Integrand size = 21, antiderivative size = 223 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}-\frac {b \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}+\frac {b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}-\frac {\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}-\frac {b \cos ^6(c+d x)}{6 a^2 d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {b \left (a^2-b^2\right )^3 \log (b+a \cos (c+d x))}{a^8 d} \] Output:
-(a^2-b^2)^3*cos(d*x+c)/a^7/d-1/2*b*(3*a^4-3*a^2*b^2+b^4)*cos(d*x+c)^2/a^6 /d+1/3*(3*a^4-3*a^2*b^2+b^4)*cos(d*x+c)^3/a^5/d+1/4*b*(3*a^2-b^2)*cos(d*x+ c)^4/a^4/d-1/5*(3*a^2-b^2)*cos(d*x+c)^5/a^3/d-1/6*b*cos(d*x+c)^6/a^2/d+1/7 *cos(d*x+c)^7/a/d+b*(a^2-b^2)^3*ln(b+a*cos(d*x+c))/a^8/d
Time = 1.34 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {-105 a \left (35 a^6-152 a^4 b^2+176 a^2 b^4-64 b^6\right ) \cos (c+d x)-105 \left (29 a^6 b-40 a^4 b^3+16 a^2 b^5\right ) \cos (2 (c+d x))+735 a^7 \cos (3 (c+d x))-1260 a^5 b^2 \cos (3 (c+d x))+560 a^3 b^4 \cos (3 (c+d x))+420 a^6 b \cos (4 (c+d x))-210 a^4 b^3 \cos (4 (c+d x))-147 a^7 \cos (5 (c+d x))+84 a^5 b^2 \cos (5 (c+d x))-35 a^6 b \cos (6 (c+d x))+15 a^7 \cos (7 (c+d x))+6720 a^6 b \log (b+a \cos (c+d x))-20160 a^4 b^3 \log (b+a \cos (c+d x))+20160 a^2 b^5 \log (b+a \cos (c+d x))-6720 b^7 \log (b+a \cos (c+d x))}{6720 a^8 d} \] Input:
Integrate[Sin[c + d*x]^7/(a + b*Sec[c + d*x]),x]
Output:
(-105*a*(35*a^6 - 152*a^4*b^2 + 176*a^2*b^4 - 64*b^6)*Cos[c + d*x] - 105*( 29*a^6*b - 40*a^4*b^3 + 16*a^2*b^5)*Cos[2*(c + d*x)] + 735*a^7*Cos[3*(c + d*x)] - 1260*a^5*b^2*Cos[3*(c + d*x)] + 560*a^3*b^4*Cos[3*(c + d*x)] + 420 *a^6*b*Cos[4*(c + d*x)] - 210*a^4*b^3*Cos[4*(c + d*x)] - 147*a^7*Cos[5*(c + d*x)] + 84*a^5*b^2*Cos[5*(c + d*x)] - 35*a^6*b*Cos[6*(c + d*x)] + 15*a^7 *Cos[7*(c + d*x)] + 6720*a^6*b*Log[b + a*Cos[c + d*x]] - 20160*a^4*b^3*Log [b + a*Cos[c + d*x]] + 20160*a^2*b^5*Log[b + a*Cos[c + d*x]] - 6720*b^7*Lo g[b + a*Cos[c + d*x]])/(6720*a^8*d)
Time = 0.52 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.91, 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.
| \(\displaystyle \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^7}{a-b \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sin ^7(c+d x) \cos (c+d x)}{-a \cos (c+d x)-b}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) \sin ^7(c+d x)}{b+a \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sin ^7(c+d x) \cos (c+d x)}{a \cos (c+d x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \cos \left (c+d x+\frac {\pi }{2}\right )^7}{a \sin \left (c+d x+\frac {\pi }{2}\right )+b}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )}{b+a \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x) \left (a^2-a^2 \cos ^2(c+d x)\right )^3}{b+a \cos (c+d x)}d(a \cos (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {a \cos (c+d x) \left (a^2-a^2 \cos ^2(c+d x)\right )^3}{b+a \cos (c+d x)}d(a \cos (c+d x))}{a^8 d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (-a^6 \cos ^6(c+d x)+a^5 b \cos ^5(c+d x)+a^4 \left (3 a^2-b^2\right ) \cos ^4(c+d x)+a^3 b \left (b^2-3 a^2\right ) \cos ^3(c+d x)-a^2 \left (3 a^4-3 b^2 a^2+b^4\right ) \cos ^2(c+d x)+a b \left (3 a^4-3 b^2 a^2+b^4\right ) \cos (c+d x)+\left (a^2-b^2\right )^3+\frac {b \left (b^2-a^2\right )^3}{b+a \cos (c+d x)}\right )d(a \cos (c+d x))}{a^8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)+\frac {1}{6} a^6 b \cos ^6(c+d x)+a \left (a^2-b^2\right )^3 \cos (c+d x)-b \left (a^2-b^2\right )^3 \log (a \cos (c+d x)+b)+\frac {1}{5} a^5 \left (3 a^2-b^2\right ) \cos ^5(c+d x)-\frac {1}{4} a^4 b \left (3 a^2-b^2\right ) \cos ^4(c+d x)+\frac {1}{2} a^2 b \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^2(c+d x)-\frac {1}{3} a^3 \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^3(c+d x)}{a^8 d}\) |
Input:
Int[Sin[c + d*x]^7/(a + b*Sec[c + d*x]),x]
Output:
-((a*(a^2 - b^2)^3*Cos[c + d*x] + (a^2*b*(3*a^4 - 3*a^2*b^2 + b^4)*Cos[c + d*x]^2)/2 - (a^3*(3*a^4 - 3*a^2*b^2 + b^4)*Cos[c + d*x]^3)/3 - (a^4*b*(3* a^2 - b^2)*Cos[c + d*x]^4)/4 + (a^5*(3*a^2 - b^2)*Cos[c + d*x]^5)/5 + (a^6 *b*Cos[c + d*x]^6)/6 - (a^7*Cos[c + d*x]^7)/7 - b*(a^2 - b^2)^3*Log[b + a* Cos[c + d*x]])/(a^8*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 = 1.80 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{6}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{5} b \cos \left (d x +c \right )^{4}}{4}-\frac {a^{3} b^{3} \cos \left (d x +c \right )^{4}}{4}+a^{6} \cos \left (d x +c \right )^{3}-a^{4} b^{2} \cos \left (d x +c \right )^{3}+\frac {a^{2} b^{4} \cos \left (d x +c \right )^{3}}{3}-\frac {3 a^{5} b \cos \left (d x +c \right )^{2}}{2}+\frac {3 a^{3} b^{3} \cos \left (d x +c \right )^{2}}{2}-\frac {a \,b^{5} \cos \left (d x +c \right )^{2}}{2}-a^{6} \cos \left (d x +c \right )+3 \cos \left (d x +c \right ) a^{4} b^{2}-3 \cos \left (d x +c \right ) a^{2} b^{4}+\cos \left (d x +c \right ) b^{6}}{a^{7}}+\frac {b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{8}}}{d}\) | \(275\) |
| default | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{6}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{5} b \cos \left (d x +c \right )^{4}}{4}-\frac {a^{3} b^{3} \cos \left (d x +c \right )^{4}}{4}+a^{6} \cos \left (d x +c \right )^{3}-a^{4} b^{2} \cos \left (d x +c \right )^{3}+\frac {a^{2} b^{4} \cos \left (d x +c \right )^{3}}{3}-\frac {3 a^{5} b \cos \left (d x +c \right )^{2}}{2}+\frac {3 a^{3} b^{3} \cos \left (d x +c \right )^{2}}{2}-\frac {a \,b^{5} \cos \left (d x +c \right )^{2}}{2}-a^{6} \cos \left (d x +c \right )+3 \cos \left (d x +c \right ) a^{4} b^{2}-3 \cos \left (d x +c \right ) a^{2} b^{4}+\cos \left (d x +c \right ) b^{6}}{a^{7}}+\frac {b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{8}}}{d}\) | \(275\) |
| parallelrisch | \(\frac {320 b \left (a -b \right )^{3} \left (a +b \right )^{3} \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-320 b \left (a -b \right )^{3} \left (a +b \right )^{3} \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-7 a \left (\frac {5 \left (29 a^{5} b -40 a^{3} b^{3}+16 a \,b^{5}\right ) \cos \left (2 d x +2 c \right )}{7}+5 \left (-a^{6}+\frac {12}{7} a^{4} b^{2}-\frac {16}{21} a^{2} b^{4}\right ) \cos \left (3 d x +3 c \right )+\frac {10 \left (-2 a^{5} b +a^{3} b^{3}\right ) \cos \left (4 d x +4 c \right )}{7}+\left (a^{6}-\frac {4}{7} a^{4} b^{2}\right ) \cos \left (5 d x +5 c \right )+\frac {5 a^{5} b \cos \left (6 d x +6 c \right )}{21}-\frac {5 a^{6} \cos \left (7 d x +7 c \right )}{49}+5 \left (5 a^{6}-\frac {152}{7} a^{4} b^{2}+\frac {176}{7} a^{2} b^{4}-\frac {64}{7} b^{6}\right ) \cos \left (d x +c \right )+\frac {1024 a^{6}}{49}-\frac {380 a^{5} b}{21}-\frac {704 a^{4} b^{2}}{7}+\frac {190 a^{3} b^{3}}{7}+\frac {2560 a^{2} b^{4}}{21}-\frac {80 a \,b^{5}}{7}-\frac {320 b^{6}}{7}\right )}{320 a^{8} d}\) | \(300\) |
| norman | \(\frac {\frac {\left (2 a^{5} b +2 a^{4} b^{2}-4 a^{3} b^{3}-4 a^{2} b^{4}+2 a \,b^{5}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{a^{7} d}+\frac {\left (14 a^{5} b +16 a^{4} b^{2}-24 a^{3} b^{3}-28 a^{2} b^{4}+10 a \,b^{5}+12 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a^{7} d}+\frac {-96 a^{6}+462 a^{4} b^{2}-560 a^{2} b^{4}+210 b^{6}}{105 a^{7} d}+\frac {\left (128 a^{5} b +174 a^{4} b^{2}-156 a^{3} b^{3}-232 a^{2} b^{4}+60 a \,b^{5}+90 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 a^{7} d}+\frac {\left (-96 a^{6}+30 a^{5} b +432 a^{4} b^{2}-60 a^{3} b^{3}-500 a^{2} b^{4}+30 a \,b^{5}+180 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 a^{7} d}+\frac {\left (-96 a^{6}+70 a^{5} b +382 a^{4} b^{2}-120 a^{3} b^{3}-420 a^{2} b^{4}+50 a \,b^{5}+150 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 a^{7} d}+\frac {\left (-96 a^{6}+128 a^{5} b +288 a^{4} b^{2}-156 a^{3} b^{3}-328 a^{2} b^{4}+60 a \,b^{5}+120 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 a^{7} d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {\left (a +b \right ) b \left (a^{5}-a^{4} b -2 a^{3} b^{2}+2 a^{2} b^{3}+a \,b^{4}-b^{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{a^{8} d}-\frac {\left (a +b \right ) b \left (a^{5}-a^{4} b -2 a^{3} b^{2}+2 a^{2} b^{3}+a \,b^{4}-b^{5}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{8} d}\) | \(577\) |
| risch | \(-\frac {2 i b c}{a^{2} d}+\frac {6 i b^{3} c}{a^{4} d}-\frac {6 i b^{5} c}{a^{6} d}+\frac {2 i b^{7} c}{a^{8} d}-\frac {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^{8} d}+\frac {i b^{7} x}{a^{8}}-\frac {29 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 a^{2} d}+\frac {5 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 a^{4} d}-\frac {b^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{6} d}+\frac {19 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 a^{3} d}-\frac {11 \,{\mathrm e}^{i \left (d x +c \right )} b^{4}}{8 a^{5} d}+\frac {{\mathrm e}^{i \left (d x +c \right )} b^{6}}{2 a^{7} d}-\frac {i b x}{a^{2}}+\frac {3 i b^{3} x}{a^{4}}-\frac {3 i b^{5} x}{a^{6}}+\frac {19 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 a^{3} d}-\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{8 a^{5} d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{6}}{2 a^{7} d}-\frac {29 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 a^{2} d}+\frac {5 b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 a^{4} d}-\frac {b^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{6} d}+\frac {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^{2} d}-\frac {3 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^{4} d}+\frac {3 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^{6} d}-\frac {b \cos \left (6 d x +6 c \right )}{192 d \,a^{2}}-\frac {35 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 d a}-\frac {35 \,{\mathrm e}^{i \left (d x +c \right )}}{128 d a}+\frac {\cos \left (5 d x +5 c \right ) b^{2}}{80 d \,a^{3}}+\frac {b \cos \left (4 d x +4 c \right )}{16 a^{2} d}-\frac {b^{3} \cos \left (4 d x +4 c \right )}{32 a^{4} d}-\frac {3 \cos \left (3 d x +3 c \right ) b^{2}}{16 d \,a^{3}}+\frac {\cos \left (3 d x +3 c \right ) b^{4}}{12 d \,a^{5}}+\frac {7 \cos \left (3 d x +3 c \right )}{64 a d}-\frac {7 \cos \left (5 d x +5 c \right )}{320 a d}+\frac {\cos \left (7 d x +7 c \right )}{448 a d}\) | \(676\) |
Input:
int(sin(d*x+c)^7/(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/a^7*(1/7*cos(d*x+c)^7*a^6-1/6*b*cos(d*x+c)^6*a^5-3/5*a^6*cos(d*x+c) ^5+1/5*a^4*b^2*cos(d*x+c)^5+3/4*a^5*b*cos(d*x+c)^4-1/4*a^3*b^3*cos(d*x+c)^ 4+a^6*cos(d*x+c)^3-a^4*b^2*cos(d*x+c)^3+1/3*a^2*b^4*cos(d*x+c)^3-3/2*a^5*b *cos(d*x+c)^2+3/2*a^3*b^3*cos(d*x+c)^2-1/2*a*b^5*cos(d*x+c)^2-a^6*cos(d*x+ c)+3*cos(d*x+c)*a^4*b^2-3*cos(d*x+c)*a^2*b^4+cos(d*x+c)*b^6)+b*(a^6-3*a^4* b^2+3*a^2*b^4-b^6)/a^8*ln(b+a*cos(d*x+c)))
Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {60 \, a^{7} \cos \left (d x + c\right )^{7} - 70 \, a^{6} b \cos \left (d x + c\right )^{6} - 84 \, {\left (3 \, a^{7} - a^{5} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{6} b - a^{4} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (3 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{6} b - 3 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) + 420 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{420 \, a^{8} d} \] Input:
integrate(sin(d*x+c)^7/(a+b*sec(d*x+c)),x, algorithm="fricas")
Output:
1/420*(60*a^7*cos(d*x + c)^7 - 70*a^6*b*cos(d*x + c)^6 - 84*(3*a^7 - a^5*b ^2)*cos(d*x + c)^5 + 105*(3*a^6*b - a^4*b^3)*cos(d*x + c)^4 + 140*(3*a^7 - 3*a^5*b^2 + a^3*b^4)*cos(d*x + c)^3 - 210*(3*a^6*b - 3*a^4*b^3 + a^2*b^5) *cos(d*x + c)^2 - 420*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c) + 420*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*log(a*cos(d*x + c) + b))/(a^8*d )
Timed out. \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(sin(d*x+c)**7/(a+b*sec(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {60 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 84 \, {\left (3 \, a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (3 \, a^{6} - 3 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )}{a^{7}} + \frac {420 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8}}}{420 \, d} \] Input:
integrate(sin(d*x+c)^7/(a+b*sec(d*x+c)),x, algorithm="maxima")
Output:
1/420*((60*a^6*cos(d*x + c)^7 - 70*a^5*b*cos(d*x + c)^6 - 84*(3*a^6 - a^4* b^2)*cos(d*x + c)^5 + 105*(3*a^5*b - a^3*b^3)*cos(d*x + c)^4 + 140*(3*a^6 - 3*a^4*b^2 + a^2*b^4)*cos(d*x + c)^3 - 210*(3*a^5*b - 3*a^3*b^3 + a*b^5)* cos(d*x + c)^2 - 420*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(d*x + c))/a^7 + 420*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*log(a*cos(d*x + c) + b)/a^8)/ d
Time = 0.13 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.48 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left ({\left | a \cos \left (d x + c\right ) + b \right |}\right )}{a^{8} d} + \frac {60 \, a^{6} d^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b d^{6} \cos \left (d x + c\right )^{6} - 252 \, a^{6} d^{6} \cos \left (d x + c\right )^{5} + 84 \, a^{4} b^{2} d^{6} \cos \left (d x + c\right )^{5} + 315 \, a^{5} b d^{6} \cos \left (d x + c\right )^{4} - 105 \, a^{3} b^{3} d^{6} \cos \left (d x + c\right )^{4} + 420 \, a^{6} d^{6} \cos \left (d x + c\right )^{3} - 420 \, a^{4} b^{2} d^{6} \cos \left (d x + c\right )^{3} + 140 \, a^{2} b^{4} d^{6} \cos \left (d x + c\right )^{3} - 630 \, a^{5} b d^{6} \cos \left (d x + c\right )^{2} + 630 \, a^{3} b^{3} d^{6} \cos \left (d x + c\right )^{2} - 210 \, a b^{5} d^{6} \cos \left (d x + c\right )^{2} - 420 \, a^{6} d^{6} \cos \left (d x + c\right ) + 1260 \, a^{4} b^{2} d^{6} \cos \left (d x + c\right ) - 1260 \, a^{2} b^{4} d^{6} \cos \left (d x + c\right ) + 420 \, b^{6} d^{6} \cos \left (d x + c\right )}{420 \, a^{7} d^{7}} \] Input:
integrate(sin(d*x+c)^7/(a+b*sec(d*x+c)),x, algorithm="giac")
Output:
(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*log(abs(a*cos(d*x + c) + b))/(a^8*d) + 1/420*(60*a^6*d^6*cos(d*x + c)^7 - 70*a^5*b*d^6*cos(d*x + c)^6 - 252*a^ 6*d^6*cos(d*x + c)^5 + 84*a^4*b^2*d^6*cos(d*x + c)^5 + 315*a^5*b*d^6*cos(d *x + c)^4 - 105*a^3*b^3*d^6*cos(d*x + c)^4 + 420*a^6*d^6*cos(d*x + c)^3 - 420*a^4*b^2*d^6*cos(d*x + c)^3 + 140*a^2*b^4*d^6*cos(d*x + c)^3 - 630*a^5* b*d^6*cos(d*x + c)^2 + 630*a^3*b^3*d^6*cos(d*x + c)^2 - 210*a*b^5*d^6*cos( d*x + c)^2 - 420*a^6*d^6*cos(d*x + c) + 1260*a^4*b^2*d^6*cos(d*x + c) - 12 60*a^2*b^4*d^6*cos(d*x + c) + 420*b^6*d^6*cos(d*x + c))/(a^7*d^7)
Time = 0.15 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a}-\frac {b^2\,\left (\frac {1}{a}-\frac {b^2}{3\,a^3}\right )}{a^2}\right )-{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a}-\frac {b^2}{5\,a^3}\right )-\cos \left (c+d\,x\right )\,\left (\frac {1}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{a^2}\right )}{a^2}\right )+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (a^6\,b-3\,a^4\,b^3+3\,a^2\,b^5-b^7\right )}{a^8}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{6\,a^2}-\frac {b\,{\cos \left (c+d\,x\right )}^2\,\left (\frac {3}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{a^2}\right )}{2\,a}+\frac {b\,{\cos \left (c+d\,x\right )}^4\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{4\,a}}{d} \] Input:
int(sin(c + d*x)^7/(a + b/cos(c + d*x)),x)
Output:
(cos(c + d*x)^3*(1/a - (b^2*(1/a - b^2/(3*a^3)))/a^2) - cos(c + d*x)^5*(3/ (5*a) - b^2/(5*a^3)) - cos(c + d*x)*(1/a - (b^2*(3/a - (b^2*(3/a - b^2/a^3 ))/a^2))/a^2) + cos(c + d*x)^7/(7*a) + (log(b + a*cos(c + d*x))*(a^6*b - b ^7 + 3*a^2*b^5 - 3*a^4*b^3))/a^8 - (b*cos(c + d*x)^6)/(6*a^2) - (b*cos(c + d*x)^2*(3/a - (b^2*(3/a - b^2/a^3))/a^2))/(2*a) + (b*cos(c + d*x)^4*(3/a - b^2/a^3))/(4*a))/d
Time = 0.19 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.47 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {-60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{7}-72 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{7}-96 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{7}+924 \cos \left (d x +c \right ) a^{5} b^{2}-1120 \cos \left (d x +c \right ) a^{3} b^{4}+420 \cos \left (d x +c \right ) a \,b^{6}-420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{6} b +1260 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{4} b^{3}-1260 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{2} b^{5}+420 \,\mathrm {log}\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^{6} b -1260 \,\mathrm {log}\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^{4} b^{3}+1260 \,\mathrm {log}\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^{2} b^{5}+70 \sin \left (d x +c \right )^{6} a^{6} b +105 \sin \left (d x +c \right )^{4} a^{6} b -105 \sin \left (d x +c \right )^{4} a^{4} b^{3}+210 \sin \left (d x +c \right )^{2} a^{6} b -420 \sin \left (d x +c \right )^{2} a^{4} b^{3}+210 \sin \left (d x +c \right )^{2} a^{2} b^{5}+84 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{5} b^{2}+252 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{5} b^{2}-140 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3} b^{4}+192 a^{7}-192 \cos \left (d x +c \right ) a^{7}+420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) b^{7}-420 \,\mathrm {log}\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 ) b^{7}-924 a^{5} b^{2}+1120 a^{3} b^{4}-420 a \,b^{6}}{420 a^{8} d} \] Input:
int(sin(d*x+c)^7/(a+b*sec(d*x+c)),x)
Output:
( - 60*cos(c + d*x)*sin(c + d*x)**6*a**7 - 72*cos(c + d*x)*sin(c + d*x)**4 *a**7 + 84*cos(c + d*x)*sin(c + d*x)**4*a**5*b**2 - 96*cos(c + d*x)*sin(c + d*x)**2*a**7 + 252*cos(c + d*x)*sin(c + d*x)**2*a**5*b**2 - 140*cos(c + d*x)*sin(c + d*x)**2*a**3*b**4 - 192*cos(c + d*x)*a**7 + 924*cos(c + d*x)* a**5*b**2 - 1120*cos(c + d*x)*a**3*b**4 + 420*cos(c + d*x)*a*b**6 - 420*lo g(tan((c + d*x)/2)**2 + 1)*a**6*b + 1260*log(tan((c + d*x)/2)**2 + 1)*a**4 *b**3 - 1260*log(tan((c + d*x)/2)**2 + 1)*a**2*b**5 + 420*log(tan((c + d*x )/2)**2 + 1)*b**7 + 420*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**6*b - 1260*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**4*b**3 + 1260*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**2*b**5 - 420*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2* b - a - b)*b**7 + 70*sin(c + d*x)**6*a**6*b + 105*sin(c + d*x)**4*a**6*b - 105*sin(c + d*x)**4*a**4*b**3 + 210*sin(c + d*x)**2*a**6*b - 420*sin(c + d*x)**2*a**4*b**3 + 210*sin(c + d*x)**2*a**2*b**5 + 192*a**7 - 924*a**5*b* *2 + 1120*a**3*b**4 - 420*a*b**6)/(420*a**8*d)