Integrand size = 28, antiderivative size = 254 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \tan (c+d x)}{d}+\frac {2 a^3 b \tan ^2(c+d x)}{d}+\frac {a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{d}+\frac {a b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{d}+\frac {\left (3 a^4+18 a^2 b^2+b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {2 a b \left (a^2+b^2\right ) \tan ^6(c+d x)}{d}+\frac {\left (a^4+18 a^2 b^2+3 b^4\right ) \tan ^7(c+d x)}{7 d}+\frac {a b \left (a^2+3 b^2\right ) \tan ^8(c+d x)}{2 d}+\frac {b^2 \left (2 a^2+b^2\right ) \tan ^9(c+d x)}{3 d}+\frac {2 a b^3 \tan ^{10}(c+d x)}{5 d}+\frac {b^4 \tan ^{11}(c+d x)}{11 d} \]
a^4*tan(d*x+c)/d+2*a^3*b*tan(d*x+c)^2/d+a^2*(a^2+2*b^2)*tan(d*x+c)^3/d+a*b *(3*a^2+b^2)*tan(d*x+c)^4/d+1/5*(3*a^4+18*a^2*b^2+b^4)*tan(d*x+c)^5/d+2*a* b*(a^2+b^2)*tan(d*x+c)^6/d+1/7*(a^4+18*a^2*b^2+3*b^4)*tan(d*x+c)^7/d+1/2*a *b*(a^2+3*b^2)*tan(d*x+c)^8/d+1/3*b^2*(2*a^2+b^2)*tan(d*x+c)^9/d+2/5*a*b^3 *tan(d*x+c)^10/d+1/11*b^4*tan(d*x+c)^11/d
Time = 1.84 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.69 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {\frac {1}{5} \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^5-a \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^6+\frac {3}{7} \left (a^2+b^2\right ) \left (5 a^2+b^2\right ) (a+b \tan (c+d x))^7-\frac {1}{2} a \left (5 a^2+3 b^2\right ) (a+b \tan (c+d x))^8+\frac {1}{3} \left (5 a^2+b^2\right ) (a+b \tan (c+d x))^9-\frac {3}{5} a (a+b \tan (c+d x))^{10}+\frac {1}{11} (a+b \tan (c+d x))^{11}}{b^7 d} \]
(((a^2 + b^2)^3*(a + b*Tan[c + d*x])^5)/5 - a*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^6 + (3*(a^2 + b^2)*(5*a^2 + b^2)*(a + b*Tan[c + d*x])^7)/7 - (a*(5* a^2 + 3*b^2)*(a + b*Tan[c + d*x])^8)/2 + ((5*a^2 + b^2)*(a + b*Tan[c + d*x ])^9)/3 - (3*a*(a + b*Tan[c + d*x])^10)/5 + (a + b*Tan[c + d*x])^11/11)/(b ^7*d)
Time = 0.43 (sec) , antiderivative size = 229, 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.
| \(\displaystyle \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^4}{\cos (c+d x)^{12}}dx\) |
| \(\Big \downarrow \) 3567 |
| \(\displaystyle -\frac {\int (b+a \cot (c+d x))^4 \left (\cot ^2(c+d x)+1\right )^3 \tan ^{12}(c+d x)d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (b^4 \tan ^{12}(c+d x)+4 a b^3 \tan ^{11}(c+d x)+3 \left (b^4+2 a^2 b^2\right ) \tan ^{10}(c+d x)+4 a b \left (a^2+3 b^2\right ) \tan ^9(c+d x)+\left (a^4+18 b^2 a^2+3 b^4\right ) \tan ^8(c+d x)+12 a b \left (a^2+b^2\right ) \tan ^7(c+d x)+\left (3 a^4+18 b^2 a^2+b^4\right ) \tan ^6(c+d x)+4 a b \left (3 a^2+b^2\right ) \tan ^5(c+d x)+3 \left (a^4+2 b^2 a^2\right ) \tan ^4(c+d x)+4 a^3 b \tan ^3(c+d x)+a^4 \tan ^2(c+d x)\right )d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-a^4 \tan (c+d x)-2 a^3 b \tan ^2(c+d x)-\frac {1}{3} b^2 \left (2 a^2+b^2\right ) \tan ^9(c+d x)-\frac {1}{2} a b \left (a^2+3 b^2\right ) \tan ^8(c+d x)-2 a b \left (a^2+b^2\right ) \tan ^6(c+d x)-a b \left (3 a^2+b^2\right ) \tan ^4(c+d x)-a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)-\frac {1}{7} \left (a^4+18 a^2 b^2+3 b^4\right ) \tan ^7(c+d x)-\frac {1}{5} \left (3 a^4+18 a^2 b^2+b^4\right ) \tan ^5(c+d x)-\frac {2}{5} a b^3 \tan ^{10}(c+d x)-\frac {1}{11} b^4 \tan ^{11}(c+d x)}{d}\) |
-((-(a^4*Tan[c + d*x]) - 2*a^3*b*Tan[c + d*x]^2 - a^2*(a^2 + 2*b^2)*Tan[c + d*x]^3 - a*b*(3*a^2 + b^2)*Tan[c + d*x]^4 - ((3*a^4 + 18*a^2*b^2 + b^4)* Tan[c + d*x]^5)/5 - 2*a*b*(a^2 + b^2)*Tan[c + d*x]^6 - ((a^4 + 18*a^2*b^2 + 3*b^4)*Tan[c + d*x]^7)/7 - (a*b*(a^2 + 3*b^2)*Tan[c + d*x]^8)/2 - (b^2*( 2*a^2 + b^2)*Tan[c + d*x]^9)/3 - (2*a*b^3*Tan[c + d*x]^10)/5 - (b^4*Tan[c + d*x]^11)/11)/d)
3.1.91.3.1 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 = 259, normalized size of antiderivative = 1.02
| method | result | size |
| parts | \(-\frac {a^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \sin \left (d x +c \right )^{5}}{33 \cos \left (d x +c \right )^{9}}+\frac {8 \sin \left (d x +c \right )^{5}}{231 \cos \left (d x +c \right )^{7}}+\frac {16 \sin \left (d x +c \right )^{5}}{1155 \cos \left (d x +c \right )^{5}}\right )}{d}+\frac {a^{3} b \sec \left (d x +c \right )^{8}}{2 d}+\frac {4 a \,b^{3} \left (\frac {\sec \left (d x +c \right )^{10}}{10}-\frac {\sec \left (d x +c \right )^{8}}{8}\right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )}{d}\) | \(259\) |
| derivativedivides | \(\frac {-a^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+\frac {a^{3} b}{2 \cos \left (d x +c \right )^{8}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{4}}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \sin \left (d x +c \right )^{5}}{33 \cos \left (d x +c \right )^{9}}+\frac {8 \sin \left (d x +c \right )^{5}}{231 \cos \left (d x +c \right )^{7}}+\frac {16 \sin \left (d x +c \right )^{5}}{1155 \cos \left (d x +c \right )^{5}}\right )}{d}\) | \(300\) |
| default | \(\frac {-a^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+\frac {a^{3} b}{2 \cos \left (d x +c \right )^{8}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{4}}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \sin \left (d x +c \right )^{5}}{33 \cos \left (d x +c \right )^{9}}+\frac {8 \sin \left (d x +c \right )^{5}}{231 \cos \left (d x +c \right )^{7}}+\frac {16 \sin \left (d x +c \right )^{5}}{1155 \cos \left (d x +c \right )^{5}}\right )}{d}\) | \(300\) |
| risch | \(\frac {32 i \left (33 a^{4}+b^{4}-22 a^{2} b^{2}+9933 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}+11 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+165 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+2541 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-9702 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-6930 a^{2} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-924 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-13860 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}-462 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-4620 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+4620 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-924 i a \,b^{3} {\mathrm e}^{12 i \left (d x +c \right )}-13860 i a^{3} b \,{\mathrm e}^{12 i \left (d x +c \right )}+1155 a^{4} {\mathrm e}^{14 i \left (d x +c \right )}+9735 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-825 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+5445 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+1815 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+55 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+363 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+4620 i a \,b^{3} {\mathrm e}^{14 i \left (d x +c \right )}-4620 i a^{3} b \,{\mathrm e}^{14 i \left (d x +c \right )}-330 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-3630 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-1210 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-242 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+5313 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}+1155 b^{4} {\mathrm e}^{14 i \left (d x +c \right )}-2079 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}\right )}{1155 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{11}}\) | \(480\) |
| parallelrisch | \(-\frac {2 \left (a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{20}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{19} a^{3} b +2 \left (-3 a^{4}+4 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{18}+4 \left (3 a^{3} b -2 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}+\frac {\left (113 a^{4}-32 a^{2} b^{2}+16 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{5}+8 \left (-5 a^{3} b -a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}+\frac {8 \left (-247 a^{4}+188 a^{2} b^{2}+36 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{35}+8 \left (11 a^{3} b -5 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\frac {2 \left (-\frac {752}{3} a^{2} b^{2}+\frac {248}{3} b^{4}+327 a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{7}+56 \left (\frac {1}{5} a \,b^{3}-2 a^{3} b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\frac {4 \left (-191 a^{4}+\frac {284}{3} a^{2} b^{2}+\frac {1328}{33} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{7}+56 \left (2 a^{3} b -\frac {1}{5} a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\frac {2 \left (-\frac {752}{3} a^{2} b^{2}+\frac {248}{3} b^{4}+327 a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{7}+8 \left (-11 a^{3} b +5 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\frac {8 \left (-247 a^{4}+188 a^{2} b^{2}+36 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{35}+8 \left (5 a^{3} b +a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\frac {\left (113 a^{4}-32 a^{2} b^{2}+16 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5}+4 \left (-3 a^{3} b +2 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (-3 a^{4}+4 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b +a^{4}\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 )^{11}}\) | \(566\) |
-a^4/d*(-16/35-1/7*sec(d*x+c)^6-6/35*sec(d*x+c)^4-8/35*sec(d*x+c)^2)*tan(d *x+c)+b^4/d*(1/11*sin(d*x+c)^5/cos(d*x+c)^11+2/33*sin(d*x+c)^5/cos(d*x+c)^ 9+8/231*sin(d*x+c)^5/cos(d*x+c)^7+16/1155*sin(d*x+c)^5/cos(d*x+c)^5)+1/2*a ^3*b/d*sec(d*x+c)^8+4*a*b^3/d*(1/10*sec(d*x+c)^10-1/8*sec(d*x+c)^8)+6*a^2* b^2/d*(1/9*sin(d*x+c)^3/cos(d*x+c)^9+2/21*sin(d*x+c)^3/cos(d*x+c)^7+8/105* sin(d*x+c)^3/cos(d*x+c)^5+16/315*sin(d*x+c)^3/cos(d*x+c)^3)
Time = 0.27 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.76 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {924 \, a b^{3} \cos \left (d x + c\right ) + 1155 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (16 \, {\left (33 \, a^{4} - 22 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{10} + 8 \, {\left (33 \, a^{4} - 22 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} + 6 \, {\left (33 \, a^{4} - 22 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} + 5 \, {\left (33 \, a^{4} - 22 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 105 \, b^{4} + 70 \, {\left (11 \, a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{2310 \, d \cos \left (d x + c\right )^{11}} \]
1/2310*(924*a*b^3*cos(d*x + c) + 1155*(a^3*b - a*b^3)*cos(d*x + c)^3 + 2*( 16*(33*a^4 - 22*a^2*b^2 + b^4)*cos(d*x + c)^10 + 8*(33*a^4 - 22*a^2*b^2 + b^4)*cos(d*x + c)^8 + 6*(33*a^4 - 22*a^2*b^2 + b^4)*cos(d*x + c)^6 + 5*(33 *a^4 - 22*a^2*b^2 + b^4)*cos(d*x + c)^4 + 105*b^4 + 70*(11*a^2*b^2 - 2*b^4 )*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^11)
Timed out. \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.92 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {66 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{4} + 44 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a^{2} b^{2} + 2 \, {\left (105 \, \tan \left (d x + c\right )^{11} + 385 \, \tan \left (d x + c\right )^{9} + 495 \, \tan \left (d x + c\right )^{7} + 231 \, \tan \left (d x + c\right )^{5}\right )} b^{4} - \frac {231 \, {\left (5 \, \sin \left (d x + c\right )^{2} - 1\right )} a b^{3}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} + \frac {1155 \, a^{3} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{4}}}{2310 \, d} \]
1/2310*(66*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35* tan(d*x + c))*a^4 + 44*(35*tan(d*x + c)^9 + 135*tan(d*x + c)^7 + 189*tan(d *x + c)^5 + 105*tan(d*x + c)^3)*a^2*b^2 + 2*(105*tan(d*x + c)^11 + 385*tan (d*x + c)^9 + 495*tan(d*x + c)^7 + 231*tan(d*x + c)^5)*b^4 - 231*(5*sin(d* x + c)^2 - 1)*a*b^3/(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) + 1155*a^3*b/(sin(d*x + c)^2 - 1)^4)/d
Time = 0.48 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.12 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {210 \, b^{4} \tan \left (d x + c\right )^{11} + 924 \, a b^{3} \tan \left (d x + c\right )^{10} + 1540 \, a^{2} b^{2} \tan \left (d x + c\right )^{9} + 770 \, b^{4} \tan \left (d x + c\right )^{9} + 1155 \, a^{3} b \tan \left (d x + c\right )^{8} + 3465 \, a b^{3} \tan \left (d x + c\right )^{8} + 330 \, a^{4} \tan \left (d x + c\right )^{7} + 5940 \, a^{2} b^{2} \tan \left (d x + c\right )^{7} + 990 \, b^{4} \tan \left (d x + c\right )^{7} + 4620 \, a^{3} b \tan \left (d x + c\right )^{6} + 4620 \, a b^{3} \tan \left (d x + c\right )^{6} + 1386 \, a^{4} \tan \left (d x + c\right )^{5} + 8316 \, a^{2} b^{2} \tan \left (d x + c\right )^{5} + 462 \, b^{4} \tan \left (d x + c\right )^{5} + 6930 \, a^{3} b \tan \left (d x + c\right )^{4} + 2310 \, a b^{3} \tan \left (d x + c\right )^{4} + 2310 \, a^{4} \tan \left (d x + c\right )^{3} + 4620 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 4620 \, a^{3} b \tan \left (d x + c\right )^{2} + 2310 \, a^{4} \tan \left (d x + c\right )}{2310 \, d} \]
1/2310*(210*b^4*tan(d*x + c)^11 + 924*a*b^3*tan(d*x + c)^10 + 1540*a^2*b^2 *tan(d*x + c)^9 + 770*b^4*tan(d*x + c)^9 + 1155*a^3*b*tan(d*x + c)^8 + 346 5*a*b^3*tan(d*x + c)^8 + 330*a^4*tan(d*x + c)^7 + 5940*a^2*b^2*tan(d*x + c )^7 + 990*b^4*tan(d*x + c)^7 + 4620*a^3*b*tan(d*x + c)^6 + 4620*a*b^3*tan( d*x + c)^6 + 1386*a^4*tan(d*x + c)^5 + 8316*a^2*b^2*tan(d*x + c)^5 + 462*b ^4*tan(d*x + c)^5 + 6930*a^3*b*tan(d*x + c)^4 + 2310*a*b^3*tan(d*x + c)^4 + 2310*a^4*tan(d*x + c)^3 + 4620*a^2*b^2*tan(d*x + c)^3 + 4620*a^3*b*tan(d *x + c)^2 + 2310*a^4*tan(d*x + c))/d
Time = 26.40 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.20 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {226\,a^4}{5}-\frac {64\,a^2\,b^2}{5}+\frac {32\,b^4}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {226\,a^4}{5}-\frac {64\,a^2\,b^2}{5}+\frac {32\,b^4}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {1308\,a^4}{7}-\frac {3008\,a^2\,b^2}{21}+\frac {992\,b^4}{21}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {1308\,a^4}{7}-\frac {3008\,a^2\,b^2}{21}+\frac {992\,b^4}{21}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (-\frac {3952\,a^4}{35}+\frac {3008\,a^2\,b^2}{35}+\frac {576\,b^4}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (-\frac {3952\,a^4}{35}+\frac {3008\,a^2\,b^2}{35}+\frac {576\,b^4}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-\frac {1528\,a^4}{7}+\frac {2272\,a^2\,b^2}{21}+\frac {10624\,b^4}{231}\right )+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^4-16\,a^2\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}\,\left (12\,a^4-16\,a^2\,b^2\right )+2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (16\,a\,b^3-24\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (16\,a\,b^3-24\,a^3\,b\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (80\,a^3\,b+16\,a\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (80\,a^3\,b+16\,a\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (80\,a\,b^3-176\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (80\,a\,b^3-176\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {112\,a\,b^3}{5}-224\,a^3\,b\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {112\,a\,b^3}{5}-224\,a^3\,b\right )+8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^{11}} \]
-(tan(c/2 + (d*x)/2)^5*((226*a^4)/5 + (32*b^4)/5 - (64*a^2*b^2)/5) + tan(c /2 + (d*x)/2)^17*((226*a^4)/5 + (32*b^4)/5 - (64*a^2*b^2)/5) + tan(c/2 + ( d*x)/2)^9*((1308*a^4)/7 + (992*b^4)/21 - (3008*a^2*b^2)/21) + tan(c/2 + (d *x)/2)^13*((1308*a^4)/7 + (992*b^4)/21 - (3008*a^2*b^2)/21) + tan(c/2 + (d *x)/2)^7*((576*b^4)/35 - (3952*a^4)/35 + (3008*a^2*b^2)/35) + tan(c/2 + (d *x)/2)^15*((576*b^4)/35 - (3952*a^4)/35 + (3008*a^2*b^2)/35) + tan(c/2 + ( d*x)/2)^11*((10624*b^4)/231 - (1528*a^4)/7 + (2272*a^2*b^2)/21) + 2*a^4*ta n(c/2 + (d*x)/2)^21 - tan(c/2 + (d*x)/2)^3*(12*a^4 - 16*a^2*b^2) - tan(c/2 + (d*x)/2)^19*(12*a^4 - 16*a^2*b^2) + 2*a^4*tan(c/2 + (d*x)/2) + tan(c/2 + (d*x)/2)^4*(16*a*b^3 - 24*a^3*b) - tan(c/2 + (d*x)/2)^18*(16*a*b^3 - 24* a^3*b) + tan(c/2 + (d*x)/2)^6*(16*a*b^3 + 80*a^3*b) - tan(c/2 + (d*x)/2)^1 6*(16*a*b^3 + 80*a^3*b) + tan(c/2 + (d*x)/2)^8*(80*a*b^3 - 176*a^3*b) - ta n(c/2 + (d*x)/2)^14*(80*a*b^3 - 176*a^3*b) - tan(c/2 + (d*x)/2)^10*((112*a *b^3)/5 - 224*a^3*b) + tan(c/2 + (d*x)/2)^12*((112*a*b^3)/5 - 224*a^3*b) + 8*a^3*b*tan(c/2 + (d*x)/2)^2 - 8*a^3*b*tan(c/2 + (d*x)/2)^20)/(d*(tan(c/2 + (d*x)/2)^2 - 1)^11)