Integrand size = 39, antiderivative size = 92 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} (A b+a B+2 b C) x+\frac {(2 a A+3 b B+3 a C) \sin (c+d x)}{3 d}+\frac {(A b+a B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d} \] Output:
1/2*(A*b+B*a+2*C*b)*x+1/3*(2*A*a+3*B*b+3*C*a)*sin(d*x+c)/d+1/2*(A*b+B*a)*c os(d*x+c)*sin(d*x+c)/d+1/3*a*A*cos(d*x+c)^2*sin(d*x+c)/d
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 A b c+6 a B c+6 A b d x+6 a B d x+12 b C d x+3 (3 a A+4 b B+4 a C) \sin (c+d x)+3 (A b+a B) \sin (2 (c+d x))+a A \sin (3 (c+d x))}{12 d} \] Input:
Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[ c + d*x]^2),x]
Output:
(6*A*b*c + 6*a*B*c + 6*A*b*d*x + 6*a*B*d*x + 12*b*C*d*x + 3*(3*a*A + 4*b*B + 4*a*C)*Sin[c + d*x] + 3*(A*b + a*B)*Sin[2*(c + d*x)] + a*A*Sin[3*(c + d *x)])/(12*d)
Time = 0.59 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 4562, 25, 3042, 4535, 3042, 3117, 4533, 24}
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.
| \(\displaystyle \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (3 b C \sec ^2(c+d x)+(2 a A+3 b B+3 a C) \sec (c+d x)+3 (A b+a B)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \int \cos ^2(c+d x) \left (3 b C \sec ^2(c+d x)+(2 a A+3 b B+3 a C) \sec (c+d x)+3 (A b+a B)\right )dx+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {3 b C \csc \left (c+d x+\frac {\pi }{2}\right )^2+(2 a A+3 b B+3 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 (A b+a B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{3} \left ((2 a A+3 a C+3 b B) \int \cos (c+d x)dx+\int \cos ^2(c+d x) \left (3 b C \sec ^2(c+d x)+3 (A b+a B)\right )dx\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left ((2 a A+3 a C+3 b B) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int \frac {3 b C \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 (A b+a B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {3 b C \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 (A b+a B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {\sin (c+d x) (2 a A+3 a C+3 b B)}{d}\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} (a B+A b+2 b C) \int 1dx+\frac {\sin (c+d x) (2 a A+3 a C+3 b B)}{d}+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{3} \left (\frac {\sin (c+d x) (2 a A+3 a C+3 b B)}{d}+\frac {3 (a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3}{2} x (a B+A b+2 b C)\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
Input:
Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d* x]^2),x]
Output:
(a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((3*(A*b + a*B + 2*b*C)*x)/2 + ( (2*a*A + 3*b*B + 3*a*C)*Sin[c + d*x])/d + (3*(A*b + a*B)*Cos[c + d*x]*Sin[ c + d*x])/(2*d))/3
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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]
Time = 0.73 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.80
| method | result | size |
| parallelrisch | \(\frac {3 \left (A b +B a \right ) \sin \left (2 d x +2 c \right )+a A \sin \left (3 d x +3 c \right )+3 \left (a \left (3 A +4 C \right )+4 B b \right ) \sin \left (d x +c \right )+6 x d \left (B a +b \left (A +2 C \right )\right )}{12 d}\) | \(74\) |
| risch | \(\frac {A b x}{2}+\frac {a B x}{2}+x C b +\frac {3 a A \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) B b}{d}+\frac {\sin \left (d x +c \right ) C a}{d}+\frac {a A \sin \left (3 d x +3 c \right )}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A b}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B a}{4 d}\) | \(101\) |
| derivativedivides | \(\frac {\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B b \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C b \left (d x +c \right )}{d}\) | \(102\) |
| default | \(\frac {\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B b \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C b \left (d x +c \right )}{d}\) | \(102\) |
| norman | \(\frac {\left (\frac {1}{2} A b +\frac {1}{2} B a +C b \right ) x +\left (\frac {1}{2} A b +\frac {1}{2} B a +C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} A b +\frac {1}{2} B a +C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {1}{2} A b +\frac {1}{2} B a +C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-A b -B a -2 C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-A b -B a -2 C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {\left (2 a A -A b -B a +2 B b +2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {\left (2 a A +A b +B a +2 B b +2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 \left (4 a A -3 A b -3 B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {2 \left (4 a A +3 A b +3 B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {4 \left (a A -3 B b -3 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(328\) |
Input:
int(cos(d*x+c)^3*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method =_RETURNVERBOSE)
Output:
1/12*(3*(A*b+B*a)*sin(2*d*x+2*c)+a*A*sin(3*d*x+3*c)+3*(a*(3*A+4*C)+4*B*b)* sin(d*x+c)+6*x*d*(B*a+b*(A+2*C)))/d
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.76 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (B a + {\left (A + 2 \, C\right )} b\right )} d x + {\left (2 \, A a \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, A + 3 \, C\right )} a + 6 \, B b + 3 \, {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
Output:
1/6*(3*(B*a + (A + 2*C)*b)*d*x + (2*A*a*cos(d*x + c)^2 + 2*(2*A + 3*C)*a + 6*B*b + 3*(B*a + A*b)*cos(d*x + c))*sin(d*x + c))/d
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(cos(d*x+c)**3*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)**2), x)
Output:
Integral((a + b*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)*cos (c + d*x)**3, x)
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b - 12 \, {\left (d x + c\right )} C b - 12 \, C a \sin \left (d x + c\right ) - 12 \, B b \sin \left (d x + c\right )}{12 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))*(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*(2*d*x + 2*c + sin(2*d* x + 2*c))*B*a - 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*b - 12*(d*x + c)*C*b - 12*C*a*sin(d*x + c) - 12*B*b*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (84) = 168\).
Time = 0.22 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.47 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (B a + A b + 2 \, C b\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")
Output:
1/6*(3*(B*a + A*b + 2*C*b)*(d*x + c) + 2*(6*A*a*tan(1/2*d*x + 1/2*c)^5 - 3 *B*a*tan(1/2*d*x + 1/2*c)^5 + 6*C*a*tan(1/2*d*x + 1/2*c)^5 - 3*A*b*tan(1/2 *d*x + 1/2*c)^5 + 6*B*b*tan(1/2*d*x + 1/2*c)^5 + 4*A*a*tan(1/2*d*x + 1/2*c )^3 + 12*C*a*tan(1/2*d*x + 1/2*c)^3 + 12*B*b*tan(1/2*d*x + 1/2*c)^3 + 6*A* a*tan(1/2*d*x + 1/2*c) + 3*B*a*tan(1/2*d*x + 1/2*c) + 6*C*a*tan(1/2*d*x + 1/2*c) + 3*A*b*tan(1/2*d*x + 1/2*c) + 6*B*b*tan(1/2*d*x + 1/2*c))/(tan(1/2 *d*x + 1/2*c)^2 + 1)^3)/d
Time = 12.60 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.09 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {A\,b\,x}{2}+\frac {B\,a\,x}{2}+C\,b\,x+\frac {3\,A\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,b\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {A\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \] Input:
int(cos(c + d*x)^3*(a + b/cos(c + d*x))*(A + B/cos(c + d*x) + C/cos(c + d* x)^2),x)
Output:
(A*b*x)/2 + (B*a*x)/2 + C*b*x + (3*A*a*sin(c + d*x))/(4*d) + (B*b*sin(c + d*x))/d + (C*a*sin(c + d*x))/d + (A*a*sin(3*c + 3*d*x))/(12*d) + (A*b*sin( 2*c + 2*d*x))/(4*d) + (B*a*sin(2*c + 2*d*x))/(4*d)
Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.86 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -\sin \left (d x +c \right )^{3} a^{2}+3 \sin \left (d x +c \right ) a^{2}+3 \sin \left (d x +c \right ) a c +3 \sin \left (d x +c \right ) b^{2}+3 a b d x +3 b c d x}{3 d} \] Input:
int(cos(d*x+c)^3*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
(3*cos(c + d*x)*sin(c + d*x)*a*b - sin(c + d*x)**3*a**2 + 3*sin(c + d*x)*a **2 + 3*sin(c + d*x)*a*c + 3*sin(c + d*x)*b**2 + 3*a*b*d*x + 3*b*c*d*x)/(3 *d)