Integrand size = 28, antiderivative size = 213 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {a^3 \tan (c+d x)}{d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {a \left (a^2+b^2\right ) \tan ^3(c+d x)}{d}+\frac {b \left (9 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {3 a \left (a^2+3 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {b \left (3 a^2+b^2\right ) \tan ^6(c+d x)}{2 d}+\frac {a \left (a^2+9 b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {3 b \left (a^2+b^2\right ) \tan ^8(c+d x)}{8 d}+\frac {a b^2 \tan ^9(c+d x)}{3 d}+\frac {b^3 \tan ^{10}(c+d x)}{10 d} \] Output:
a^3*tan(d*x+c)/d+3/2*a^2*b*tan(d*x+c)^2/d+a*(a^2+b^2)*tan(d*x+c)^3/d+1/4*b *(9*a^2+b^2)*tan(d*x+c)^4/d+3/5*a*(a^2+3*b^2)*tan(d*x+c)^5/d+1/2*b*(3*a^2+ b^2)*tan(d*x+c)^6/d+1/7*a*(a^2+9*b^2)*tan(d*x+c)^7/d+3/8*b*(a^2+b^2)*tan(d *x+c)^8/d+1/3*a*b^2*tan(d*x+c)^9/d+1/10*b^3*tan(d*x+c)^10/d
Time = 1.45 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.83 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {\frac {1}{4} \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^4-\frac {6}{5} a \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^5+\frac {1}{2} \left (a^2+b^2\right ) \left (5 a^2+b^2\right ) (a+b \tan (c+d x))^6-\frac {4}{7} a \left (5 a^2+3 b^2\right ) (a+b \tan (c+d x))^7+\frac {3}{8} \left (5 a^2+b^2\right ) (a+b \tan (c+d x))^8-\frac {2}{3} a (a+b \tan (c+d x))^9+\frac {1}{10} (a+b \tan (c+d x))^{10}}{b^7 d} \] Input:
Integrate[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]
Output:
(((a^2 + b^2)^3*(a + b*Tan[c + d*x])^4)/4 - (6*a*(a^2 + b^2)^2*(a + b*Tan[ c + d*x])^5)/5 + ((a^2 + b^2)*(5*a^2 + b^2)*(a + b*Tan[c + d*x])^6)/2 - (4 *a*(5*a^2 + 3*b^2)*(a + b*Tan[c + d*x])^7)/7 + (3*(5*a^2 + b^2)*(a + b*Tan [c + d*x])^8)/8 - (2*a*(a + b*Tan[c + d*x])^9)/3 + (a + b*Tan[c + d*x])^10 /10)/(b^7*d)
Time = 0.37 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.89, 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.
| \(\displaystyle \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^3}{\cos (c+d x)^{11}}dx\) |
| \(\Big \downarrow \) 3567 |
| \(\displaystyle -\frac {\int (b+a \cot (c+d x))^3 \left (\cot ^2(c+d x)+1\right )^3 \tan ^{11}(c+d x)d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (b^3 \tan ^{11}(c+d x)+3 a b^2 \tan ^{10}(c+d x)+3 b \left (a^2+b^2\right ) \tan ^9(c+d x)+\left (a^3+9 b^2 a\right ) \tan ^8(c+d x)+3 \left (b^3+3 a^2 b\right ) \tan ^7(c+d x)+3 \left (a^3+3 b^2 a\right ) \tan ^6(c+d x)+\left (b^3+9 a^2 b\right ) \tan ^5(c+d x)+3 a \left (a^2+b^2\right ) \tan ^4(c+d x)+3 a^2 b \tan ^3(c+d x)+a^3 \tan ^2(c+d x)\right )d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-a^3 \tan (c+d x)-\frac {3}{8} b \left (a^2+b^2\right ) \tan ^8(c+d x)-\frac {1}{7} a \left (a^2+9 b^2\right ) \tan ^7(c+d x)-\frac {1}{2} b \left (3 a^2+b^2\right ) \tan ^6(c+d x)-\frac {3}{5} a \left (a^2+3 b^2\right ) \tan ^5(c+d x)-\frac {1}{4} b \left (9 a^2+b^2\right ) \tan ^4(c+d x)-a \left (a^2+b^2\right ) \tan ^3(c+d x)-\frac {3}{2} a^2 b \tan ^2(c+d x)-\frac {1}{3} a b^2 \tan ^9(c+d x)-\frac {1}{10} b^3 \tan ^{10}(c+d x)}{d}\) |
Input:
Int[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]
Output:
-((-(a^3*Tan[c + d*x]) - (3*a^2*b*Tan[c + d*x]^2)/2 - a*(a^2 + b^2)*Tan[c + d*x]^3 - (b*(9*a^2 + b^2)*Tan[c + d*x]^4)/4 - (3*a*(a^2 + 3*b^2)*Tan[c + d*x]^5)/5 - (b*(3*a^2 + b^2)*Tan[c + d*x]^6)/2 - (a*(a^2 + 9*b^2)*Tan[c + d*x]^7)/7 - (3*b*(a^2 + b^2)*Tan[c + d*x]^8)/8 - (a*b^2*Tan[c + d*x]^9)/3 - (b^3*Tan[c + d*x]^10)/10)/d)
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 = 1.18 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.82
| method | result | size |
| parts | \(-\frac {a^{3} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{3} \left (\frac {\sec \left (d x +c \right )^{10}}{10}-\frac {\sec \left (d x +c \right )^{8}}{8}\right )}{d}+\frac {3 a^{2} b \sec \left (d x +c \right )^{8}}{8 d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )}{d}\) | \(175\) |
| derivativedivides | \(\frac {-a^{3} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{8 \cos \left (d x +c \right )^{8}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(219\) |
| default | \(\frac {-a^{3} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{8 \cos \left (d x +c \right )^{8}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(219\) |
| risch | \(-\frac {32 \left (i a \,b^{2}+10 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-315 a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{12 i \left (d x +c \right )}-30 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-360 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-630 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}-126 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-3 i a^{3}+126 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-315 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+315 i a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-105 i a^{3} {\mathrm e}^{12 i \left (d x +c \right )}-525 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-135 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-105 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+45 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+120 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-378 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}\) | \(306\) |
| parallelrisch | \(-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-105 a^{3}+105 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{18}-525 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a^{3}-210 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15} b^{3}-420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13} b^{3}-2520 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13} a^{2} b +2340 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} a \,b^{2}+4410 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} a^{2} b -1420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a \,b^{2}-2520 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2} b -84 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a \,b^{2}+630 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b -420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{2}-315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17} a^{2} b +420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a \,b^{2}+630 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15} a^{2} b +84 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14} a \,b^{2}+1848 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}-4080 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} a^{3}-1470 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} b^{3}-1176 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b^{3}+4080 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{3}-1848 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}-210 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{3}+525 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}-4410 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} a^{2} b +1420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a \,b^{2}+4410 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{2} b -2340 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \,b^{2}+5730 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a^{3}-5730 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{3}-1470 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{3}-420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}\right )}{105 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{10}}\) | \(579\) |
Input:
int(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-a^3/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^3/d*(1/10*sec(d*x+c)^10-1/8*sec(d*x+c)^8)+3/8*a^2*b/d*sec(d*x+c)^8 +3*a*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 = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.70 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {84 \, b^{3} + 105 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (16 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{9} + 8 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} + 6 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{5} + 35 \, a b^{2} \cos \left (d x + c\right ) + 5 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{10}} \] Input:
integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="fricas ")
Output:
1/840*(84*b^3 + 105*(3*a^2*b - b^3)*cos(d*x + c)^2 + 8*(16*(3*a^3 - a*b^2) *cos(d*x + c)^9 + 8*(3*a^3 - a*b^2)*cos(d*x + c)^7 + 6*(3*a^3 - a*b^2)*cos (d*x + c)^5 + 35*a*b^2*cos(d*x + c) + 5*(3*a^3 - a*b^2)*cos(d*x + c)^3)*si n(d*x + c))/(d*cos(d*x + c)^10)
Timed out. \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**11*(a*cos(d*x+c)+b*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.86 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {24 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{3} + 8 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac {21 \, {\left (5 \, \sin \left (d x + c\right )^{2} - 1\right )} b^{3}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} + \frac {315 \, a^{2} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{4}}}{840 \, d} \] Input:
integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="maxima ")
Output:
1/840*(24*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*t an(d*x + c))*a^3 + 8*(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*b^2 - 21*(5*sin(d*x + c)^2 - 1)*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) + 315*a^2*b/(sin(d*x + c)^2 - 1)^4)/d
Time = 0.19 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.03 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {84 \, b^{3} \tan \left (d x + c\right )^{10} + 280 \, a b^{2} \tan \left (d x + c\right )^{9} + 315 \, a^{2} b \tan \left (d x + c\right )^{8} + 315 \, b^{3} \tan \left (d x + c\right )^{8} + 120 \, a^{3} \tan \left (d x + c\right )^{7} + 1080 \, a b^{2} \tan \left (d x + c\right )^{7} + 1260 \, a^{2} b \tan \left (d x + c\right )^{6} + 420 \, b^{3} \tan \left (d x + c\right )^{6} + 504 \, a^{3} \tan \left (d x + c\right )^{5} + 1512 \, a b^{2} \tan \left (d x + c\right )^{5} + 1890 \, a^{2} b \tan \left (d x + c\right )^{4} + 210 \, b^{3} \tan \left (d x + c\right )^{4} + 840 \, a^{3} \tan \left (d x + c\right )^{3} + 840 \, a b^{2} \tan \left (d x + c\right )^{3} + 1260 \, a^{2} b \tan \left (d x + c\right )^{2} + 840 \, a^{3} \tan \left (d x + c\right )}{840 \, d} \] Input:
integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/840*(84*b^3*tan(d*x + c)^10 + 280*a*b^2*tan(d*x + c)^9 + 315*a^2*b*tan(d *x + c)^8 + 315*b^3*tan(d*x + c)^8 + 120*a^3*tan(d*x + c)^7 + 1080*a*b^2*t an(d*x + c)^7 + 1260*a^2*b*tan(d*x + c)^6 + 420*b^3*tan(d*x + c)^6 + 504*a ^3*tan(d*x + c)^5 + 1512*a*b^2*tan(d*x + c)^5 + 1890*a^2*b*tan(d*x + c)^4 + 210*b^3*tan(d*x + c)^4 + 840*a^3*tan(d*x + c)^3 + 840*a*b^2*tan(d*x + c) ^3 + 1260*a^2*b*tan(d*x + c)^2 + 840*a^3*tan(d*x + c))/d
Time = 17.81 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.89 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {a^3\,\sin \left (c+d\,x\right )}{7}-\frac {a\,b^2\,\sin \left (c+d\,x\right )}{21}\right )+{\cos \left (c+d\,x\right )}^5\,\left (\frac {6\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {2\,a\,b^2\,\sin \left (c+d\,x\right )}{35}\right )+{\cos \left (c+d\,x\right )}^7\,\left (\frac {8\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )}{105}\right )+{\cos \left (c+d\,x\right )}^9\,\left (\frac {16\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {16\,a\,b^2\,\sin \left (c+d\,x\right )}{105}\right )+{\cos \left (c+d\,x\right )}^2\,\left (\frac {3\,a^2\,b}{8}-\frac {b^3}{8}\right )+\frac {b^3}{10}+\frac {a\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{3}}{d\,{\cos \left (c+d\,x\right )}^{10}} \] Input:
int((a*cos(c + d*x) + b*sin(c + d*x))^3/cos(c + d*x)^11,x)
Output:
(cos(c + d*x)^3*((a^3*sin(c + d*x))/7 - (a*b^2*sin(c + d*x))/21) + cos(c + d*x)^5*((6*a^3*sin(c + d*x))/35 - (2*a*b^2*sin(c + d*x))/35) + cos(c + d* x)^7*((8*a^3*sin(c + d*x))/35 - (8*a*b^2*sin(c + d*x))/105) + cos(c + d*x) ^9*((16*a^3*sin(c + d*x))/35 - (16*a*b^2*sin(c + d*x))/105) + cos(c + d*x) ^2*((3*a^2*b)/8 - b^3/8) + b^3/10 + (a*b^2*cos(c + d*x)*sin(c + d*x))/3)/( d*cos(c + d*x)^10)
Time = 0.18 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.65 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {\sin \left (d x +c \right ) \left (-384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} a^{3}+128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} a \,b^{2}+1728 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{3}-576 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a \,b^{2}-3024 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3}+1008 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{2}+2520 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3}-840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{2}-840 \cos \left (d x +c \right ) a^{3}-315 \sin \left (d x +c \right )^{9} a^{2} b +21 \sin \left (d x +c \right )^{9} b^{3}+1575 \sin \left (d x +c \right )^{7} a^{2} b -105 \sin \left (d x +c \right )^{7} b^{3}-3150 \sin \left (d x +c \right )^{5} a^{2} b +210 \sin \left (d x +c \right )^{5} b^{3}+3150 \sin \left (d x +c \right )^{3} a^{2} b -210 \sin \left (d x +c \right )^{3} b^{3}-1260 \sin \left (d x +c \right ) a^{2} b \right )}{840 d \left (\sin \left (d x +c \right )^{10}-5 \sin \left (d x +c \right )^{8}+10 \sin \left (d x +c \right )^{6}-10 \sin \left (d x +c \right )^{4}+5 \sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^3,x)
Output:
(sin(c + d*x)*( - 384*cos(c + d*x)*sin(c + d*x)**8*a**3 + 128*cos(c + d*x) *sin(c + d*x)**8*a*b**2 + 1728*cos(c + d*x)*sin(c + d*x)**6*a**3 - 576*cos (c + d*x)*sin(c + d*x)**6*a*b**2 - 3024*cos(c + d*x)*sin(c + d*x)**4*a**3 + 1008*cos(c + d*x)*sin(c + d*x)**4*a*b**2 + 2520*cos(c + d*x)*sin(c + d*x )**2*a**3 - 840*cos(c + d*x)*sin(c + d*x)**2*a*b**2 - 840*cos(c + d*x)*a** 3 - 315*sin(c + d*x)**9*a**2*b + 21*sin(c + d*x)**9*b**3 + 1575*sin(c + d* x)**7*a**2*b - 105*sin(c + d*x)**7*b**3 - 3150*sin(c + d*x)**5*a**2*b + 21 0*sin(c + d*x)**5*b**3 + 3150*sin(c + d*x)**3*a**2*b - 210*sin(c + d*x)**3 *b**3 - 1260*sin(c + d*x)*a**2*b))/(840*d*(sin(c + d*x)**10 - 5*sin(c + d* x)**8 + 10*sin(c + d*x)**6 - 10*sin(c + d*x)**4 + 5*sin(c + d*x)**2 - 1))