Integrand size = 21, antiderivative size = 170 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {a \left (a^2-6 b^2\right ) \cos (c+d x)}{d}+\frac {b \left (6 a^2-b^2\right ) \cos ^2(c+d x)}{2 d}+\frac {a \left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 d}-\frac {3 a^2 b \cos ^4(c+d x)}{4 d}-\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {b \left (3 a^2-2 b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
-a*(a^2-6*b^2)*cos(d*x+c)/d+1/2*b*(6*a^2-b^2)*cos(d*x+c)^2/d+1/3*a*(2*a^2- 3*b^2)*cos(d*x+c)^3/d-3/4*a^2*b*cos(d*x+c)^4/d-1/5*a^3*cos(d*x+c)^5/d-b*(3 *a^2-2*b^2)*ln(cos(d*x+c))/d+3*a*b^2*sec(d*x+c)/d+1/2*b^3*sec(d*x+c)^2/d
Time = 1.39 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=\frac {-60 a \left (5 a^2-42 b^2\right ) \cos (c+d x)+60 \left (9 a^2 b-2 b^3\right ) \cos (2 (c+d x))+50 a^3 \cos (3 (c+d x))-120 a b^2 \cos (3 (c+d x))-45 a^2 b \cos (4 (c+d x))-6 a^3 \cos (5 (c+d x))-1440 a^2 b \log (\cos (c+d x))+960 b^3 \log (\cos (c+d x))+1440 a b^2 \sec (c+d x)+240 b^3 \sec ^2(c+d x)}{480 d} \]
(-60*a*(5*a^2 - 42*b^2)*Cos[c + d*x] + 60*(9*a^2*b - 2*b^3)*Cos[2*(c + d*x )] + 50*a^3*Cos[3*(c + d*x)] - 120*a*b^2*Cos[3*(c + d*x)] - 45*a^2*b*Cos[4 *(c + d*x)] - 6*a^3*Cos[5*(c + d*x)] - 1440*a^2*b*Log[Cos[c + d*x]] + 960* b^3*Log[Cos[c + d*x]] + 1440*a*b^2*Sec[c + d*x] + 240*b^3*Sec[c + d*x]^2)/ (480*d)
Time = 0.54 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int \sin ^5(c+d x) (a+b \sec (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^2(c+d x) \tan ^3(c+d x) \left (-(-a \cos (c+d x)-b)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(b+a \cos (c+d x))^3 \sin ^2(c+d x) \tan ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin ^2(c+d x) \tan ^3(c+d x) (a \cos (c+d x)+b)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^5 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \left (b+a \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^3}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle -\frac {\int (b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2 \sec ^3(c+d x)d(a \cos (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2 \sec ^3(c+d x)}{a^3}d(a \cos (c+d x))}{a^2 d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (\cos ^4(c+d x) a^4+\left (1-\frac {6 b^2}{a^2}\right ) a^4+3 b \cos ^3(c+d x) a^3-\left (2 a^2-3 b^2\right ) \cos ^2(c+d x) a^2+3 b^2 \sec ^2(c+d x) a^2+b^3 \sec ^3(c+d x) a+b \left (b^2-6 a^2\right ) \cos (c+d x) a+\frac {\left (3 a^4 b-2 a^2 b^3\right ) \sec (c+d x)}{a}\right )d(a \cos (c+d x))}{a^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{5} a^5 \cos ^5(c+d x)+\frac {3}{4} a^4 b \cos ^4(c+d x)-3 a^3 b^2 \sec (c+d x)-\frac {1}{2} a^2 b^3 \sec ^2(c+d x)-\frac {1}{2} a^2 b \left (6 a^2-b^2\right ) \cos ^2(c+d x)+a^2 b \left (3 a^2-2 b^2\right ) \log (a \cos (c+d x))-\frac {1}{3} a^3 \left (2 a^2-3 b^2\right ) \cos ^3(c+d x)+a^3 \left (a^2-6 b^2\right ) \cos (c+d x)}{a^2 d}\) |
-((a^3*(a^2 - 6*b^2)*Cos[c + d*x] - (a^2*b*(6*a^2 - b^2)*Cos[c + d*x]^2)/2 - (a^3*(2*a^2 - 3*b^2)*Cos[c + d*x]^3)/3 + (3*a^4*b*Cos[c + d*x]^4)/4 + ( a^5*Cos[c + d*x]^5)/5 + a^2*b*(3*a^2 - 2*b^2)*Log[a*Cos[c + d*x]] - 3*a^3* b^2*Sec[c + d*x] - (a^2*b^3*Sec[c + d*x]^2)/2)/(a^2*d))
3.2.85.3.1 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 = 2.45 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{3} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(174\) |
| default | \(\frac {-\frac {a^{3} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(174\) |
| parts | \(-\frac {a^{3} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5 d}+\frac {b^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(182\) |
| parallelrisch | \(\frac {2880 \left (a^{2}-\frac {2 b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-2880 \left (a^{2}-\frac {2 b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2880 \left (a^{2}-\frac {2 b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-512 a^{3}+45 a^{2} b +7680 a \,b^{2}-480 b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (-206 a^{3}+2280 a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (450 a^{2} b -120 b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (38 a^{3}-120 a \,b^{2}\right ) \cos \left (5 d x +5 c \right )-45 a^{2} b \cos \left (6 d x +6 c \right )-6 a^{3} \cos \left (7 d x +7 c \right )+\left (-850 a^{3}+13200 a \,b^{2}\right ) \cos \left (d x +c \right )-512 a^{3}-450 a^{2} b +7680 a \,b^{2}+600 b^{3}}{960 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(294\) |
| norman | \(\frac {\frac {\left (16 a^{3}+24 a^{2} b -48 a \,b^{2}+16 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {16 a^{3}-240 a \,b^{2}}{15 d}-\frac {\left (6 a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}-\frac {\left (18 a^{2} b -12 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {\left (16 a^{3}+30 a^{2} b -240 a \,b^{2}-20 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {\left (16 a^{3}+270 a^{2} b -240 a \,b^{2}-180 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{15 d}-\frac {\left (32 a^{3}-72 a^{2} b +96 a \,b^{2}-48 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {b \left (3 a^{2}-2 b^{2}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(350\) |
| risch | \(\frac {6 i b \,a^{2} c}{d}-\frac {4 i b^{3} c}{d}+\frac {5 \,{\mathrm e}^{3 i \left (d x +c \right )} a^{3}}{96 d}-\frac {{\mathrm e}^{3 i \left (d x +c \right )} a \,b^{2}}{8 d}+\frac {9 \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2} b}{16 d}-\frac {{\mathrm e}^{2 i \left (d x +c \right )} b^{3}}{8 d}-\frac {5 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {21 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{2}}{8 d}-\frac {5 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {21 \,{\mathrm e}^{-i \left (d x +c \right )} a \,b^{2}}{8 d}+\frac {9 \,{\mathrm e}^{-2 i \left (d x +c \right )} a^{2} b}{16 d}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )} b^{3}}{8 d}+\frac {5 \,{\mathrm e}^{-3 i \left (d x +c \right )} a^{3}}{96 d}-\frac {{\mathrm e}^{-3 i \left (d x +c \right )} a \,b^{2}}{8 d}+3 i a^{2} b x -2 i b^{3} x +\frac {2 b^{2} \left (3 a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )} a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}+\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {a^{3} \cos \left (5 d x +5 c \right )}{80 d}-\frac {3 a^{2} b \cos \left (4 d x +4 c \right )}{32 d}\) | \(381\) |
1/d*(-1/5*a^3*(8/3+sin(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c)+3*a^2*b*(-1/4 *sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))+3*a*b^2*(sin(d*x+c)^6/cos(d *x+c)+(8/3+sin(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c))+b^3*(1/2*sin(d*x+c)^ 6/cos(d*x+c)^2+1/2*sin(d*x+c)^4+sin(d*x+c)^2+2*ln(cos(d*x+c))))
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.03 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {96 \, a^{3} \cos \left (d x + c\right )^{7} + 360 \, a^{2} b \cos \left (d x + c\right )^{6} - 160 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 240 \, {\left (6 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{4} - 1440 \, a b^{2} \cos \left (d x + c\right ) + 480 \, {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 480 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 240 \, b^{3} + 15 \, {\left (39 \, a^{2} b - 8 \, b^{3}\right )} \cos \left (d x + c\right )^{2}}{480 \, d \cos \left (d x + c\right )^{2}} \]
-1/480*(96*a^3*cos(d*x + c)^7 + 360*a^2*b*cos(d*x + c)^6 - 160*(2*a^3 - 3* a*b^2)*cos(d*x + c)^5 - 240*(6*a^2*b - b^3)*cos(d*x + c)^4 - 1440*a*b^2*co s(d*x + c) + 480*(a^3 - 6*a*b^2)*cos(d*x + c)^3 + 480*(3*a^2*b - 2*b^3)*co s(d*x + c)^2*log(-cos(d*x + c)) - 240*b^3 + 15*(39*a^2*b - 8*b^3)*cos(d*x + c)^2)/(d*cos(d*x + c)^2)
Timed out. \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.84 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {12 \, a^{3} \cos \left (d x + c\right )^{5} + 45 \, a^{2} b \cos \left (d x + c\right )^{4} - 20 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (6 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right ) + 60 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \log \left (\cos \left (d x + c\right )\right ) - \frac {30 \, {\left (6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{60 \, d} \]
-1/60*(12*a^3*cos(d*x + c)^5 + 45*a^2*b*cos(d*x + c)^4 - 20*(2*a^3 - 3*a*b ^2)*cos(d*x + c)^3 - 30*(6*a^2*b - b^3)*cos(d*x + c)^2 + 60*(a^3 - 6*a*b^2 )*cos(d*x + c) + 60*(3*a^2*b - 2*b^3)*log(cos(d*x + c)) - 30*(6*a*b^2*cos( d*x + c) + b^3)/cos(d*x + c)^2)/d
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (160) = 320\).
Time = 0.44 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.09 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=\text {Too large to display} \]
1/60*(60*(3*a^2*b - 2*b^3)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 60*(3*a^2*b - 2*b^3)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1 ) - 1)) + 30*(9*a^2*b + 12*a*b^2 - 6*b^3 + 18*a^2*b*(cos(d*x + c) - 1)/(co s(d*x + c) + 1) + 12*a*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 16*b^3* (cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*a^2*b*(cos(d*x + c) - 1)^2/(cos( d*x + c) + 1)^2 - 6*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/((cos(d *x + c) - 1)/(cos(d*x + c) + 1) + 1)^2 + (64*a^3 + 411*a^2*b - 600*a*b^2 - 274*b^3 - 320*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2415*a^2*b*(cos (d*x + c) - 1)/(cos(d*x + c) + 1) + 2640*a*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1490*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 640*a^3*(cos (d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 5910*a^2*b*(cos(d*x + c) - 1)^2/(c os(d*x + c) + 1)^2 - 3840*a*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 3100*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 5910*a^2*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 2160*a*b^2*(cos(d*x + c) - 1)^3/(cos(d *x + c) + 1)^3 + 3100*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 2415 *a^2*b*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 360*a*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 1490*b^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 411*a^2*b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 274*b^3*(c os(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^5)/d
Time = 13.58 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.84 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {{\cos \left (c+d\,x\right )}^3\,\left (a\,b^2-\frac {2\,a^3}{3}\right )-{\cos \left (c+d\,x\right )}^2\,\left (3\,a^2\,b-\frac {b^3}{2}\right )+\ln \left (\cos \left (c+d\,x\right )\right )\,\left (3\,a^2\,b-2\,b^3\right )-\frac {\frac {b^3}{2}+3\,a\,\cos \left (c+d\,x\right )\,b^2}{{\cos \left (c+d\,x\right )}^2}-\cos \left (c+d\,x\right )\,\left (6\,a\,b^2-a^3\right )+\frac {a^3\,{\cos \left (c+d\,x\right )}^5}{5}+\frac {3\,a^2\,b\,{\cos \left (c+d\,x\right )}^4}{4}}{d} \]