Integrand size = 28, antiderivative size = 242 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {a^5 \tan (c+d x)}{d}+\frac {5 a^4 b \tan ^2(c+d x)}{2 d}+\frac {2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac {a \left (a^4+20 a^2 b^2+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {b \left (5 a^4+20 a^2 b^2+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac {10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac {5 a b^4 \tan ^9(c+d x)}{9 d}+\frac {b^5 \tan ^{10}(c+d x)}{10 d} \]
a^5*tan(d*x+c)/d+5/2*a^4*b*tan(d*x+c)^2/d+2/3*a^3*(a^2+5*b^2)*tan(d*x+c)^3 /d+5/2*a^2*b*(a^2+b^2)*tan(d*x+c)^4/d+1/5*a*(a^4+20*a^2*b^2+5*b^4)*tan(d*x +c)^5/d+1/6*b*(5*a^4+20*a^2*b^2+b^4)*tan(d*x+c)^6/d+10/7*a*b^2*(a^2+b^2)*t an(d*x+c)^7/d+1/4*b^3*(5*a^2+b^2)*tan(d*x+c)^8/d+5/9*a*b^4*tan(d*x+c)^9/d+ 1/10*b^5*tan(d*x+c)^10/d
Time = 1.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.48 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {\frac {1}{6} \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^6-\frac {4}{7} a \left (a^2+b^2\right ) (a+b \tan (c+d x))^7+\frac {1}{4} \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^8-\frac {4}{9} a (a+b \tan (c+d x))^9+\frac {1}{10} (a+b \tan (c+d x))^{10}}{b^5 d} \]
(((a^2 + b^2)^2*(a + b*Tan[c + d*x])^6)/6 - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^7)/7 + ((3*a^2 + b^2)*(a + b*Tan[c + d*x])^8)/4 - (4*a*(a + b*Tan[ c + d*x])^9)/9 + (a + b*Tan[c + d*x])^10/10)/(b^5*d)
Time = 0.41 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3567, 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.
| \(\displaystyle \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^5}{\cos (c+d x)^{11}}dx\) |
| \(\Big \downarrow \) 3567 |
| \(\displaystyle -\frac {\int (b+a \cot (c+d x))^5 \left (\cot ^2(c+d x)+1\right )^2 \tan ^{11}(c+d x)d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (b^5 \tan ^{11}(c+d x)+5 a b^4 \tan ^{10}(c+d x)+2 \left (b^5+5 a^2 b^3\right ) \tan ^9(c+d x)+10 a b^2 \left (a^2+b^2\right ) \tan ^8(c+d x)+\left (b^5+20 a^2 b^3+5 a^4 b\right ) \tan ^7(c+d x)+\left (a^5+20 b^2 a^3+5 b^4 a\right ) \tan ^6(c+d x)+10 a^2 b \left (a^2+b^2\right ) \tan ^5(c+d x)+2 \left (a^5+5 b^2 a^3\right ) \tan ^4(c+d x)+5 a^4 b \tan ^3(c+d x)+a^5 \tan ^2(c+d x)\right )d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-a^5 \tan (c+d x)-\frac {5}{2} a^4 b \tan ^2(c+d x)-\frac {10}{7} a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)-\frac {5}{2} a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)-\frac {1}{4} b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)-\frac {1}{6} b \left (5 a^4+20 a^2 b^2+b^4\right ) \tan ^6(c+d x)-\frac {1}{5} a \left (a^4+20 a^2 b^2+5 b^4\right ) \tan ^5(c+d x)-\frac {2}{3} a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)-\frac {5}{9} a b^4 \tan ^9(c+d x)-\frac {1}{10} b^5 \tan ^{10}(c+d x)}{d}\) |
-((-(a^5*Tan[c + d*x]) - (5*a^4*b*Tan[c + d*x]^2)/2 - (2*a^3*(a^2 + 5*b^2) *Tan[c + d*x]^3)/3 - (5*a^2*b*(a^2 + b^2)*Tan[c + d*x]^4)/2 - (a*(a^4 + 20 *a^2*b^2 + 5*b^4)*Tan[c + d*x]^5)/5 - (b*(5*a^4 + 20*a^2*b^2 + b^4)*Tan[c + d*x]^6)/6 - (10*a*b^2*(a^2 + b^2)*Tan[c + d*x]^7)/7 - (b^3*(5*a^2 + b^2) *Tan[c + d*x]^8)/4 - (5*a*b^4*Tan[c + d*x]^9)/9 - (b^5*Tan[c + d*x]^10)/10 )/d)
3.2.8.3.1 Defintions of rubi rules used
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[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[x^m*((b + a*x)^n/(1 + x^2)^((m + n + 2)/2)), x], x, Cot[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && !(GtQ[ n, 0] && GtQ[m, 1])
Time = 2.36 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.05
| method | result | size |
| parts | \(-\frac {a^{5} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{5} \left (\frac {\sec \left (d x +c \right )^{10}}{10}-\frac {\sec \left (d x +c \right )^{8}}{4}+\frac {\sec \left (d x +c \right )^{6}}{6}\right )}{d}+\frac {10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )}{d}+\frac {5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )}{d}+\frac {5 a^{4} b \sec \left (d x +c \right )^{6}}{6 d}+\frac {10 a^{2} b^{3} \left (\frac {\sec \left (d x +c \right )^{8}}{8}-\frac {\sec \left (d x +c \right )^{6}}{6}\right )}{d}\) | \(255\) |
| derivativedivides | \(\frac {-a^{5} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+\frac {5 a^{4} b}{6 \cos \left (d x +c \right )^{6}}+10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )+10 a^{2} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{24 \cos \left (d x +c \right )^{4}}\right )+5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{6}}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{6}}{60 \cos \left (d x +c \right )^{6}}\right )}{d}\) | \(299\) |
| default | \(\frac {-a^{5} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+\frac {5 a^{4} b}{6 \cos \left (d x +c \right )^{6}}+10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )+10 a^{2} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{24 \cos \left (d x +c \right )^{4}}\right )+5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{6}}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{6}}{60 \cos \left (d x +c \right )^{6}}\right )}{d}\) | \(299\) |
| parallelrisch | \(-\frac {2 \left (a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{18}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17} a^{4} b +\frac {\left (-19 a^{5}+40 a^{3} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{3}+20 \left (a^{4} b -a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}+4 \left (-\frac {22}{3} a^{3} b^{2}+4 a \,b^{4}+\frac {77}{15} a^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\frac {4 \left (-35 a^{4} b +10 a^{2} b^{3}-4 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{3}+4 \left (\frac {20}{7} a \,b^{4}+\frac {90}{7} a^{3} b^{2}-11 a^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\frac {4 \left (65 a^{4} b -25 a^{2} b^{3}-8 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3}+\frac {2 \left (-\frac {1060}{7} a^{3} b^{2}+\frac {880}{21} a \,b^{4}+97 a^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3}+2 \left (-55 a^{4} b +40 a^{2} b^{3}-\frac {48}{5} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\frac {2 \left (\frac {1060}{7} a^{3} b^{2}-\frac {880}{21} a \,b^{4}-97 a^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3}+\frac {4 \left (65 a^{4} b -25 a^{2} b^{3}-8 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3}+4 \left (-\frac {90}{7} a^{3} b^{2}-\frac {20}{7} a \,b^{4}+11 a^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {4 \left (-35 a^{4} b +10 a^{2} b^{3}-4 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3}+4 \left (\frac {22}{3} a^{3} b^{2}-4 a \,b^{4}-\frac {77}{15} a^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+20 \left (a^{4} b -a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {\left (19 a^{5}-40 a^{3} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b -a^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{10}}\) | \(554\) |
| risch | \(\frac {-\frac {320 a^{2} b^{3} {\mathrm e}^{12 i \left (d x +c \right )}}{3}+\frac {640 a^{4} b \,{\mathrm e}^{12 i \left (d x +c \right )}}{3}+\frac {32 i a^{5} {\mathrm e}^{14 i \left (d x +c \right )}}{3}-\frac {1600 i a^{3} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{21}+\frac {160 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{63}-192 i a^{3} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-\frac {64 b^{5} {\mathrm e}^{12 i \left (d x +c \right )}}{3}+32 i a \,b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-\frac {160 i a^{3} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}}{3}+80 i a \,b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-\frac {480 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}}{7}+\frac {80 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{7}-\frac {320 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{21}-\frac {160 i a \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}}{7}+\frac {192 b^{5} {\mathrm e}^{10 i \left (d x +c \right )}}{5}+\frac {32 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {16 i a^{5}}{15}-\frac {64 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}}{3}+\frac {176 i a^{5} {\mathrm e}^{12 i \left (d x +c \right )}}{3}-\frac {320 i a^{3} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}}{3}+\frac {160 i a \,b^{4} {\mathrm e}^{14 i \left (d x +c \right )}}{3}+\frac {160 a^{4} b \,{\mathrm e}^{14 i \left (d x +c \right )}}{3}-\frac {320 a^{2} b^{3} {\mathrm e}^{14 i \left (d x +c \right )}}{3}+\frac {32 b^{5} {\mathrm e}^{14 i \left (d x +c \right )}}{3}+\frac {672 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}}{5}+\frac {496 i a^{5} {\mathrm e}^{8 i \left (d x +c \right )}}{3}+\frac {352 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+48 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+\frac {32 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{3}-\frac {32 i a^{3} b^{2}}{21}+\frac {16 i a \,b^{4}}{63}-\frac {320 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}}{3}-\frac {320 a^{2} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}}{3}+\frac {160 a^{4} b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {640 a^{4} b \,{\mathrm e}^{8 i \left (d x +c \right )}}{3}+320 a^{4} b \,{\mathrm e}^{10 i \left (d x +c \right )}-\frac {800 i a^{3} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}}{3}-\frac {80 i a \,b^{4} {\mathrm e}^{12 i \left (d x +c \right )}}{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}\) | \(598\) |
-a^5/d*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+b^5/d*(1/10*s ec(d*x+c)^10-1/4*sec(d*x+c)^8+1/6*sec(d*x+c)^6)+10*a^3*b^2/d*(1/7*sin(d*x+ c)^3/cos(d*x+c)^7+4/35*sin(d*x+c)^3/cos(d*x+c)^5+8/105*sin(d*x+c)^3/cos(d* x+c)^3)+5*a*b^4/d*(1/9*sin(d*x+c)^5/cos(d*x+c)^9+4/63*sin(d*x+c)^5/cos(d*x +c)^7+8/315*sin(d*x+c)^5/cos(d*x+c)^5)+5/6*a^4*b/d*sec(d*x+c)^6+10*a^2*b^3 /d*(1/8*sec(d*x+c)^8-1/6*sec(d*x+c)^6)
Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.86 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {126 \, b^{5} + 210 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 315 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (8 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 4 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 175 \, a b^{4} \cos \left (d x + c\right ) + 3 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 50 \, {\left (9 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1260 \, d \cos \left (d x + c\right )^{10}} \]
1/1260*(126*b^5 + 210*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^4 + 315*(5 *a^2*b^3 - b^5)*cos(d*x + c)^2 + 4*(8*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos( d*x + c)^9 + 4*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^7 + 175*a*b^4* cos(d*x + c) + 3*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^5 + 50*(9*a^ 3*b^2 - 5*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^10)
Timed out. \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.14 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {84 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{5} + 120 \, {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{3} b^{2} + 20 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} a b^{4} + \frac {525 \, {\left (4 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - \frac {21 \, {\left (10 \, \sin \left (d x + c\right )^{4} - 5 \, \sin \left (d x + c\right )^{2} + 1\right )} b^{5}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - \frac {1050 \, a^{4} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{1260 \, d} \]
1/1260*(84*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*a^5 + 120*(15*tan(d*x + c)^7 + 42*tan(d*x + c)^5 + 35*tan(d*x + c)^3)*a^3*b^2 + 20*(35*tan(d*x + c)^9 + 90*tan(d*x + c)^7 + 63*tan(d*x + c)^5)*a*b^4 + 525 *(4*sin(d*x + c)^2 - 1)*a^2*b^3/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin (d*x + c)^4 - 4*sin(d*x + c)^2 + 1) - 21*(10*sin(d*x + c)^4 - 5*sin(d*x + c)^2 + 1)*b^5/(sin(d*x + c)^10 - 5*sin(d*x + c)^8 + 10*sin(d*x + c)^6 - 10 *sin(d*x + c)^4 + 5*sin(d*x + c)^2 - 1) - 1050*a^4*b/(sin(d*x + c)^2 - 1)^ 3)/d
Time = 0.62 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.08 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {126 \, b^{5} \tan \left (d x + c\right )^{10} + 700 \, a b^{4} \tan \left (d x + c\right )^{9} + 1575 \, a^{2} b^{3} \tan \left (d x + c\right )^{8} + 315 \, b^{5} \tan \left (d x + c\right )^{8} + 1800 \, a^{3} b^{2} \tan \left (d x + c\right )^{7} + 1800 \, a b^{4} \tan \left (d x + c\right )^{7} + 1050 \, a^{4} b \tan \left (d x + c\right )^{6} + 4200 \, a^{2} b^{3} \tan \left (d x + c\right )^{6} + 210 \, b^{5} \tan \left (d x + c\right )^{6} + 252 \, a^{5} \tan \left (d x + c\right )^{5} + 5040 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 1260 \, a b^{4} \tan \left (d x + c\right )^{5} + 3150 \, a^{4} b \tan \left (d x + c\right )^{4} + 3150 \, a^{2} b^{3} \tan \left (d x + c\right )^{4} + 840 \, a^{5} \tan \left (d x + c\right )^{3} + 4200 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 3150 \, a^{4} b \tan \left (d x + c\right )^{2} + 1260 \, a^{5} \tan \left (d x + c\right )}{1260 \, d} \]
1/1260*(126*b^5*tan(d*x + c)^10 + 700*a*b^4*tan(d*x + c)^9 + 1575*a^2*b^3* tan(d*x + c)^8 + 315*b^5*tan(d*x + c)^8 + 1800*a^3*b^2*tan(d*x + c)^7 + 18 00*a*b^4*tan(d*x + c)^7 + 1050*a^4*b*tan(d*x + c)^6 + 4200*a^2*b^3*tan(d*x + c)^6 + 210*b^5*tan(d*x + c)^6 + 252*a^5*tan(d*x + c)^5 + 5040*a^3*b^2*t an(d*x + c)^5 + 1260*a*b^4*tan(d*x + c)^5 + 3150*a^4*b*tan(d*x + c)^4 + 31 50*a^2*b^3*tan(d*x + c)^4 + 840*a^5*tan(d*x + c)^3 + 4200*a^3*b^2*tan(d*x + c)^3 + 3150*a^4*b*tan(d*x + c)^2 + 1260*a^5*tan(d*x + c))/d
Time = 27.27 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.26 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {616\,a^5}{15}-\frac {176\,a^3\,b^2}{3}+32\,a\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {616\,a^5}{15}-\frac {176\,a^3\,b^2}{3}+32\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (-88\,a^5+\frac {720\,a^3\,b^2}{7}+\frac {160\,a\,b^4}{7}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (-88\,a^5+\frac {720\,a^3\,b^2}{7}+\frac {160\,a\,b^4}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {388\,a^5}{3}-\frac {4240\,a^3\,b^2}{21}+\frac {3520\,a\,b^4}{63}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {388\,a^5}{3}-\frac {4240\,a^3\,b^2}{21}+\frac {3520\,a\,b^4}{63}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {280\,a^4\,b}{3}-\frac {80\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {280\,a^4\,b}{3}-\frac {80\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (220\,a^4\,b-160\,a^2\,b^3+\frac {192\,b^5}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (-\frac {520\,a^4\,b}{3}+\frac {200\,a^2\,b^3}{3}+\frac {64\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (-\frac {520\,a^4\,b}{3}+\frac {200\,a^2\,b^3}{3}+\frac {64\,b^5}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {38\,a^5}{3}-\frac {80\,a^3\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {38\,a^5}{3}-\frac {80\,a^3\,b^2}{3}\right )+2\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^{10}} \]
(tan(c/2 + (d*x)/2)^5*(32*a*b^4 + (616*a^5)/15 - (176*a^3*b^2)/3) - tan(c/ 2 + (d*x)/2)^4*(40*a^4*b - 40*a^2*b^3) - tan(c/2 + (d*x)/2)^16*(40*a^4*b - 40*a^2*b^3) - 2*a^5*tan(c/2 + (d*x)/2)^19 - tan(c/2 + (d*x)/2)^15*(32*a*b ^4 + (616*a^5)/15 - (176*a^3*b^2)/3) + tan(c/2 + (d*x)/2)^7*((160*a*b^4)/7 - 88*a^5 + (720*a^3*b^2)/7) - tan(c/2 + (d*x)/2)^13*((160*a*b^4)/7 - 88*a ^5 + (720*a^3*b^2)/7) + tan(c/2 + (d*x)/2)^9*((3520*a*b^4)/63 + (388*a^5)/ 3 - (4240*a^3*b^2)/21) - tan(c/2 + (d*x)/2)^11*((3520*a*b^4)/63 + (388*a^5 )/3 - (4240*a^3*b^2)/21) + tan(c/2 + (d*x)/2)^6*((280*a^4*b)/3 + (32*b^5)/ 3 - (80*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^14*((280*a^4*b)/3 + (32*b^5)/3 - (80*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^10*(220*a^4*b + (192*b^5)/5 - 160*a^2 *b^3) + tan(c/2 + (d*x)/2)^8*((64*b^5)/3 - (520*a^4*b)/3 + (200*a^2*b^3)/3 ) + tan(c/2 + (d*x)/2)^12*((64*b^5)/3 - (520*a^4*b)/3 + (200*a^2*b^3)/3) - tan(c/2 + (d*x)/2)^3*((38*a^5)/3 - (80*a^3*b^2)/3) + tan(c/2 + (d*x)/2)^1 7*((38*a^5)/3 - (80*a^3*b^2)/3) + 2*a^5*tan(c/2 + (d*x)/2) + 10*a^4*b*tan( c/2 + (d*x)/2)^2 + 10*a^4*b*tan(c/2 + (d*x)/2)^18)/(d*(tan(c/2 + (d*x)/2)^ 2 - 1)^10)