Integrand size = 39, antiderivative size = 82 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a (A+B+2 C) x+\frac {a (2 A+3 (B+C)) \sin (c+d x)}{3 d}+\frac {a (A+B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d} \]
1/2*a*(A+B+2*C)*x+1/3*a*(2*A+3*B+3*C)*sin(d*x+c)/d+1/2*a*(A+B)*cos(d*x+c)* sin(d*x+c)/d+1/3*a*A*cos(d*x+c)^2*sin(d*x+c)/d
Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (6 A d x+6 B d x+12 C d x+3 (3 A+4 (B+C)) \sin (c+d x)+3 (A+B) \sin (2 (c+d x))+A \sin (3 (c+d x)))}{12 d} \]
(a*(6*A*d*x + 6*B*d*x + 12*C*d*x + 3*(3*A + 4*(B + C))*Sin[c + d*x] + 3*(A + B)*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(12*d)
Time = 0.57 (sec) , antiderivative size = 84, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a \sec (c+d x)+a) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right ) \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (3 a C \sec ^2(c+d x)+a (2 A+3 (B+C)) \sec (c+d x)+3 a (A+B)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \int \cos ^2(c+d x) \left (3 a C \sec ^2(c+d x)+a (2 A+3 (B+C)) \sec (c+d x)+3 a (A+B)\right )dx+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {3 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+a (2 A+3 (B+C)) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (A+B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{3} \left (a (2 A+3 (B+C)) \int \cos (c+d x)dx+\int \cos ^2(c+d x) \left (3 a C \sec ^2(c+d x)+3 a (A+B)\right )dx\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (a (2 A+3 (B+C)) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int \frac {3 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 a (A+B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {3 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 a (A+B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a (2 A+3 (B+C)) \sin (c+d x)}{d}\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} a (A+B+2 C) \int 1dx+\frac {a (2 A+3 (B+C)) \sin (c+d x)}{d}+\frac {3 a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{3} \left (\frac {a (2 A+3 (B+C)) \sin (c+d x)}{d}+\frac {3 a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3}{2} a x (A+B+2 C)\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}\) |
(a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((3*a*(A + B + 2*C)*x)/2 + (a*(2 *A + 3*(B + C))*Sin[c + d*x])/d + (3*a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/ (2*d))/3
3.5.14.3.1 Defintions of rubi rules used
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 = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74
| method | result | size |
| parallelrisch | \(\frac {a \left (3 \left (A +B \right ) \sin \left (2 d x +2 c \right )+A \sin \left (3 d x +3 c \right )+3 \left (3 A +4 B +4 C \right ) \sin \left (d x +c \right )+6 \left (A +B +2 C \right ) x d \right )}{12 d}\) | \(61\) |
| risch | \(\frac {a A x}{2}+\frac {a B x}{2}+a x C +\frac {3 a A \sin \left (d x +c \right )}{4 d}+\frac {a B \sin \left (d x +c \right )}{d}+\frac {\sin \left (d x +c \right ) C a}{d}+\frac {a A \sin \left (3 d x +3 c \right )}{12 d}+\frac {a A \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a B}{4 d}\) | \(101\) |
| derivativedivides | \(\frac {\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C a \left (d x +c \right )}{d}\) | \(102\) |
| default | \(\frac {\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C a \left (d x +c \right )}{d}\) | \(102\) |
| norman | \(\frac {\frac {a \left (A +B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {a \left (3 A +3 B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (A +B +2 C \right ) x}{2}-\frac {2 a \left (A -3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {2 a \left (7 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {4 a \left (A -3 B -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}+\frac {a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(265\) |
1/12*a*(3*(A+B)*sin(2*d*x+2*c)+A*sin(3*d*x+3*c)+3*(3*A+4*B+4*C)*sin(d*x+c) +6*(A+B+2*C)*x*d)/d
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (A + B + 2 \, C\right )} a d x + {\left (2 \, A a \cos \left (d x + c\right )^{2} + 3 \, {\left (A + B\right )} a \cos \left (d x + c\right ) + 2 \, {\left (2 \, A + 3 \, B + 3 \, C\right )} a\right )} \sin \left (d x + c\right )}{6 \, d} \]
integrate(cos(d*x+c)^3*(a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
1/6*(3*(A + B + 2*C)*a*d*x + (2*A*a*cos(d*x + c)^2 + 3*(A + B)*a*cos(d*x + c) + 2*(2*A + 3*B + 3*C)*a)*sin(d*x + c))/d
\[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a \left (\int A \cos ^{3}{\left (c + d x \right )}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
a*(Integral(A*cos(c + d*x)**3, x) + Integral(A*cos(c + d*x)**3*sec(c + d*x ), x) + Integral(B*cos(c + d*x)**3*sec(c + d*x), x) + Integral(B*cos(c + d *x)**3*sec(c + d*x)**2, x) + Integral(C*cos(c + d*x)**3*sec(c + d*x)**2, x ) + Integral(C*cos(c + d*x)**3*sec(c + d*x)**3, x))
Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.20 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 12 \, {\left (d x + c\right )} C a - 12 \, B a \sin \left (d x + c\right ) - 12 \, C a \sin \left (d x + c\right )}{12 \, d} \]
integrate(cos(d*x+c)^3*(a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
-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))*A*a - 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a - 12*(d*x + c)*C*a - 12*B*a*sin(d*x + c) - 12*C*a*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (75) = 150\).
Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.09 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (A a + B a + 2 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
1/6*(3*(A*a + B*a + 2*C*a)*(d*x + c) + 2*(3*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 + 4*A*a*tan(1/2 *d*x + 1/2*c)^3 + 12*B*a*tan(1/2*d*x + 1/2*c)^3 + 12*C*a*tan(1/2*d*x + 1/2 *c)^3 + 9*A*a*tan(1/2*d*x + 1/2*c) + 9*B*a*tan(1/2*d*x + 1/2*c) + 6*C*a*ta n(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 16.68 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.22 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {A\,a\,x}{2}+\frac {B\,a\,x}{2}+C\,a\,x+\frac {3\,A\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]