (c+d x))+152 a^9 \cos (5 (c+d x))-840 a^7 b^2 \cos (5 (c+d x))+672 a^5 b^4 \cos (5 (c+d x))+174 a^8 b \cos (6 (c+d x))-168 a^6 b^3 \cos (6 (c+d x))-39 a^9 \cos (7 (c+d x))+48 a^7 b^2 \cos (7 (c+d x))-15 a^8 b \cos (8 (c+d x))+5 a^9 \cos (9 (c+d x))+13440 a^8 b \log (b+a \cos (c+d x))-107520 a^6 b^3 \log (b+a \cos (c+d x))+13440 a^4 b^5 \log (b+a \cos (c+d x))+403200 a^2 b^7 \log (b+a \cos (c+d x))-322560 b^9 \log (b+a \cos (c+d x))+70 a^2 b \cos (2 (c+d x)) \left (-137 a^6+1896 a^4 b^2-4656 a^2 b^4+2912 b^6+192 \left (a^6-10 a^4 b^2+21 a^2 b^4-12 b^6\right ) \log (b+a \cos (c+d x))\right )-70 a \cos (c+d x) \left (49 a^8-1472 a^6 b^2+3216 a^4 b^4+576 a^2 b^6-2432 b^8-768 b^2 \left (a^6-10 a^4 b^2+21 a^2 b^4-12 b^6\right ) \log (b+a \cos (c+d x))\right )}{8960 a^{10} d (b+a \cos (c+d x))^2} \] Input:
Integrate[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^3,x]
Output:
(-7945*a^8*b + 164080*a^6*b^3 - 502320*a^4*b^5 + 425600*a^2*b^7 - 76160*b^ 9 - 784*a^9*Cos[3*(c + d*x)] + 17528*a^7*b^2*Cos[3*(c + d*x)] - 43680*a^5* b^4*Cos[3*(c + d*x)] + 26880*a^3*b^6*Cos[3*(c + d*x)] - 1456*a^8*b*Cos[4*( c + d*x)] + 4872*a^6*b^3*Cos[4*(c + d*x)] - 3360*a^4*b^5*Cos[4*(c + d*x)] + 152*a^9*Cos[5*(c + d*x)] - 840*a^7*b^2*Cos[5*(c + d*x)] + 672*a^5*b^4*Co s[5*(c + d*x)] + 174*a^8*b*Cos[6*(c + d*x)] - 168*a^6*b^3*Cos[6*(c + d*x)] - 39*a^9*Cos[7*(c + d*x)] + 48*a^7*b^2*Cos[7*(c + d*x)] - 15*a^8*b*Cos[8* (c + d*x)] + 5*a^9*Cos[9*(c + d*x)] + 13440*a^8*b*Log[b + a*Cos[c + d*x]] - 107520*a^6*b^3*Log[b + a*Cos[c + d*x]] + 13440*a^4*b^5*Log[b + a*Cos[c + d*x]] + 403200*a^2*b^7*Log[b + a*Cos[c + d*x]] - 322560*b^9*Log[b + a*Cos [c + d*x]] + 70*a^2*b*Cos[2*(c + d*x)]*(-137*a^6 + 1896*a^4*b^2 - 4656*a^2 *b^4 + 2912*b^6 + 192*(a^6 - 10*a^4*b^2 + 21*a^2*b^4 - 12*b^6)*Log[b + a*C os[c + d*x]]) - 70*a*Cos[c + d*x]*(49*a^8 - 1472*a^6*b^2 + 3216*a^4*b^4 + 576*a^2*b^6 - 2432*b^8 - 768*b^2*(a^6 - 10*a^4*b^2 + 21*a^2*b^4 - 12*b^6)* Log[b + a*Cos[c + d*x]]))/(8960*a^10*d*(b + a*Cos[c + d*x])^2)
Time = 0.70 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.90, 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, 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.
| \(\displaystyle \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^7}{\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sin ^7(c+d x) \cos ^3(c+d x)}{(-a \cos (c+d x)-b)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos ^3(c+d x) \sin ^7(c+d x)}{(b+a \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sin ^7(c+d x) \cos ^3(c+d x)}{(a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \cos \left (c+d x+\frac {\pi }{2}\right )^7}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}{\left (b+a \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle -\frac {\int \frac {\cos ^3(c+d x) \left (a^2-a^2 \cos ^2(c+d x)\right )^3}{(b+a \cos (c+d x))^3}d(a \cos (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {a^3 \cos ^3(c+d x) \left (a^2-a^2 \cos ^2(c+d x)\right )^3}{(b+a \cos (c+d x))^3}d(a \cos (c+d x))}{a^{10} d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (-\cos ^6(c+d x) a^6+\left (\frac {-28 b^6+45 a^2 b^4-18 a^4 b^2}{a^6}+1\right ) a^6+3 b \cos ^5(c+d x) a^5+3 \left (a^2-2 b^2\right ) \cos ^4(c+d x) a^4+b \left (10 b^2-9 a^2\right ) \cos ^3(c+d x) a^3-3 \left (a^4-6 b^2 a^2+5 b^4\right ) \cos ^2(c+d x) a^2+3 b \left (3 a^4-10 b^2 a^2+7 b^4\right ) \cos (c+d x) a+\frac {3 b \left (-a^6+10 b^2 a^4-21 b^4 a^2+12 b^6\right )}{b+a \cos (c+d x)}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{(b+a \cos (c+d x))^2}+\frac {b^3 \left (b^2-a^2\right )^3}{(b+a \cos (c+d x))^3}\right )d(a \cos (c+d x))}{a^{10} d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)+\frac {1}{2} a^6 b \cos ^6(c+d x)-\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{a \cos (c+d x)+b}+\frac {b^3 \left (a^2-b^2\right )^3}{2 (a \cos (c+d x)+b)^2}+\frac {3}{5} a^5 \left (a^2-2 b^2\right ) \cos ^5(c+d x)-\frac {1}{4} a^4 b \left (9 a^2-10 b^2\right ) \cos ^4(c+d x)+\frac {3}{2} a^2 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) \cos ^2(c+d x)-3 b \left (a^2-b^2\right ) \left (a^4-9 a^2 b^2+12 b^4\right ) \log (a \cos (c+d x)+b)+a \left (a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right ) \cos (c+d x)-a^3 \left (a^4-6 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^{10} d}\) |
Input:
Int[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^3,x]
Output:
-((a*(a^6 - 18*a^4*b^2 + 45*a^2*b^4 - 28*b^6)*Cos[c + d*x] + (3*a^2*b*(3*a ^4 - 10*a^2*b^2 + 7*b^4)*Cos[c + d*x]^2)/2 - a^3*(a^4 - 6*a^2*b^2 + 5*b^4) *Cos[c + d*x]^3 - (a^4*b*(9*a^2 - 10*b^2)*Cos[c + d*x]^4)/4 + (3*a^5*(a^2 - 2*b^2)*Cos[c + d*x]^5)/5 + (a^6*b*Cos[c + d*x]^6)/2 - (a^7*Cos[c + d*x]^ 7)/7 + (b^3*(a^2 - b^2)^3)/(2*(b + a*Cos[c + d*x])^2) - (3*b^2*(a^2 - 3*b^ 2)*(a^2 - b^2)^2)/(b + a*Cos[c + d*x]) - 3*b*(a^2 - b^2)*(a^4 - 9*a^2*b^2 + 12*b^4)*Log[b + a*Cos[c + d*x]])/(a^10*d))
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 = 3.64 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{2}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {6 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {9 a^{5} b \cos \left (d x +c \right )^{4}}{4}-\frac {5 a^{3} b^{3} \cos \left (d x +c \right )^{4}}{2}+a^{6} \cos \left (d x +c \right )^{3}-6 a^{4} b^{2} \cos \left (d x +c \right )^{3}+5 a^{2} b^{4} \cos \left (d x +c \right )^{3}-\frac {9 a^{5} b \cos \left (d x +c \right )^{2}}{2}+15 a^{3} b^{3} \cos \left (d x +c \right )^{2}-\frac {21 a \,b^{5} \cos \left (d x +c \right )^{2}}{2}-a^{6} \cos \left (d x +c \right )+18 \cos \left (d x +c \right ) a^{4} b^{2}-45 \cos \left (d x +c \right ) a^{2} b^{4}+28 \cos \left (d x +c \right ) b^{6}}{a^{9}}+\frac {3 b \left (a^{6}-10 a^{4} b^{2}+21 a^{2} b^{4}-12 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{10}}-\frac {b^{3} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{2 a^{10} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 b^{2} \left (a^{6}-5 a^{4} b^{2}+7 a^{2} b^{4}-3 b^{6}\right )}{a^{10} \left (b +a \cos \left (d x +c \right )\right )}}{d}\) | \(367\) |
| default | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{2}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {6 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {9 a^{5} b \cos \left (d x +c \right )^{4}}{4}-\frac {5 a^{3} b^{3} \cos \left (d x +c \right )^{4}}{2}+a^{6} \cos \left (d x +c \right )^{3}-6 a^{4} b^{2} \cos \left (d x +c \right )^{3}+5 a^{2} b^{4} \cos \left (d x +c \right )^{3}-\frac {9 a^{5} b \cos \left (d x +c \right )^{2}}{2}+15 a^{3} b^{3} \cos \left (d x +c \right )^{2}-\frac {21 a \,b^{5} \cos \left (d x +c \right )^{2}}{2}-a^{6} \cos \left (d x +c \right )+18 \cos \left (d x +c \right ) a^{4} b^{2}-45 \cos \left (d x +c \right ) a^{2} b^{4}+28 \cos \left (d x +c \right ) b^{6}}{a^{9}}+\frac {3 b \left (a^{6}-10 a^{4} b^{2}+21 a^{2} b^{4}-12 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{10}}-\frac {b^{3} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{2 a^{10} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 b^{2} \left (a^{6}-5 a^{4} b^{2}+7 a^{2} b^{4}-3 b^{6}\right )}{a^{10} \left (b +a \cos \left (d x +c \right )\right )}}{d}\) | \(367\) |
| parallelrisch | \(\frac {1680 b \left (a -b \right ) \left (a +b \right ) \left (a^{4}-9 a^{2} b^{2}+12 b^{4}\right ) \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-1680 b \left (a -b \right ) \left (a +b \right ) \left (a^{4}-9 a^{2} b^{2}+12 b^{4}\right ) \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+19 \left (-\frac {256 \left (a^{7}+\frac {2135}{1024} a^{6} b -\frac {567}{16} a^{5} b^{2}-\frac {15225}{256} a^{4} b^{3}+\frac {1785}{16} a^{3} b^{4}+\frac {5565}{32} a^{2} b^{5}-\frac {315}{4} a \,b^{6}-\frac {945}{8} b^{7}\right ) a \cos \left (2 d x +2 c \right )}{19}-\frac {98 a^{2} \left (a^{6}-\frac {313}{14} a^{4} b^{2}+\frac {390}{7} a^{2} b^{4}-\frac {240}{7} b^{6}\right ) \cos \left (3 d x +3 c \right )}{19}-\frac {182 b \,a^{3} \left (a^{4}-\frac {87}{26} a^{2} b^{2}+\frac {30}{13} b^{4}\right ) \cos \left (4 d x +4 c \right )}{19}+a^{4} \left (a^{4}-\frac {105}{19} a^{2} b^{2}+\frac {84}{19} b^{4}\right ) \cos \left (5 d x +5 c \right )+\frac {3 \left (\frac {29}{4} a^{7} b -7 a^{5} b^{3}\right ) \cos \left (6 d x +6 c \right )}{19}+\frac {3 \left (-\frac {13}{8} a^{8}+2 a^{6} b^{2}\right ) \cos \left (7 d x +7 c \right )}{19}-\frac {15 \cos \left (8 d x +8 c \right ) a^{7} b}{152}+\frac {5 a^{8} \cos \left (9 d x +9 c \right )}{152}+\frac {\left (36288 a^{5} b^{3}-33600 a^{4} b^{4}-114240 a^{3} b^{5}-20160 a^{2} b^{6}+80640 a \,b^{7}-1024 a^{7} b +15540 a^{6} b^{2}-\frac {1715}{4} a^{8}+40320 b^{8}\right ) \cos \left (d x +c \right )}{19}+\frac {40320 b^{8}}{19}-\frac {256 a^{8}}{19}+\frac {20475 a^{5} b^{3}}{19}-\frac {10416 a^{4} b^{4}}{19}-\frac {69300 a^{3} b^{5}}{19}-\frac {36960 a^{2} b^{6}}{19}+\frac {50400 a \,b^{7}}{19}-\frac {2625 a^{7} b}{152}+\frac {8560 a^{6} b^{2}}{19}\right ) a}{560 \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right ) a^{10} d}\) | \(581\) |
| risch | \(\text {Expression too large to display}\) | \(962\) |
Input:
int(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/a^9*(1/7*cos(d*x+c)^7*a^6-1/2*b*cos(d*x+c)^6*a^5-3/5*a^6*cos(d*x+c) ^5+6/5*a^4*b^2*cos(d*x+c)^5+9/4*a^5*b*cos(d*x+c)^4-5/2*a^3*b^3*cos(d*x+c)^ 4+a^6*cos(d*x+c)^3-6*a^4*b^2*cos(d*x+c)^3+5*a^2*b^4*cos(d*x+c)^3-9/2*a^5*b *cos(d*x+c)^2+15*a^3*b^3*cos(d*x+c)^2-21/2*a*b^5*cos(d*x+c)^2-a^6*cos(d*x+ c)+18*cos(d*x+c)*a^4*b^2-45*cos(d*x+c)*a^2*b^4+28*cos(d*x+c)*b^6)+3/a^10*b *(a^6-10*a^4*b^2+21*a^2*b^4-12*b^6)*ln(b+a*cos(d*x+c))-1/2*b^3*(a^6-3*a^4* b^2+3*a^2*b^4-b^6)/a^10/(b+a*cos(d*x+c))^2+3/a^10*b^2*(a^6-5*a^4*b^2+7*a^2 *b^4-3*b^6)/(b+a*cos(d*x+c)))
Time = 0.19 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.36 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {80 \, a^{9} \cos \left (d x + c\right )^{9} - 120 \, a^{8} b \cos \left (d x + c\right )^{8} + 2275 \, a^{6} b^{3} - 11235 \, a^{4} b^{5} + 13860 \, a^{2} b^{7} - 4760 \, b^{9} - 48 \, {\left (7 \, a^{9} - 4 \, a^{7} b^{2}\right )} \cos \left (d x + c\right )^{7} + 84 \, {\left (7 \, a^{8} b - 4 \, a^{6} b^{3}\right )} \cos \left (d x + c\right )^{6} + 56 \, {\left (10 \, a^{9} - 21 \, a^{7} b^{2} + 12 \, a^{5} b^{4}\right )} \cos \left (d x + c\right )^{5} - 140 \, {\left (10 \, a^{8} b - 21 \, a^{6} b^{3} + 12 \, a^{4} b^{5}\right )} \cos \left (d x + c\right )^{4} - 560 \, {\left (a^{9} - 10 \, a^{7} b^{2} + 21 \, a^{5} b^{4} - 12 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{3} - 35 \, {\left (7 \, a^{8} b - 399 \, a^{6} b^{3} + 1116 \, a^{4} b^{5} - 728 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 70 \, {\left (41 \, a^{7} b^{2} - 81 \, a^{5} b^{4} - 108 \, a^{3} b^{6} + 152 \, a b^{8}\right )} \cos \left (d x + c\right ) + 1680 \, {\left (a^{6} b^{3} - 10 \, a^{4} b^{5} + 21 \, a^{2} b^{7} - 12 \, b^{9} + {\left (a^{8} b - 10 \, a^{6} b^{3} + 21 \, a^{4} b^{5} - 12 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{2} - 10 \, a^{5} b^{4} + 21 \, a^{3} b^{6} - 12 \, a b^{8}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{560 \, {\left (a^{12} d \cos \left (d x + c\right )^{2} + 2 \, a^{11} b d \cos \left (d x + c\right ) + a^{10} b^{2} d\right )}} \] Input:
integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/560*(80*a^9*cos(d*x + c)^9 - 120*a^8*b*cos(d*x + c)^8 + 2275*a^6*b^3 - 1 1235*a^4*b^5 + 13860*a^2*b^7 - 4760*b^9 - 48*(7*a^9 - 4*a^7*b^2)*cos(d*x + c)^7 + 84*(7*a^8*b - 4*a^6*b^3)*cos(d*x + c)^6 + 56*(10*a^9 - 21*a^7*b^2 + 12*a^5*b^4)*cos(d*x + c)^5 - 140*(10*a^8*b - 21*a^6*b^3 + 12*a^4*b^5)*co s(d*x + c)^4 - 560*(a^9 - 10*a^7*b^2 + 21*a^5*b^4 - 12*a^3*b^6)*cos(d*x + c)^3 - 35*(7*a^8*b - 399*a^6*b^3 + 1116*a^4*b^5 - 728*a^2*b^7)*cos(d*x + c )^2 + 70*(41*a^7*b^2 - 81*a^5*b^4 - 108*a^3*b^6 + 152*a*b^8)*cos(d*x + c) + 1680*(a^6*b^3 - 10*a^4*b^5 + 21*a^2*b^7 - 12*b^9 + (a^8*b - 10*a^6*b^3 + 21*a^4*b^5 - 12*a^2*b^7)*cos(d*x + c)^2 + 2*(a^7*b^2 - 10*a^5*b^4 + 21*a^ 3*b^6 - 12*a*b^8)*cos(d*x + c))*log(a*cos(d*x + c) + b))/(a^12*d*cos(d*x + c)^2 + 2*a^11*b*d*cos(d*x + c) + a^10*b^2*d)
Timed out. \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(sin(d*x+c)**7/(a+b*sec(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.99 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {70 \, {\left (5 \, a^{6} b^{3} - 27 \, a^{4} b^{5} + 39 \, a^{2} b^{7} - 17 \, b^{9} + 6 \, {\left (a^{7} b^{2} - 5 \, a^{5} b^{4} + 7 \, a^{3} b^{6} - 3 \, a b^{8}\right )} \cos \left (d x + c\right )\right )}}{a^{12} \cos \left (d x + c\right )^{2} + 2 \, a^{11} b \cos \left (d x + c\right ) + a^{10} b^{2}} + \frac {20 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 84 \, {\left (a^{6} - 2 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 35 \, {\left (9 \, a^{5} b - 10 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (a^{6} - 6 \, a^{4} b^{2} + 5 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right )^{2} - 140 \, {\left (a^{6} - 18 \, a^{4} b^{2} + 45 \, a^{2} b^{4} - 28 \, b^{6}\right )} \cos \left (d x + c\right )}{a^{9}} + \frac {420 \, {\left (a^{6} b - 10 \, a^{4} b^{3} + 21 \, a^{2} b^{5} - 12 \, b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{10}}}{140 \, d} \] Input:
integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/140*(70*(5*a^6*b^3 - 27*a^4*b^5 + 39*a^2*b^7 - 17*b^9 + 6*(a^7*b^2 - 5*a ^5*b^4 + 7*a^3*b^6 - 3*a*b^8)*cos(d*x + c))/(a^12*cos(d*x + c)^2 + 2*a^11* b*cos(d*x + c) + a^10*b^2) + (20*a^6*cos(d*x + c)^7 - 70*a^5*b*cos(d*x + c )^6 - 84*(a^6 - 2*a^4*b^2)*cos(d*x + c)^5 + 35*(9*a^5*b - 10*a^3*b^3)*cos( d*x + c)^4 + 140*(a^6 - 6*a^4*b^2 + 5*a^2*b^4)*cos(d*x + c)^3 - 210*(3*a^5 *b - 10*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^2 - 140*(a^6 - 18*a^4*b^2 + 45*a^2 *b^4 - 28*b^6)*cos(d*x + c))/a^9 + 420*(a^6*b - 10*a^4*b^3 + 21*a^2*b^5 - 12*b^7)*log(a*cos(d*x + c) + b)/a^10)/d
Time = 0.15 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {3 \, {\left (a^{6} b - 10 \, a^{4} b^{3} + 21 \, a^{2} b^{5} - 12 \, b^{7}\right )} \log \left ({\left | a \cos \left (d x + c\right ) + b \right |}\right )}{a^{10} d} + \frac {5 \, a^{6} b^{3} - 27 \, a^{4} b^{5} + 39 \, a^{2} b^{7} - 17 \, b^{9} + 6 \, {\left (a^{7} b^{2} - 5 \, a^{5} b^{4} + 7 \, a^{3} b^{6} - 3 \, a b^{8}\right )} \cos \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right ) + b\right )}^{2} a^{10} d} + \frac {20 \, a^{18} d^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{17} b d^{6} \cos \left (d x + c\right )^{6} - 84 \, a^{18} d^{6} \cos \left (d x + c\right )^{5} + 168 \, a^{16} b^{2} d^{6} \cos \left (d x + c\right )^{5} + 315 \, a^{17} b d^{6} \cos \left (d x + c\right )^{4} - 350 \, a^{15} b^{3} d^{6} \cos \left (d x + c\right )^{4} + 140 \, a^{18} d^{6} \cos \left (d x + c\right )^{3} - 840 \, a^{16} b^{2} d^{6} \cos \left (d x + c\right )^{3} + 700 \, a^{14} b^{4} d^{6} \cos \left (d x + c\right )^{3} - 630 \, a^{17} b d^{6} \cos \left (d x + c\right )^{2} + 2100 \, a^{15} b^{3} d^{6} \cos \left (d x + c\right )^{2} - 1470 \, a^{13} b^{5} d^{6} \cos \left (d x + c\right )^{2} - 140 \, a^{18} d^{6} \cos \left (d x + c\right ) + 2520 \, a^{16} b^{2} d^{6} \cos \left (d x + c\right ) - 6300 \, a^{14} b^{4} d^{6} \cos \left (d x + c\right ) + 3920 \, a^{12} b^{6} d^{6} \cos \left (d x + c\right )}{140 \, a^{21} d^{7}} \] Input:
integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x, algorithm="giac")
Output:
3*(a^6*b - 10*a^4*b^3 + 21*a^2*b^5 - 12*b^7)*log(abs(a*cos(d*x + c) + b))/ (a^10*d) + 1/2*(5*a^6*b^3 - 27*a^4*b^5 + 39*a^2*b^7 - 17*b^9 + 6*(a^7*b^2 - 5*a^5*b^4 + 7*a^3*b^6 - 3*a*b^8)*cos(d*x + c))/((a*cos(d*x + c) + b)^2*a ^10*d) + 1/140*(20*a^18*d^6*cos(d*x + c)^7 - 70*a^17*b*d^6*cos(d*x + c)^6 - 84*a^18*d^6*cos(d*x + c)^5 + 168*a^16*b^2*d^6*cos(d*x + c)^5 + 315*a^17* b*d^6*cos(d*x + c)^4 - 350*a^15*b^3*d^6*cos(d*x + c)^4 + 140*a^18*d^6*cos( d*x + c)^3 - 840*a^16*b^2*d^6*cos(d*x + c)^3 + 700*a^14*b^4*d^6*cos(d*x + c)^3 - 630*a^17*b*d^6*cos(d*x + c)^2 + 2100*a^15*b^3*d^6*cos(d*x + c)^2 - 1470*a^13*b^5*d^6*cos(d*x + c)^2 - 140*a^18*d^6*cos(d*x + c) + 2520*a^16*b ^2*d^6*cos(d*x + c) - 6300*a^14*b^4*d^6*cos(d*x + c) + 3920*a^12*b^6*d^6*c os(d*x + c))/(a^21*d^7)
Time = 0.23 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.32 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(sin(c + d*x)^7/(a + b/cos(c + d*x))^3,x)
Output:
(cos(c + d*x)^4*((2*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/(4*a)))/d - ((1 7*b^9 - 39*a^2*b^7 + 27*a^4*b^5 - 5*a^6*b^3)/(2*a) + cos(c + d*x)*(9*b^8 - 21*a^2*b^6 + 15*a^4*b^4 - 3*a^6*b^2))/(d*(a^11*cos(c + d*x)^2 + a^9*b^2 + 2*a^10*b*cos(c + d*x))) - (cos(c + d*x)^5*(3/(5*a^3) - (6*b^2)/(5*a^5)))/ d - (cos(c + d*x)^2*((3*b^2*((8*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a)) /(2*a^2) - (b^3*(3/a^3 - (6*b^2)/a^5))/(2*a^3) + (3*b*(3/a^3 + (3*b^4)/a^7 + (3*b^2*(3/a^3 - (6*b^2)/a^5))/a^2 - (3*b*((8*b^3)/a^6 + (3*b*(3/a^3 - ( 6*b^2)/a^5))/a))/a))/(2*a)))/d + cos(c + d*x)^7/(7*a^3*d) - (cos(c + d*x)* (1/a^3 + (b^3*((8*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/a^3 + (3*b^2* (3/a^3 + (3*b^4)/a^7 + (3*b^2*(3/a^3 - (6*b^2)/a^5))/a^2 - (3*b*((8*b^3)/a ^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/a))/a^2 - (3*b*((3*b^2*((8*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/a^2 - (b^3*(3/a^3 - (6*b^2)/a^5))/a^3 + ( 3*b*(3/a^3 + (3*b^4)/a^7 + (3*b^2*(3/a^3 - (6*b^2)/a^5))/a^2 - (3*b*((8*b^ 3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/a))/a))/a))/d + (cos(c + d*x)^3*( 1/a^3 + b^4/a^7 + (b^2*(3/a^3 - (6*b^2)/a^5))/a^2 - (b*((8*b^3)/a^6 + (3*b *(3/a^3 - (6*b^2)/a^5))/a))/a))/d - (b*cos(c + d*x)^6)/(2*a^4*d) + (log(b + a*cos(c + d*x))*(3*a^6*b - 36*b^7 + 63*a^2*b^5 - 30*a^4*b^3))/(a^10*d)
Time = 0.26 (sec) , antiderivative size = 1585, normalized size of antiderivative = 4.82 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x)
Output:
( - 840*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a**7*b**2 + 8400*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a**5*b**4 - 17640*cos(c + d*x)*log(tan( (c + d*x)/2)**2 + 1)*a**3*b**6 + 10080*cos(c + d*x)*log(tan((c + d*x)/2)** 2 + 1)*a*b**8 + 840*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x) /2)**2*b - a - b)*a**7*b**2 - 8400*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**5*b**4 + 17640*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**3*b**6 - 10080*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a*b**8 + 20*cos(c + d*x)*sin(c + d*x)**8*a**9 + 4*cos(c + d*x)*sin(c + d*x)**6*a**9 - 48*cos(c + d*x)*sin(c + d*x)**6*a**7*b**2 + 8*cos(c + d*x)*sin(c + d*x) **4*a**9 - 150*cos(c + d*x)*sin(c + d*x)**4*a**7*b**2 + 168*cos(c + d*x)*s in(c + d*x)**4*a**5*b**4 + 32*cos(c + d*x)*sin(c + d*x)**2*a**9 - 956*cos( c + d*x)*sin(c + d*x)**2*a**7*b**2 + 2604*cos(c + d*x)*sin(c + d*x)**2*a** 5*b**4 - 1680*cos(c + d*x)*sin(c + d*x)**2*a**3*b**6 - 64*cos(c + d*x)*a** 9 + 128*cos(c + d*x)*a**8*b + 2204*cos(c + d*x)*a**7*b**2 - 4536*cos(c + d *x)*a**6*b**3 - 4872*cos(c + d*x)*a**5*b**4 + 14280*cos(c + d*x)*a**4*b**5 - 2100*cos(c + d*x)*a**3*b**6 - 10080*cos(c + d*x)*a**2*b**7 + 5040*cos(c + d*x)*a*b**8 + 420*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**8*b - 4200*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**6*b**3 + 8820*log(ta n((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**4*b**5 - 5040*log(tan((c + d*...