Integrand size = 41, antiderivative size = 196 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} \left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) x+\frac {b^2 (b B+3 a C) \text {arctanh}(\sin (c+d x))}{d}+\frac {a \left (3 A b^2+6 a b B+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}+\frac {(A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 (5 A b+3 a B-6 b C) \tan (c+d x)}{6 d} \] Output:
1/2*(2*A*b^3+B*a^3+6*B*a*b^2+3*a^2*b*(A+2*C))*x+b^2*(B*b+3*C*a)*arctanh(si n(d*x+c))/d+1/3*a*(3*A*b^2+6*B*a*b+a^2*(2*A+3*C))*sin(d*x+c)/d+1/2*(A*b+B* a)*cos(d*x+c)*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/3*A*cos(d*x+c)^2*(a+b*sec( d*x+c))^3*sin(d*x+c)/d-1/6*b^2*(5*A*b+3*B*a-6*C*b)*tan(d*x+c)/d
Time = 3.00 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.34 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 \left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) (c+d x)-12 b^2 (b B+3 a C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 b^2 (b B+3 a C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {12 b^3 C \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {12 b^3 C \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+3 a \left (12 A b^2+12 a b B+a^2 (3 A+4 C)\right ) \sin (c+d x)+3 a^2 (3 A b+a B) \sin (2 (c+d x))+a^3 A \sin (3 (c+d x))}{12 d} \] Input:
Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(6*(2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*(c + d*x) - 12*b^2*(b *B + 3*a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^2*(b*B + 3*a*C )*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (12*b^3*C*Sin[(c + d*x)/2])/( Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (12*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*a*(12*A*b^2 + 12*a*b*B + a^2*(3*A + 4*C ))*Sin[c + d*x] + 3*a^2*(3*A*b + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*d)
Time = 1.46 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {3042, 4582, 3042, 4582, 3042, 4564, 3042, 4535, 24, 3042, 4533, 3042, 4257}
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.
| \(\displaystyle \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (-b (A-3 C) \sec ^2(c+d x)+(2 a A+3 b B+3 a C) \sec (c+d x)+3 (A b+a B)\right )dx+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-b (A-3 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(2 a A+3 b B+3 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 (A b+a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \cos (c+d x) (a+b \sec (c+d x)) \left (-b (5 A b-6 C b+3 a B) \sec ^2(c+d x)+\left (3 B a^2+b (5 A+12 C) a+6 b^2 B\right ) \sec (c+d x)+2 \left (\frac {1}{2} (4 A+6 C) a^2+6 b B a+3 A b^2\right )\right )dx+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b (5 A b-6 C b+3 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 B a^2+b (5 A+12 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (\frac {1}{2} (4 A+6 C) a^2+6 b B a+3 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (6 b^2 (b B+3 a C) \sec ^2(c+d x)+3 \left (B a^3+3 b (A+2 C) a^2+6 b^2 B a+2 A b^3\right ) \sec (c+d x)+2 a \left ((2 A+3 C) a^2+6 b B a+3 A b^2\right )\right )dx-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \frac {6 b^2 (b B+3 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 \left (B a^3+3 b (A+2 C) a^2+6 b^2 B a+2 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a \left ((2 A+3 C) a^2+6 b B a+3 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (6 b^2 (b B+3 a C) \sec ^2(c+d x)+2 a \left ((2 A+3 C) a^2+6 b B a+3 A b^2\right )\right )dx+3 \left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right ) \int 1dx-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (6 b^2 (b B+3 a C) \sec ^2(c+d x)+2 a \left ((2 A+3 C) a^2+6 b B a+3 A b^2\right )\right )dx+3 x \left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right )-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \frac {6 b^2 (b B+3 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left ((2 A+3 C) a^2+6 b B a+3 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+3 x \left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right )-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (6 b^2 (3 a C+b B) \int \sec (c+d x)dx+\frac {2 a \sin (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}+3 x \left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right )-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (6 b^2 (3 a C+b B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {2 a \sin (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}+3 x \left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right )-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\frac {2 a \sin (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}+3 x \left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right )-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{d}+\frac {6 b^2 (3 a C+b B) \text {arctanh}(\sin (c+d x))}{d}\right )+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\) |
Input:
Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((3*(A*b + a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (3*(2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*x + (6*b^2*(b*B + 3*a*C)*ArcTanh [Sin[c + d*x]])/d + (2*a*(3*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Sin[c + d*x ])/d - (b^2*(5*A*b + 3*a*B - 6*b*C)*Tan[c + d*x])/d)/2)/3
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
Time = 1.52 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{2} b \sin \left (d x +c \right )+3 a^{2} b C \left (d x +c \right )+3 a A \,b^{2} \sin \left (d x +c \right )+3 B a \,b^{2} \left (d x +c \right )+3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{3} \left (d x +c \right )+B \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{3} \tan \left (d x +c \right )}{d}\) | \(206\) |
| default | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{2} b \sin \left (d x +c \right )+3 a^{2} b C \left (d x +c \right )+3 a A \,b^{2} \sin \left (d x +c \right )+3 B a \,b^{2} \left (d x +c \right )+3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{3} \left (d x +c \right )+B \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{3} \tan \left (d x +c \right )}{d}\) | \(206\) |
| parallelrisch | \(\frac {-24 b^{2} \cos \left (d x +c \right ) \left (B b +3 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+24 b^{2} \cos \left (d x +c \right ) \left (B b +3 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (10 A +12 C \right ) a^{3}+36 B \,a^{2} b +36 a A \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (9 A \,a^{2} b +3 B \,a^{3}\right ) \sin \left (3 d x +3 c \right )+a^{3} A \sin \left (4 d x +4 c \right )+36 x \left (\frac {B \,a^{3}}{3}+a^{2} b \left (A +2 C \right )+2 B a \,b^{2}+\frac {2 A \,b^{3}}{3}\right ) d \cos \left (d x +c \right )+9 \left (A \,a^{2} b +\frac {1}{3} B \,a^{3}+\frac {8}{3} C \,b^{3}\right ) \sin \left (d x +c \right )}{24 d \cos \left (d x +c \right )}\) | \(218\) |
| risch | \(\frac {3 a^{2} A b x}{2}+A \,b^{3} x +\frac {a^{3} B x}{2}+3 x B a \,b^{2}+3 C \,a^{2} b x -\frac {3 i A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,a^{3}}{8 d}+\frac {2 i C \,b^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {3 i a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{2} b}{2 d}-\frac {3 i a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a A \,b^{2}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{3} C}{2 d}+\frac {3 i A \,a^{2} b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{3} C}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{2} b}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a A \,b^{2}}{2 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{3}}{8 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,b^{3}}{d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,b^{3}}{d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{2}}{d}+\frac {a^{3} A \sin \left (3 d x +3 c \right )}{12 d}\) | \(403\) |
| norman | \(\frac {\left (\frac {3}{2} A \,a^{2} b +A \,b^{3}+\frac {1}{2} B \,a^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) x +\left (-\frac {9}{2} A \,a^{2} b -3 A \,b^{3}-\frac {3}{2} B \,a^{3}-9 B a \,b^{2}-9 a^{2} b C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {9}{2} A \,a^{2} b -3 A \,b^{3}-\frac {3}{2} B \,a^{3}-9 B a \,b^{2}-9 a^{2} b C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-\frac {3}{2} A \,a^{2} b -A \,b^{3}-\frac {1}{2} B \,a^{3}-3 B a \,b^{2}-3 a^{2} b C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {3}{2} A \,a^{2} b -A \,b^{3}-\frac {1}{2} B \,a^{3}-3 B a \,b^{2}-3 a^{2} b C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {3}{2} A \,a^{2} b +A \,b^{3}+\frac {1}{2} B \,a^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {9}{2} A \,a^{2} b +3 A \,b^{3}+\frac {3}{2} B \,a^{3}+9 B a \,b^{2}+9 a^{2} b C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {9}{2} A \,a^{2} b +3 A \,b^{3}+\frac {3}{2} B \,a^{3}+9 B a \,b^{2}+9 a^{2} b C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {\left (2 a^{3} A -3 A \,a^{2} b +6 a A \,b^{2}-B \,a^{3}+6 B \,a^{2} b +2 a^{3} C -2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}+\frac {\left (2 a^{3} A +3 A \,a^{2} b +6 a A \,b^{2}+B \,a^{3}+6 B \,a^{2} b +2 a^{3} C +2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (26 a^{3} A -45 A \,a^{2} b -18 a A \,b^{2}-15 B \,a^{3}-18 B \,a^{2} b -6 a^{3} C +18 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}+\frac {\left (26 a^{3} A +45 A \,a^{2} b -18 a A \,b^{2}+15 B \,a^{3}-18 B \,a^{2} b -6 a^{3} C -18 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}-\frac {8 a \left (a^{2} A -3 A \,b^{2}-3 B a b -C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {4 a \left (5 a^{2} A -9 a A b +9 A \,b^{2}-3 B \,a^{2}+9 B a b +3 C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {4 a \left (5 a^{2} A +9 a A b +9 A \,b^{2}+3 B \,a^{2}+9 B a b +3 C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}+\frac {b^{2} \left (B b +3 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {b^{2} \left (B b +3 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(839\) |
Input:
int(cos(d*x+c)^3*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,meth od=_RETURNVERBOSE)
Output:
1/d*(1/3*a^3*A*(2+cos(d*x+c)^2)*sin(d*x+c)+B*a^3*(1/2*cos(d*x+c)*sin(d*x+c )+1/2*d*x+1/2*c)+a^3*C*sin(d*x+c)+3*A*a^2*b*(1/2*cos(d*x+c)*sin(d*x+c)+1/2 *d*x+1/2*c)+3*B*a^2*b*sin(d*x+c)+3*a^2*b*C*(d*x+c)+3*a*A*b^2*sin(d*x+c)+3* B*a*b^2*(d*x+c)+3*C*a*b^2*ln(sec(d*x+c)+tan(d*x+c))+A*b^3*(d*x+c)+B*b^3*ln (sec(d*x+c)+tan(d*x+c))+C*b^3*tan(d*x+c))
Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.03 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (B a^{3} + 3 \, {\left (A + 2 \, C\right )} a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{3} + 6 \, C b^{3} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} + 9 \, B a^{2} b + 9 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
Output:
1/6*(3*(B*a^3 + 3*(A + 2*C)*a^2*b + 6*B*a*b^2 + 2*A*b^3)*d*x*cos(d*x + c) + 3*(3*C*a*b^2 + B*b^3)*cos(d*x + c)*log(sin(d*x + c) + 1) - 3*(3*C*a*b^2 + B*b^3)*cos(d*x + c)*log(-sin(d*x + c) + 1) + (2*A*a^3*cos(d*x + c)^3 + 6 *C*b^3 + 3*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^2 + 2*((2*A + 3*C)*a^3 + 9*B*a ^2*b + 9*A*a*b^2)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c))
Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)** 2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.10 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 36 \, {\left (d x + c\right )} C a^{2} b - 36 \, {\left (d x + c\right )} B a b^{2} - 12 \, {\left (d x + c\right )} A b^{3} - 18 \, C a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, B b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{3} \sin \left (d x + c\right ) - 36 \, B a^{2} b \sin \left (d x + c\right ) - 36 \, A a b^{2} \sin \left (d x + c\right ) - 12 \, C b^{3} \tan \left (d x + c\right )}{12 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
Output:
-1/12*(4*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 - 3*(2*d*x + 2*c + sin(2* d*x + 2*c))*B*a^3 - 9*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^2*b - 36*(d*x + c)*C*a^2*b - 36*(d*x + c)*B*a*b^2 - 12*(d*x + c)*A*b^3 - 18*C*a*b^2*(log( sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) - 6*B*b^3*(log(sin(d*x + c) + 1 ) - log(sin(d*x + c) - 1)) - 12*C*a^3*sin(d*x + c) - 36*B*a^2*b*sin(d*x + c) - 36*A*a*b^2*sin(d*x + c) - 12*C*b^3*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (186) = 372\).
Time = 0.30 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.13 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {\frac {12 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, C a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} {\left (d x + c\right )} - 6 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
Output:
-1/6*(12*C*b^3*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - 3*(B*a^ 3 + 3*A*a^2*b + 6*C*a^2*b + 6*B*a*b^2 + 2*A*b^3)*(d*x + c) - 6*(3*C*a*b^2 + B*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) + 6*(3*C*a*b^2 + B*b^3)*log(ab s(tan(1/2*d*x + 1/2*c) - 1)) - 2*(6*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 3*B*a^3 *tan(1/2*d*x + 1/2*c)^5 + 6*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 9*A*a^2*b*tan(1 /2*d*x + 1/2*c)^5 + 18*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^2*tan(1/2 *d*x + 1/2*c)^5 + 4*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 12*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 36*A*a*b^2*tan(1/2*d*x + 1/ 2*c)^3 + 6*A*a^3*tan(1/2*d*x + 1/2*c) + 3*B*a^3*tan(1/2*d*x + 1/2*c) + 6*C *a^3*tan(1/2*d*x + 1/2*c) + 9*A*a^2*b*tan(1/2*d*x + 1/2*c) + 18*B*a^2*b*ta n(1/2*d*x + 1/2*c) + 18*A*a*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c )^2 + 1)^3)/d
Time = 15.61 (sec) , antiderivative size = 2470, normalized size of antiderivative = 12.60 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
int(cos(c + d*x)^3*(a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
Output:
(tan(c/2 + (d*x)/2)*(2*A*a^3 + B*a^3 + 2*C*a^3 + 2*C*b^3 + 6*A*a*b^2 + 3*A *a^2*b + 6*B*a^2*b) - tan(c/2 + (d*x)/2)^7*(2*A*a^3 - B*a^3 + 2*C*a^3 - 2* C*b^3 + 6*A*a*b^2 - 3*A*a^2*b + 6*B*a^2*b) + tan(c/2 + (d*x)/2)^3*(2*C*a^3 - B*a^3 - (2*A*a^3)/3 + 6*C*b^3 + 6*A*a*b^2 - 3*A*a^2*b + 6*B*a^2*b) - ta n(c/2 + (d*x)/2)^5*(B*a^3 - (2*A*a^3)/3 + 2*C*a^3 - 6*C*b^3 + 6*A*a*b^2 + 3*A*a^2*b + 6*B*a^2*b))/(d*(2*tan(c/2 + (d*x)/2)^2 - 2*tan(c/2 + (d*x)/2)^ 6 - tan(c/2 + (d*x)/2)^8 + 1)) - (atan((((B*b^3 + 3*C*a*b^2)*(32*A*b^3 + 1 6*B*a^3 + 32*B*b^3 + 48*A*a^2*b + 96*B*a*b^2 + 96*C*a*b^2 + 96*C*a^2*b) + tan(c/2 + (d*x)/2)*(32*A^2*b^6 + 8*B^2*a^6 + 32*B^2*b^6 + 96*A^2*a^2*b^4 + 72*A^2*a^4*b^2 + 288*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 288 *C^2*a^4*b^2 + 192*A*B*a*b^5 + 48*A*B*a^5*b + 192*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 192*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 576*B*C*a^3*b^3)) *(B*b^3 + 3*C*a*b^2)*1i - ((B*b^3 + 3*C*a*b^2)*(32*A*b^3 + 16*B*a^3 + 32*B *b^3 + 48*A*a^2*b + 96*B*a*b^2 + 96*C*a*b^2 + 96*C*a^2*b) - tan(c/2 + (d*x )/2)*(32*A^2*b^6 + 8*B^2*a^6 + 32*B^2*b^6 + 96*A^2*a^2*b^4 + 72*A^2*a^4*b^ 2 + 288*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 288*C^2*a^4*b^2 + 192*A*B*a*b^5 + 48*A*B*a^5*b + 192*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3 *b^3 + 192*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 576*B*C*a^3*b^3))*(B*b^3 + 3*C* a*b^2)*1i)/(((B*b^3 + 3*C*a*b^2)*(32*A*b^3 + 16*B*a^3 + 32*B*b^3 + 48*A*a^ 2*b + 96*B*a*b^2 + 96*C*a*b^2 + 96*C*a^2*b) + tan(c/2 + (d*x)/2)*(32*A^...
Time = 0.15 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.56 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {-9 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a \,b^{2} c -3 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{4}+9 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a \,b^{2} c +3 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{4}-\cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{4}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} c +18 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{2}+6 \cos \left (d x +c \right ) a^{3} b c +6 \cos \left (d x +c \right ) a^{3} b d x +9 \cos \left (d x +c \right ) a^{2} b \,c^{2}+9 \cos \left (d x +c \right ) a^{2} b c d x +12 \cos \left (d x +c \right ) a \,b^{3} c +12 \cos \left (d x +c \right ) a \,b^{3} d x -6 \sin \left (d x +c \right )^{3} a^{3} b +6 \sin \left (d x +c \right ) a^{3} b +3 \sin \left (d x +c \right ) b^{3} c}{3 \cos \left (d x +c \right ) d} \] Input:
int(cos(d*x+c)^3*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
( - 9*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a*b**2*c - 3*cos(c + d*x)*log (tan((c + d*x)/2) - 1)*b**4 + 9*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a*b **2*c + 3*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*b**4 - cos(c + d*x)*sin(c + d*x)**3*a**4 + 3*cos(c + d*x)*sin(c + d*x)*a**4 + 3*cos(c + d*x)*sin(c + d*x)*a**3*c + 18*cos(c + d*x)*sin(c + d*x)*a**2*b**2 + 6*cos(c + d*x)*a* *3*b*c + 6*cos(c + d*x)*a**3*b*d*x + 9*cos(c + d*x)*a**2*b*c**2 + 9*cos(c + d*x)*a**2*b*c*d*x + 12*cos(c + d*x)*a*b**3*c + 12*cos(c + d*x)*a*b**3*d* x - 6*sin(c + d*x)**3*a**3*b + 6*sin(c + d*x)*a**3*b + 3*sin(c + d*x)*b**3 *c)/(3*cos(c + d*x)*d)