Integrand size = 23, antiderivative size = 114 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {a^2 \sin (c+d x)}{d}-\frac {2 a (2 a-b) \sin ^3(c+d x)}{3 d}+\frac {\left (6 a^2-6 a b+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac {2 (a-b) (2 a-b) \sin ^7(c+d x)}{7 d}+\frac {(a-b)^2 \sin ^9(c+d x)}{9 d} \] Output:
a^2*sin(d*x+c)/d-2/3*a*(2*a-b)*sin(d*x+c)^3/d+1/5*(6*a^2-6*a*b+b^2)*sin(d* x+c)^5/d-2/7*(a-b)*(2*a-b)*sin(d*x+c)^7/d+1/9*(a-b)^2*sin(d*x+c)^9/d
Time = 0.40 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {630 \left (63 a^2+14 a b+3 b^2\right ) \sin (c+d x)+420 \left (21 a^2-b^2\right ) \sin (3 (c+d x))+252 \left (9 a^2-4 a b-b^2\right ) \sin (5 (c+d x))+45 (a-b) (9 a-b) \sin (7 (c+d x))+35 (a-b)^2 \sin (9 (c+d x))}{80640 d} \] Input:
Integrate[Cos[c + d*x]^9*(a + b*Tan[c + d*x]^2)^2,x]
Output:
(630*(63*a^2 + 14*a*b + 3*b^2)*Sin[c + d*x] + 420*(21*a^2 - b^2)*Sin[3*(c + d*x)] + 252*(9*a^2 - 4*a*b - b^2)*Sin[5*(c + d*x)] + 45*(a - b)*(9*a - b )*Sin[7*(c + d*x)] + 35*(a - b)^2*Sin[9*(c + d*x)])/(80640*d)
Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4159, 290, 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.
| \(\displaystyle \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \tan (c+d x)^2\right )^2}{\sec (c+d x)^9}dx\) |
| \(\Big \downarrow \) 4159 |
| \(\displaystyle \frac {\int \left (1-\sin ^2(c+d x)\right )^2 \left (a-(a-b) \sin ^2(c+d x)\right )^2d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {\int \left ((a-b)^2 \sin ^8(c+d x)-2 \left (2 a^2-3 b a+b^2\right ) \sin ^6(c+d x)+\left (6 a^2-6 b a+b^2\right ) \sin ^4(c+d x)-2 a (2 a-b) \sin ^2(c+d x)+a^2\right )d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{5} \left (6 a^2-6 a b+b^2\right ) \sin ^5(c+d x)+a^2 \sin (c+d x)+\frac {1}{9} (a-b)^2 \sin ^9(c+d x)-\frac {2}{7} (a-b) (2 a-b) \sin ^7(c+d x)-\frac {2}{3} a (2 a-b) \sin ^3(c+d x)}{d}\) |
Input:
Int[Cos[c + d*x]^9*(a + b*Tan[c + d*x]^2)^2,x]
Output:
(a^2*Sin[c + d*x] - (2*a*(2*a - b)*Sin[c + d*x]^3)/3 + ((6*a^2 - 6*a*b + b ^2)*Sin[c + d*x]^5)/5 - (2*(a - b)*(2*a - b)*Sin[c + d*x]^7)/7 + ((a - b)^ 2*Sin[c + d*x]^9)/9)/d
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2 *x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Time = 315.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.61
| method | result | size |
| derivativedivides | \(\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )+2 a b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+\frac {a^{2} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(183\) |
| default | \(\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )+2 a b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+\frac {a^{2} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(183\) |
| risch | \(\frac {63 a^{2} \sin \left (d x +c \right )}{128 d}+\frac {7 \sin \left (d x +c \right ) a b}{64 d}+\frac {3 \sin \left (d x +c \right ) b^{2}}{128 d}+\frac {\sin \left (9 d x +9 c \right ) a^{2}}{2304 d}-\frac {\sin \left (9 d x +9 c \right ) a b}{1152 d}+\frac {\sin \left (9 d x +9 c \right ) b^{2}}{2304 d}+\frac {9 \sin \left (7 d x +7 c \right ) a^{2}}{1792 d}-\frac {5 \sin \left (7 d x +7 c \right ) a b}{896 d}+\frac {\sin \left (7 d x +7 c \right ) b^{2}}{1792 d}+\frac {9 \sin \left (5 d x +5 c \right ) a^{2}}{320 d}-\frac {\sin \left (5 d x +5 c \right ) a b}{80 d}-\frac {\sin \left (5 d x +5 c \right ) b^{2}}{320 d}+\frac {7 \sin \left (3 d x +3 c \right ) a^{2}}{64 d}-\frac {\sin \left (3 d x +3 c \right ) b^{2}}{192 d}\) | \(227\) |
Input:
int(cos(d*x+c)^9*(a+b*tan(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(b^2*(-1/9*sin(d*x+c)^3*cos(d*x+c)^6-1/21*sin(d*x+c)*cos(d*x+c)^6+1/10 5*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+2*a*b*(-1/9*sin(d*x+c)*c os(d*x+c)^8+1/63*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin (d*x+c))+1/9*a^2*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+ 64/35*cos(d*x+c)^2)*sin(d*x+c))
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {{\left (35 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (4 \, a^{2} + a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (16 \, a^{2} + 4 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (16 \, a^{2} + 4 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 128 \, a^{2} + 32 \, a b + 8 \, b^{2}\right )} \sin \left (d x + c\right )}{315 \, d} \] Input:
integrate(cos(d*x+c)^9*(a+b*tan(d*x+c)^2)^2,x, algorithm="fricas")
Output:
1/315*(35*(a^2 - 2*a*b + b^2)*cos(d*x + c)^8 + 10*(4*a^2 + a*b - 5*b^2)*co s(d*x + c)^6 + 3*(16*a^2 + 4*a*b + b^2)*cos(d*x + c)^4 + 4*(16*a^2 + 4*a*b + b^2)*cos(d*x + c)^2 + 128*a^2 + 32*a*b + 8*b^2)*sin(d*x + c)/d
Timed out. \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**9*(a+b*tan(d*x+c)**2)**2,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.91 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {35 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{9} - 90 \, {\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{7} + 63 \, {\left (6 \, a^{2} - 6 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{5} - 210 \, {\left (2 \, a^{2} - a b\right )} \sin \left (d x + c\right )^{3} + 315 \, a^{2} \sin \left (d x + c\right )}{315 \, d} \] Input:
integrate(cos(d*x+c)^9*(a+b*tan(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/315*(35*(a^2 - 2*a*b + b^2)*sin(d*x + c)^9 - 90*(2*a^2 - 3*a*b + b^2)*si n(d*x + c)^7 + 63*(6*a^2 - 6*a*b + b^2)*sin(d*x + c)^5 - 210*(2*a^2 - a*b) *sin(d*x + c)^3 + 315*a^2*sin(d*x + c))/d
Timed out. \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^9*(a+b*tan(d*x+c)^2)^2,x, algorithm="giac")
Output:
Timed out
Time = 8.44 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.65 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {\frac {63\,a^2\,\sin \left (c+d\,x\right )}{128}+\frac {3\,b^2\,\sin \left (c+d\,x\right )}{128}+\frac {7\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{64}+\frac {9\,a^2\,\sin \left (5\,c+5\,d\,x\right )}{320}+\frac {9\,a^2\,\sin \left (7\,c+7\,d\,x\right )}{1792}+\frac {a^2\,\sin \left (9\,c+9\,d\,x\right )}{2304}-\frac {b^2\,\sin \left (3\,c+3\,d\,x\right )}{192}-\frac {b^2\,\sin \left (5\,c+5\,d\,x\right )}{320}+\frac {b^2\,\sin \left (7\,c+7\,d\,x\right )}{1792}+\frac {b^2\,\sin \left (9\,c+9\,d\,x\right )}{2304}+\frac {7\,a\,b\,\sin \left (c+d\,x\right )}{64}-\frac {a\,b\,\sin \left (5\,c+5\,d\,x\right )}{80}-\frac {5\,a\,b\,\sin \left (7\,c+7\,d\,x\right )}{896}-\frac {a\,b\,\sin \left (9\,c+9\,d\,x\right )}{1152}}{d} \] Input:
int(cos(c + d*x)^9*(a + b*tan(c + d*x)^2)^2,x)
Output:
((63*a^2*sin(c + d*x))/128 + (3*b^2*sin(c + d*x))/128 + (7*a^2*sin(3*c + 3 *d*x))/64 + (9*a^2*sin(5*c + 5*d*x))/320 + (9*a^2*sin(7*c + 7*d*x))/1792 + (a^2*sin(9*c + 9*d*x))/2304 - (b^2*sin(3*c + 3*d*x))/192 - (b^2*sin(5*c + 5*d*x))/320 + (b^2*sin(7*c + 7*d*x))/1792 + (b^2*sin(9*c + 9*d*x))/2304 + (7*a*b*sin(c + d*x))/64 - (a*b*sin(5*c + 5*d*x))/80 - (5*a*b*sin(7*c + 7* d*x))/896 - (a*b*sin(9*c + 9*d*x))/1152)/d
Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.37 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {\sin \left (d x +c \right ) \left (35 \sin \left (d x +c \right )^{8} a^{2}-70 \sin \left (d x +c \right )^{8} a b +35 \sin \left (d x +c \right )^{8} b^{2}-180 \sin \left (d x +c \right )^{6} a^{2}+270 \sin \left (d x +c \right )^{6} a b -90 \sin \left (d x +c \right )^{6} b^{2}+378 \sin \left (d x +c \right )^{4} a^{2}-378 \sin \left (d x +c \right )^{4} a b +63 \sin \left (d x +c \right )^{4} b^{2}-420 \sin \left (d x +c \right )^{2} a^{2}+210 \sin \left (d x +c \right )^{2} a b +315 a^{2}\right )}{315 d} \] Input:
int(cos(d*x+c)^9*(a+b*tan(d*x+c)^2)^2,x)
Output:
(sin(c + d*x)*(35*sin(c + d*x)**8*a**2 - 70*sin(c + d*x)**8*a*b + 35*sin(c + d*x)**8*b**2 - 180*sin(c + d*x)**6*a**2 + 270*sin(c + d*x)**6*a*b - 90* sin(c + d*x)**6*b**2 + 378*sin(c + d*x)**4*a**2 - 378*sin(c + d*x)**4*a*b + 63*sin(c + d*x)**4*b**2 - 420*sin(c + d*x)**2*a**2 + 210*sin(c + d*x)**2 *a*b + 315*a**2))/(315*d)