Integrand size = 21, antiderivative size = 145 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {1}{8} \left (3 a^4+24 a^2 b^2+8 b^4\right ) x+\frac {4 a b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a^3 b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d} \]
1/8*(3*a^4+24*a^2*b^2+8*b^4)*x+4/3*a*b*(2*a^2+3*b^2)*sin(d*x+c)/d+1/8*a^2* (3*a^2+22*b^2)*cos(d*x+c)*sin(d*x+c)/d+5/6*a^3*b*cos(d*x+c)^2*sin(d*x+c)/d +1/4*a^2*cos(d*x+c)^3*(a+b*sec(d*x+c))^2*sin(d*x+c)/d
Time = 0.37 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.72 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {12 \left (3 a^4+24 a^2 b^2+8 b^4\right ) (c+d x)+96 a b \left (3 a^2+4 b^2\right ) \sin (c+d x)+24 a^2 \left (a^2+6 b^2\right ) \sin (2 (c+d x))+32 a^3 b \sin (3 (c+d x))+3 a^4 \sin (4 (c+d x))}{96 d} \]
(12*(3*a^4 + 24*a^2*b^2 + 8*b^4)*(c + d*x) + 96*a*b*(3*a^2 + 4*b^2)*Sin[c + d*x] + 24*a^2*(a^2 + 6*b^2)*Sin[2*(c + d*x)] + 32*a^3*b*Sin[3*(c + d*x)] + 3*a^4*Sin[4*(c + d*x)])/(96*d)
Time = 0.90 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4328, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4328 |
| \(\displaystyle \frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (10 b a^2+3 \left (a^2+4 b^2\right ) \sec (c+d x) a+b \left (a^2+4 b^2\right ) \sec ^2(c+d x)\right )dx+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (10 b a^2+3 \left (a^2+4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+b \left (a^2+4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{4} \left (\frac {10 a^3 b \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (3 \left (3 a^2+22 b^2\right ) a^2+16 b \left (2 a^2+3 b^2\right ) \sec (c+d x) a+3 b^2 \left (a^2+4 b^2\right ) \sec ^2(c+d x)\right )dx\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) \left (3 \left (3 a^2+22 b^2\right ) a^2+16 b \left (2 a^2+3 b^2\right ) \sec (c+d x) a+3 b^2 \left (a^2+4 b^2\right ) \sec ^2(c+d x)\right )dx+\frac {10 a^3 b \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {3 \left (3 a^2+22 b^2\right ) a^2+16 b \left (2 a^2+3 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+3 b^2 \left (a^2+4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {10 a^3 b \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (16 a b \left (2 a^2+3 b^2\right ) \int \cos (c+d x)dx+\int \cos ^2(c+d x) \left (3 \left (3 a^2+22 b^2\right ) a^2+3 b^2 \left (a^2+4 b^2\right ) \sec ^2(c+d x)\right )dx\right )+\frac {10 a^3 b \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (16 a b \left (2 a^2+3 b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int \frac {3 \left (3 a^2+22 b^2\right ) a^2+3 b^2 \left (a^2+4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx\right )+\frac {10 a^3 b \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\int \frac {3 \left (3 a^2+22 b^2\right ) a^2+3 b^2 \left (a^2+4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {16 a b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{d}\right )+\frac {10 a^3 b \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (3 a^4+24 a^2 b^2+8 b^4\right ) \int 1dx+\frac {16 a b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{d}+\frac {3 a^2 \left (3 a^2+22 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {10 a^3 b \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}+\frac {1}{4} \left (\frac {10 a^3 b \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{3} \left (\frac {16 a b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{d}+\frac {3 a^2 \left (3 a^2+22 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3}{2} x \left (3 a^4+24 a^2 b^2+8 b^4\right )\right )\right )\) |
(a^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d) + ((10*a^3* b*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((3*(3*a^4 + 24*a^2*b^2 + 8*b^4)*x) /2 + (16*a*b*(2*a^2 + 3*b^2)*Sin[c + d*x])/d + (3*a^2*(3*a^2 + 22*b^2)*Cos [c + d*x]*Sin[c + d*x])/(2*d))/3)/4
3.5.83.3.1 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_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[a^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)* ((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2* (n + 1))*Csc[e + f*x] - b*(b^2*n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && ((Int egerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))
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 = 0.90 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.70
| method | result | size |
| parallelrisch | \(\frac {24 \left (a^{4}+6 a^{2} b^{2}\right ) \sin \left (2 d x +2 c \right )+32 a^{3} b \sin \left (3 d x +3 c \right )+3 a^{4} \sin \left (4 d x +4 c \right )+96 \left (3 a^{3} b +4 a \,b^{3}\right ) \sin \left (d x +c \right )+36 d x \left (a^{4}+8 a^{2} b^{2}+\frac {8}{3} b^{4}\right )}{96 d}\) | \(101\) |
| derivativedivides | \(\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a \,b^{3} \sin \left (d x +c \right )+b^{4} \left (d x +c \right )}{d}\) | \(116\) |
| default | \(\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a \,b^{3} \sin \left (d x +c \right )+b^{4} \left (d x +c \right )}{d}\) | \(116\) |
| risch | \(\frac {3 a^{4} x}{8}+3 a^{2} b^{2} x +x \,b^{4}+\frac {3 a^{3} b \sin \left (d x +c \right )}{d}+\frac {4 \sin \left (d x +c \right ) a \,b^{3}}{d}+\frac {a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{3} b \sin \left (3 d x +3 c \right )}{3 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b^{2}}{2 d}\) | \(124\) |
| norman | \(\frac {\left (-\frac {3}{8} a^{4}-3 a^{2} b^{2}-b^{4}\right ) x +\left (-\frac {9}{8} a^{4}-9 a^{2} b^{2}-3 b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-\frac {9}{8} a^{4}-9 a^{2} b^{2}-3 b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-\frac {3}{8} a^{4}-3 a^{2} b^{2}-b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {3}{8} a^{4}+3 a^{2} b^{2}+b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {3}{8} a^{4}+3 a^{2} b^{2}+b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {9}{8} a^{4}+9 a^{2} b^{2}+3 b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {9}{8} a^{4}+9 a^{2} b^{2}+3 b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {a^{2} \left (7 a^{2}-24 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {a \left (5 a^{3}-32 a^{2} b +24 a \,b^{2}-32 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}-\frac {a \left (5 a^{3}+32 a^{2} b +24 a \,b^{2}+32 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a \left (81 a^{3}-32 a^{2} b -72 a \,b^{2}-288 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {a \left (81 a^{3}+32 a^{2} b -72 a \,b^{2}+288 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}+\frac {a^{2} \left (27 a^{2}-64 a b +72 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}+\frac {a^{2} \left (27 a^{2}+64 a b +72 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(529\) |
1/96*(24*(a^4+6*a^2*b^2)*sin(2*d*x+2*c)+32*a^3*b*sin(3*d*x+3*c)+3*a^4*sin( 4*d*x+4*c)+96*(3*a^3*b+4*a*b^3)*sin(d*x+c)+36*d*x*(a^4+8*a^2*b^2+8/3*b^4)) /d
Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {3 \, {\left (3 \, a^{4} + 24 \, a^{2} b^{2} + 8 \, b^{4}\right )} d x + {\left (6 \, a^{4} \cos \left (d x + c\right )^{3} + 32 \, a^{3} b \cos \left (d x + c\right )^{2} + 64 \, a^{3} b + 96 \, a b^{3} + 9 \, {\left (a^{4} + 8 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
1/24*(3*(3*a^4 + 24*a^2*b^2 + 8*b^4)*d*x + (6*a^4*cos(d*x + c)^3 + 32*a^3* b*cos(d*x + c)^2 + 64*a^3*b + 96*a*b^3 + 9*(a^4 + 8*a^2*b^2)*cos(d*x + c)) *sin(d*x + c))/d
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{4} \cos ^{4}{\left (c + d x \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.75 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} b + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b^{2} + 96 \, {\left (d x + c\right )} b^{4} + 384 \, a b^{3} \sin \left (d x + c\right )}{96 \, d} \]
1/96*(3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^4 - 128* (sin(d*x + c)^3 - 3*sin(d*x + c))*a^3*b + 144*(2*d*x + 2*c + sin(2*d*x + 2 *c))*a^2*b^2 + 96*(d*x + c)*b^4 + 384*a*b^3*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (135) = 270\).
Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.19 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {3 \, {\left (3 \, a^{4} + 24 \, a^{2} b^{2} + 8 \, b^{4}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
1/24*(3*(3*a^4 + 24*a^2*b^2 + 8*b^4)*(d*x + c) - 2*(15*a^4*tan(1/2*d*x + 1 /2*c)^7 - 96*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 72*a^2*b^2*tan(1/2*d*x + 1/2*c )^7 - 96*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 9*a^4*tan(1/2*d*x + 1/2*c)^5 - 160 *a^3*b*tan(1/2*d*x + 1/2*c)^5 + 72*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 - 288*a* b^3*tan(1/2*d*x + 1/2*c)^5 + 9*a^4*tan(1/2*d*x + 1/2*c)^3 - 160*a^3*b*tan( 1/2*d*x + 1/2*c)^3 - 72*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 288*a*b^3*tan(1/2 *d*x + 1/2*c)^3 - 15*a^4*tan(1/2*d*x + 1/2*c) - 96*a^3*b*tan(1/2*d*x + 1/2 *c) - 72*a^2*b^2*tan(1/2*d*x + 1/2*c) - 96*a*b^3*tan(1/2*d*x + 1/2*c))/(ta n(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 12.87 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.85 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {3\,a^4\,x}{8}+b^4\,x+3\,a^2\,b^2\,x+\frac {a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {a^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {3\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {4\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \]