Integrand size = 41, antiderivative size = 223 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) x+\frac {b^3 C \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \sin (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(3 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d} \] Output:
1/8*(12*B*a^2*b+8*B*b^3+12*a*b^2*(A+2*C)+a^3*(3*A+4*C))*x+b^3*C*arctanh(si n(d*x+c))/d+1/6*(3*A*b^3+4*B*a^3+16*B*a*b^2+6*a^2*b*(2*A+3*C))*sin(d*x+c)/ d+1/24*a*(6*A*b^2+20*B*a*b+3*a^2*(3*A+4*C))*cos(d*x+c)*sin(d*x+c)/d+1/12*( 3*A*b+4*B*a)*cos(d*x+c)^2*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/4*A*cos(d*x+c) ^3*(a+b*sec(d*x+c))^3*sin(d*x+c)/d
Time = 2.16 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.96 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) (c+d x)-96 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \left (4 A b^3+3 a^3 B+12 a b^2 B+3 a^2 b (3 A+4 C)\right ) \sin (c+d x)+24 a \left (3 A b^2+3 a b B+a^2 (A+C)\right ) \sin (2 (c+d x))+8 a^2 (3 A b+a B) \sin (3 (c+d x))+3 a^3 A \sin (4 (c+d x))}{96 d} \] Input:
Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(12*(12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*(c + d*x ) - 96*b^3*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*b^3*C*Log[Cos[( c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(4*A*b^3 + 3*a^3*B + 12*a*b^2*B + 3*a ^2*b*(3*A + 4*C))*Sin[c + d*x] + 24*a*(3*A*b^2 + 3*a*b*B + a^2*(A + C))*Si n[2*(c + d*x)] + 8*a^2*(3*A*b + a*B)*Sin[3*(c + d*x)] + 3*a^3*A*Sin[4*(c + d*x)])/(96*d)
Time = 1.55 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.341, Rules used = {3042, 4582, 3042, 4582, 3042, 4562, 25, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left (4 b C \sec ^2(c+d x)+(3 a A+4 b B+4 a C) \sec (c+d x)+3 A b+4 a B\right )dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (4 b C \csc \left (c+d x+\frac {\pi }{2}\right )^2+(3 a A+4 b B+4 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 A b+4 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (3 (3 A+4 C) a^2+20 b B a+6 A b^2+12 b^2 C \sec ^2(c+d x)+\left (8 B a^2+15 A b a+24 b C a+12 b^2 B\right ) \sec (c+d x)\right )dx+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (3 (3 A+4 C) a^2+20 b B a+6 A b^2+12 b^2 C \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (8 B a^2+15 A b a+24 b C a+12 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}-\frac {1}{2} \int -\cos (c+d x) \left (24 C \sec ^2(c+d x) b^3+4 \left (4 B a^3+6 b (2 A+3 C) a^2+16 b^2 B a+3 A b^3\right )+3 \left ((3 A+4 C) a^3+12 b B a^2+12 b^2 (A+2 C) a+8 b^3 B\right ) \sec (c+d x)\right )dx\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \cos (c+d x) \left (24 C \sec ^2(c+d x) b^3+4 \left (4 B a^3+6 b (2 A+3 C) a^2+16 b^2 B a+3 A b^3\right )+3 \left ((3 A+4 C) a^3+12 b B a^2+12 b^2 (A+2 C) a+8 b^3 B\right ) \sec (c+d x)\right )dx+\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {24 C \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^3+4 \left (4 B a^3+6 b (2 A+3 C) a^2+16 b^2 B a+3 A b^3\right )+3 \left ((3 A+4 C) a^3+12 b B a^2+12 b^2 (A+2 C) a+8 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (24 C \sec ^2(c+d x) b^3+4 \left (4 B a^3+6 b (2 A+3 C) a^2+16 b^2 B a+3 A b^3\right )\right )dx+3 \left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right ) \int 1dx\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (24 C \sec ^2(c+d x) b^3+4 \left (4 B a^3+6 b (2 A+3 C) a^2+16 b^2 B a+3 A b^3\right )\right )dx+3 x \left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right )\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {24 C \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^3+4 \left (4 B a^3+6 b (2 A+3 C) a^2+16 b^2 B a+3 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+3 x \left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right )\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (24 b^3 C \int \sec (c+d x)dx+\frac {4 \sin (c+d x) \left (4 a^3 B+6 a^2 b (2 A+3 C)+16 a b^2 B+3 A b^3\right )}{d}+3 x \left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right )\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (24 b^3 C \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {4 \sin (c+d x) \left (4 a^3 B+6 a^2 b (2 A+3 C)+16 a b^2 B+3 A b^3\right )}{d}+3 x \left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right )\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{2 d}+\frac {1}{2} \left (\frac {4 \sin (c+d x) \left (4 a^3 B+6 a^2 b (2 A+3 C)+16 a b^2 B+3 A b^3\right )}{d}+3 x \left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right )+\frac {24 b^3 C \text {arctanh}(\sin (c+d x))}{d}\right )\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
Input:
Int[Cos[c + d*x]^4*(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]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) + (((3*A*b + 4*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + ((a*(6* A*b^2 + 20*a*b*B + 3*a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(2*d) + ( 3*(12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*x + (24*b^ 3*C*ArcTanh[Sin[c + d*x]])/d + (4*(3*A*b^3 + 4*a^3*B + 16*a*b^2*B + 6*a^2* b*(2*A + 3*C))*Sin[c + d*x])/d)/2)/3)/4
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[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si mp[1/(d*n) Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, A, B, C}, 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 = 1.43 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.85
| method | result | size |
| parallelrisch | \(\frac {-96 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{3}+96 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{3}+24 a \left (a^{2} \left (A +C \right )+3 B a b +3 A \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (24 A \,a^{2} b +8 B \,a^{3}\right ) \sin \left (3 d x +3 c \right )+3 a^{3} A \sin \left (4 d x +4 c \right )+\left (72 B \,a^{3}+216 b \left (A +\frac {4 C}{3}\right ) a^{2}+288 B a \,b^{2}+96 A \,b^{3}\right ) \sin \left (d x +c \right )+36 x \left (\left (A +\frac {4 C}{3}\right ) a^{3}+4 B \,a^{2} b +4 a \,b^{2} \left (A +2 C \right )+\frac {8 B \,b^{3}}{3}\right ) d}{96 d}\) | \(189\) |
| derivativedivides | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{2} b \sin \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B a \,b^{2} \sin \left (d x +c \right )+3 C a \,b^{2} \left (d x +c \right )+A \,b^{3} \sin \left (d x +c \right )+B \,b^{3} \left (d x +c \right )+C \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(251\) |
| default | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{2} b \sin \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B a \,b^{2} \sin \left (d x +c \right )+3 C a \,b^{2} \left (d x +c \right )+A \,b^{3} \sin \left (d x +c \right )+B \,b^{3} \left (d x +c \right )+C \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(251\) |
| risch | \(\frac {3 a^{3} A x}{8}+\frac {a^{3} x C}{2}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{2} b}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a A \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b}{4 d}+\frac {a^{3} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{12 d}+\frac {3 A a \,b^{2} x}{2}-\frac {3 i B \,a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3}}{8 d}+3 x C a \,b^{2}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {9 i A \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {9 i A \,a^{2} b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 B \,a^{2} b x}{2}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B a \,b^{2}}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} b C}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B a \,b^{2}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{2} b C}{2 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{3}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{3}}{2 d}+x B \,b^{3}\) | \(414\) |
| norman | \(\frac {\left (\frac {3}{8} a^{3} A +\frac {3}{2} a A \,b^{2}+\frac {3}{2} B \,a^{2} b +B \,b^{3}+\frac {1}{2} a^{3} C +3 C a \,b^{2}\right ) x +\left (-\frac {3}{2} a^{3} A -6 a A \,b^{2}-6 B \,a^{2} b -4 B \,b^{3}-2 a^{3} C -12 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {3}{2} a^{3} A -6 a A \,b^{2}-6 B \,a^{2} b -4 B \,b^{3}-2 a^{3} C -12 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {3}{8} a^{3} A +\frac {3}{2} a A \,b^{2}+\frac {3}{2} B \,a^{2} b +B \,b^{3}+\frac {1}{2} a^{3} C +3 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (\frac {9}{4} a^{3} A +9 a A \,b^{2}+9 B \,a^{2} b +6 B \,b^{3}+3 a^{3} C +18 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {\left (5 a^{3} A -24 A \,a^{2} b +12 a A \,b^{2}-8 A \,b^{3}-8 B \,a^{3}+12 B \,a^{2} b -24 B a \,b^{2}+4 a^{3} C -24 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}+\frac {\left (5 a^{3} A +24 A \,a^{2} b +12 a A \,b^{2}+8 A \,b^{3}+8 B \,a^{3}+12 B \,a^{2} b +24 B a \,b^{2}+4 a^{3} C +24 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (45 a^{3} A -24 A \,a^{2} b +12 a A \,b^{2}+24 A \,b^{3}-8 B \,a^{3}+12 B \,a^{2} b +72 B a \,b^{2}+4 a^{3} C +72 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}+\frac {\left (45 a^{3} A +24 A \,a^{2} b +12 a A \,b^{2}-24 A \,b^{3}+8 B \,a^{3}+12 B \,a^{2} b -72 B a \,b^{2}+4 a^{3} C -72 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}+\frac {\left (69 a^{3} A -168 A \,a^{2} b +108 a A \,b^{2}-24 A \,b^{3}-56 B \,a^{3}+108 B \,a^{2} b -72 B a \,b^{2}+36 a^{3} C -72 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 d}-\frac {\left (69 a^{3} A +168 A \,a^{2} b +108 a A \,b^{2}+24 A \,b^{3}+56 B \,a^{3}+108 B \,a^{2} b +72 B a \,b^{2}+36 a^{3} C +72 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}-\frac {\left (165 a^{3} A -24 A \,a^{2} b -180 a A \,b^{2}-72 A \,b^{3}-8 B \,a^{3}-180 B \,a^{2} b -216 B a \,b^{2}-60 a^{3} C -216 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{12 d}+\frac {\left (165 a^{3} A +24 A \,a^{2} b -180 a A \,b^{2}+72 A \,b^{3}+8 B \,a^{3}-180 B \,a^{2} b +216 B a \,b^{2}-60 a^{3} C +216 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}+\frac {C \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {C \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(933\) |
Input:
int(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,meth od=_RETURNVERBOSE)
Output:
1/96*(-96*C*ln(tan(1/2*d*x+1/2*c)-1)*b^3+96*C*ln(tan(1/2*d*x+1/2*c)+1)*b^3 +24*a*(a^2*(A+C)+3*B*a*b+3*A*b^2)*sin(2*d*x+2*c)+(24*A*a^2*b+8*B*a^3)*sin( 3*d*x+3*c)+3*a^3*A*sin(4*d*x+4*c)+(72*B*a^3+216*b*(A+4/3*C)*a^2+288*B*a*b^ 2+96*A*b^3)*sin(d*x+c)+36*x*((A+4/3*C)*a^3+4*B*a^2*b+4*a*b^2*(A+2*C)+8/3*B *b^3)*d)/d
Time = 0.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.85 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \, C b^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, C b^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, {\left (A + 2 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} d x + {\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 16 \, B a^{3} + 24 \, {\left (2 \, A + 3 \, C\right )} a^{2} b + 72 \, B a b^{2} + 24 \, A b^{3} + 8 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \] Input:
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
Output:
1/24*(12*C*b^3*log(sin(d*x + c) + 1) - 12*C*b^3*log(-sin(d*x + c) + 1) + 3 *((3*A + 4*C)*a^3 + 12*B*a^2*b + 12*(A + 2*C)*a*b^2 + 8*B*b^3)*d*x + (6*A* a^3*cos(d*x + c)^3 + 16*B*a^3 + 24*(2*A + 3*C)*a^2*b + 72*B*a*b^2 + 24*A*b ^3 + 8*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^2 + 3*((3*A + 4*C)*a^3 + 12*B*a^2* b + 12*A*a*b^2)*cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**4*(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 = 246, normalized size of antiderivative = 1.10 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 288 \, {\left (d x + c\right )} C a b^{2} + 96 \, {\left (d x + c\right )} B b^{3} + 48 \, C b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 288 \, C a^{2} b \sin \left (d x + c\right ) + 288 \, B a b^{2} \sin \left (d x + c\right ) + 96 \, A b^{3} \sin \left (d x + c\right )}{96 \, d} \] Input:
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
Output:
1/96*(3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 - 32 *(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 + 24*(2*d*x + 2*c + sin(2*d*x + 2 *c))*C*a^3 - 96*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b + 72*(2*d*x + 2* c + sin(2*d*x + 2*c))*B*a^2*b + 72*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a*b^ 2 + 288*(d*x + c)*C*a*b^2 + 96*(d*x + c)*B*b^3 + 48*C*b^3*(log(sin(d*x + c ) + 1) - log(sin(d*x + c) - 1)) + 288*C*a^2*b*sin(d*x + c) + 288*B*a*b^2*s in(d*x + c) + 96*A*b^3*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 723 vs. \(2 (213) = 426\).
Time = 0.30 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.24 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
Output:
1/24*(24*C*b^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 24*C*b^3*log(abs(tan(1 /2*d*x + 1/2*c) - 1)) + 3*(3*A*a^3 + 4*C*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 2 4*C*a*b^2 + 8*B*b^3)*(d*x + c) - 2*(15*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 24*B *a^3*tan(1/2*d*x + 1/2*c)^7 + 12*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 72*A*a^2*b *tan(1/2*d*x + 1/2*c)^7 + 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 72*C*a^2*b*t an(1/2*d*x + 1/2*c)^7 + 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 72*B*a*b^2*tan (1/2*d*x + 1/2*c)^7 - 24*A*b^3*tan(1/2*d*x + 1/2*c)^7 - 9*A*a^3*tan(1/2*d* x + 1/2*c)^5 - 40*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 12*C*a^3*tan(1/2*d*x + 1/ 2*c)^5 - 120*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^2*b*tan(1/2*d*x + 1/2 *c)^5 - 216*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*A*a*b^2*tan(1/2*d*x + 1/2* c)^5 - 216*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 72*A*b^3*tan(1/2*d*x + 1/2*c)^ 5 + 9*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 40*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 12* C*a^3*tan(1/2*d*x + 1/2*c)^3 - 120*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*B*a ^2*b*tan(1/2*d*x + 1/2*c)^3 - 216*C*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 - 216*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 72*A*b^3 *tan(1/2*d*x + 1/2*c)^3 - 15*A*a^3*tan(1/2*d*x + 1/2*c) - 24*B*a^3*tan(1/2 *d*x + 1/2*c) - 12*C*a^3*tan(1/2*d*x + 1/2*c) - 72*A*a^2*b*tan(1/2*d*x + 1 /2*c) - 36*B*a^2*b*tan(1/2*d*x + 1/2*c) - 72*C*a^2*b*tan(1/2*d*x + 1/2*c) - 36*A*a*b^2*tan(1/2*d*x + 1/2*c) - 72*B*a*b^2*tan(1/2*d*x + 1/2*c) - 24*A *b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 15.87 (sec) , antiderivative size = 3250, normalized size of antiderivative = 14.57 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
int(cos(c + d*x)^4*(a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
Output:
(atan(((((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2* b*3i)/2 + C*a*b^2*3i)*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a* b^2 + 48*B*a^2*b + 96*C*a*b^2) + tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^ 2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2* a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3* b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b*3i)/2 + C*a*b^2*3i)* 1i - (((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b* 3i)/2 + C*a*b^2*3i)*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^ 2 + 48*B*a^2*b + 96*C*a*b^2) - tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2* b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^ 2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 9 6*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^ 3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b*3i)/2 + C*a*b^2*3i)*1i )/((((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b*3i )/2 + C*a*b^2*3i)*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2) + tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^ 6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a...
Time = 0.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.10 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{4}+15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4}+12 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} c +72 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{2}-24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{3} c +24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{3} c -32 \sin \left (d x +c \right )^{3} a^{3} b +96 \sin \left (d x +c \right ) a^{3} b +72 \sin \left (d x +c \right ) a^{2} b c +96 \sin \left (d x +c \right ) a \,b^{3}+9 a^{4} c +9 a^{4} d x +12 a^{3} c^{2}+12 a^{3} c d x +72 a^{2} b^{2} c +72 a^{2} b^{2} d x +72 a \,b^{2} c^{2}+72 a \,b^{2} c d x +24 b^{4} c +24 b^{4} d x}{24 d} \] Input:
int(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
( - 6*cos(c + d*x)*sin(c + d*x)**3*a**4 + 15*cos(c + d*x)*sin(c + d*x)*a** 4 + 12*cos(c + d*x)*sin(c + d*x)*a**3*c + 72*cos(c + d*x)*sin(c + d*x)*a** 2*b**2 - 24*log(tan((c + d*x)/2) - 1)*b**3*c + 24*log(tan((c + d*x)/2) + 1 )*b**3*c - 32*sin(c + d*x)**3*a**3*b + 96*sin(c + d*x)*a**3*b + 72*sin(c + d*x)*a**2*b*c + 96*sin(c + d*x)*a*b**3 + 9*a**4*c + 9*a**4*d*x + 12*a**3* c**2 + 12*a**3*c*d*x + 72*a**2*b**2*c + 72*a**2*b**2*d*x + 72*a*b**2*c**2 + 72*a*b**2*c*d*x + 24*b**4*c + 24*b**4*d*x)/(24*d)