Integrand size = 31, antiderivative size = 95 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a (3 A+4 C) x+\frac {a (A+C) \sin (c+d x)}{d}+\frac {a (3 A+4 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a A \sin ^3(c+d x)}{3 d} \]
1/8*a*(3*A+4*C)*x+a*(A+C)*sin(d*x+c)/d+1/8*a*(3*A+4*C)*cos(d*x+c)*sin(d*x+ c)/d+1/4*a*A*cos(d*x+c)^3*sin(d*x+c)/d-1/3*a*A*sin(d*x+c)^3/d
Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a (36 A c+48 c C+36 A d x+48 C d x+24 (3 A+4 C) \sin (c+d x)+24 (A+C) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+3 A \sin (4 (c+d x)))}{96 d} \]
(a*(36*A*c + 48*c*C + 36*A*d*x + 48*C*d*x + 24*(3*A + 4*C)*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]) )/(96*d)
Time = 0.64 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {3042, 4563, 25, 3042, 4535, 3042, 3115, 24, 4532, 3042, 3492, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a \sec (c+d x)+a) \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right ) \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4563 |
| \(\displaystyle \frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (4 a C \sec ^2(c+d x)+a (3 A+4 C) \sec (c+d x)+4 a A\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \int \cos ^3(c+d x) \left (4 a C \sec ^2(c+d x)+a (3 A+4 C) \sec (c+d x)+4 a A\right )dx+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {4 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+a (3 A+4 C) \csc \left (c+d x+\frac {\pi }{2}\right )+4 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{4} \left (a (3 A+4 C) \int \cos ^2(c+d x)dx+\int \cos ^3(c+d x) \left (4 a C \sec ^2(c+d x)+4 a A\right )dx\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (a (3 A+4 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\int \frac {4 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {4 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+a (3 A+4 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {4 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+a (3 A+4 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 4532 |
| \(\displaystyle \frac {1}{4} \left (\int \cos (c+d x) \left (4 a A \cos ^2(c+d x)+4 a C\right )dx+a (3 A+4 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (4 a A \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 a C\right )dx+a (3 A+4 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3492 |
| \(\displaystyle \frac {1}{4} \left (a (3 A+4 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\int \left (4 a (A+C)-4 a A \sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (a (3 A+4 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\frac {4}{3} a A \sin ^3(c+d x)-4 a (A+C) \sin (c+d x)}{d}\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
(a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*(3*A + 4*C)*(x/2 + (Cos[c + d *x]*Sin[c + d*x])/(2*d)) - (-4*a*(A + C)*Sin[c + d*x] + (4*a*A*Sin[c + d*x ]^3)/3)/d)/4
3.1.91.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 ), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[ {e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
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_)]^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] + Simp[1/(d*n) Int[(d*Csc[e + f*x] )^(n + 1)*Simp[A*b*n + a*(C*n + 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, C}, x] && LtQ[n, -1]
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71
| method | result | size |
| parallelrisch | \(\frac {a \left (\left (8 A +8 C \right ) \sin \left (2 d x +2 c \right )+\frac {8 A \sin \left (3 d x +3 c \right )}{3}+A \sin \left (4 d x +4 c \right )+12 \left (A +\frac {4 C}{3}\right ) \left (d x +2 \sin \left (d x +c \right )\right )\right )}{32 d}\) | \(67\) |
| derivativedivides | \(\frac {a A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \sin \left (d x +c \right )}{d}\) | \(96\) |
| default | \(\frac {a A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \sin \left (d x +c \right )}{d}\) | \(96\) |
| risch | \(\frac {3 a A x}{8}+\frac {a x C}{2}+\frac {3 a A \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) C a}{d}+\frac {a A \sin \left (4 d x +4 c \right )}{32 d}+\frac {a A \sin \left (3 d x +3 c \right )}{12 d}+\frac {a A \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C a}{4 d}\) | \(101\) |
| norman | \(\frac {\left (\frac {3}{8} a A +\frac {1}{2} C a \right ) x +\left (-\frac {3}{2} a A -2 C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {3}{8} a A -\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {3}{8} a A -\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {3}{4} a A +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {3}{4} a A +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {3}{8} a A +\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\frac {a \left (3 A +4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}+\frac {a \left (13 A -36 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 d}+\frac {a \left (13 A +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a \left (29 A +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {a \left (31 A +36 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}-\frac {a \left (47 A -12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(316\) |
1/32*a*((8*A+8*C)*sin(2*d*x+2*c)+8/3*A*sin(3*d*x+3*c)+A*sin(4*d*x+4*c)+12* (A+4/3*C)*(d*x+2*sin(d*x+c)))/d
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.80 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, A + 4 \, C\right )} a d x + {\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \, A a \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A + 4 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (2 \, A + 3 \, C\right )} a\right )} \sin \left (d x + c\right )}{24 \, d} \]
1/24*(3*(3*A + 4*C)*a*d*x + (6*A*a*cos(d*x + c)^3 + 8*A*a*cos(d*x + c)^2 + 3*(3*A + 4*C)*a*cos(d*x + c) + 8*(2*A + 3*C)*a)*sin(d*x + c))/d
\[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=a \left (\int A \cos ^{4}{\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
a*(Integral(A*cos(c + d*x)**4, x) + Integral(A*cos(c + d*x)**4*sec(c + d*x ), x) + Integral(C*cos(c + d*x)**4*sec(c + d*x)**2, x) + Integral(C*cos(c + d*x)**4*sec(c + d*x)**3, x))
Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 96 \, C a \sin \left (d x + c\right )}{96 \, d} \]
-1/96*(32*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a - 3*(12*d*x + 12*c + sin(4 *d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a - 24*(2*d*x + 2*c + sin(2*d*x + 2*c) )*C*a - 96*C*a*sin(d*x + c))/d
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.64 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, A a + 4 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 49 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 84 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 39 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
1/24*(3*(3*A*a + 4*C*a)*(d*x + c) + 2*(9*A*a*tan(1/2*d*x + 1/2*c)^7 + 12*C *a*tan(1/2*d*x + 1/2*c)^7 + 49*A*a*tan(1/2*d*x + 1/2*c)^5 + 60*C*a*tan(1/2 *d*x + 1/2*c)^5 + 31*A*a*tan(1/2*d*x + 1/2*c)^3 + 84*C*a*tan(1/2*d*x + 1/2 *c)^3 + 39*A*a*tan(1/2*d*x + 1/2*c) + 36*C*a*tan(1/2*d*x + 1/2*c))/(tan(1/ 2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 18.40 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.94 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {3\,A\,a}{4}+C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {49\,A\,a}{12}+5\,C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {31\,A\,a}{12}+7\,C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a}{4}+3\,C\,a\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,A+4\,C\right )}{4\,\left (\frac {3\,A\,a}{4}+C\,a\right )}\right )\,\left (3\,A+4\,C\right )}{4\,d} \]
(tan(c/2 + (d*x)/2)*((13*A*a)/4 + 3*C*a) + tan(c/2 + (d*x)/2)^7*((3*A*a)/4 + C*a) + tan(c/2 + (d*x)/2)^3*((31*A*a)/12 + 7*C*a) + tan(c/2 + (d*x)/2)^ 5*((49*A*a)/12 + 5*C*a))/(d*(4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2) ^4 + 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) + (a*atan((a*tan( c/2 + (d*x)/2)*(3*A + 4*C))/(4*((3*A*a)/4 + C*a)))*(3*A + 4*C))/(4*d)