Integrand size = 33, antiderivative size = 156 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {3 (5 A+4 C) x}{8 a}-\frac {(4 A+3 C) \sin (c+d x)}{a d}+\frac {3 (5 A+4 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(5 A+4 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {(4 A+3 C) \sin ^3(c+d x)}{3 a d} \] Output:
3/8*(5*A+4*C)*x/a-(4*A+3*C)*sin(d*x+c)/a/d+3/8*(5*A+4*C)*cos(d*x+c)*sin(d* x+c)/a/d+1/4*(5*A+4*C)*cos(d*x+c)^3*sin(d*x+c)/a/d-(A+C)*cos(d*x+c)^3*sin( d*x+c)/d/(a+a*sec(d*x+c))+1/3*(4*A+3*C)*sin(d*x+c)^3/a/d
Time = 1.57 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (72 (5 A+4 C) d x \cos \left (\frac {d x}{2}\right )+72 (5 A+4 C) d x \cos \left (c+\frac {d x}{2}\right )-552 A \sin \left (\frac {d x}{2}\right )-480 C \sin \left (\frac {d x}{2}\right )-168 A \sin \left (c+\frac {d x}{2}\right )-96 C \sin \left (c+\frac {d x}{2}\right )-120 A \sin \left (c+\frac {3 d x}{2}\right )-72 C \sin \left (c+\frac {3 d x}{2}\right )-120 A \sin \left (2 c+\frac {3 d x}{2}\right )-72 C \sin \left (2 c+\frac {3 d x}{2}\right )+40 A \sin \left (2 c+\frac {5 d x}{2}\right )+24 C \sin \left (2 c+\frac {5 d x}{2}\right )+40 A \sin \left (3 c+\frac {5 d x}{2}\right )+24 C \sin \left (3 c+\frac {5 d x}{2}\right )-5 A \sin \left (3 c+\frac {7 d x}{2}\right )-5 A \sin \left (4 c+\frac {7 d x}{2}\right )+3 A \sin \left (4 c+\frac {9 d x}{2}\right )+3 A \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{192 a d (1+\cos (c+d x))} \] Input:
Integrate[(Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]
Output:
(Cos[(c + d*x)/2]*Sec[c/2]*(72*(5*A + 4*C)*d*x*Cos[(d*x)/2] + 72*(5*A + 4* C)*d*x*Cos[c + (d*x)/2] - 552*A*Sin[(d*x)/2] - 480*C*Sin[(d*x)/2] - 168*A* Sin[c + (d*x)/2] - 96*C*Sin[c + (d*x)/2] - 120*A*Sin[c + (3*d*x)/2] - 72*C *Sin[c + (3*d*x)/2] - 120*A*Sin[2*c + (3*d*x)/2] - 72*C*Sin[2*c + (3*d*x)/ 2] + 40*A*Sin[2*c + (5*d*x)/2] + 24*C*Sin[2*c + (5*d*x)/2] + 40*A*Sin[3*c + (5*d*x)/2] + 24*C*Sin[3*c + (5*d*x)/2] - 5*A*Sin[3*c + (7*d*x)/2] - 5*A* Sin[4*c + (7*d*x)/2] + 3*A*Sin[4*c + (9*d*x)/2] + 3*A*Sin[5*c + (9*d*x)/2] ))/(192*a*d*(1 + Cos[c + d*x]))
Time = 0.69 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4573, 25, 3042, 4274, 3042, 3113, 2009, 3115, 3042, 3115, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )}dx\) |
| \(\Big \downarrow \) 4573 |
| \(\displaystyle -\frac {\int -\cos ^4(c+d x) (a (5 A+4 C)-a (4 A+3 C) \sec (c+d x))dx}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \cos ^4(c+d x) (a (5 A+4 C)-a (4 A+3 C) \sec (c+d x))dx}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (5 A+4 C)-a (4 A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {a (5 A+4 C) \int \cos ^4(c+d x)dx-a (4 A+3 C) \int \cos ^3(c+d x)dx}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (5 A+4 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-a (4 A+3 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {a (4 A+3 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}+a (5 A+4 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a (5 A+4 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {a (4 A+3 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a (5 A+4 C) \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {a (4 A+3 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (5 A+4 C) \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {a (4 A+3 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a (5 A+4 C) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {a (4 A+3 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {a (4 A+3 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}+a (5 A+4 C) \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}\) |
Input:
Int[(Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]
Output:
-(((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))) + ((a*(4 *A + 3*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d + a*(5*A + 4*C)*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d))) /4))/a^2
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-a) *(A + C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp[1/(a*b*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*C sc[e + f*x])^n*Simp[b*C*n + A*b*(2*m + n + 1) - (a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && EqQ[ a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.49 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.58
| method | result | size |
| parallelrisch | \(\frac {\left (\left (38 A +24 C \right ) \cos \left (2 d x +2 c \right )-2 A \cos \left (3 d x +3 c \right )+3 A \cos \left (4 d x +4 c \right )+\left (-82 A -48 C \right ) \cos \left (d x +c \right )-221 A -168 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+180 d \left (A +\frac {4 C}{5}\right ) x}{96 d a}\) | \(91\) |
| derivativedivides | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {25 A}{8}-\frac {3 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+2 \left (-\frac {115 A}{24}-\frac {7 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2 \left (-\frac {109 A}{24}-\frac {5 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (-\frac {7 A}{8}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (5 A +4 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(144\) |
| default | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {25 A}{8}-\frac {3 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+2 \left (-\frac {115 A}{24}-\frac {7 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2 \left (-\frac {109 A}{24}-\frac {5 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (-\frac {7 A}{8}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (5 A +4 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(144\) |
| risch | \(\frac {15 A x}{8 a}+\frac {3 x C}{2 a}+\frac {7 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a d}-\frac {7 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a d}-\frac {2 i A}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {2 i C}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {A \sin \left (4 d x +4 c \right )}{32 a d}-\frac {A \sin \left (3 d x +3 c \right )}{12 a d}+\frac {A \sin \left (2 d x +2 c \right )}{2 a d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 a d}\) | \(210\) |
| norman | \(\frac {-\frac {3 \left (5 A +4 C \right ) x}{8 a}-\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{a d}-\frac {9 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {3 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 a}+\frac {3 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a}+\frac {9 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a}+\frac {3 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a}+\frac {\left (5 A +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 a d}-\frac {2 \left (8 A +9 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}+\frac {\left (11 A +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (31 A +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}-\frac {\left (37 A +24 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(316\) |
Input:
int(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x,method=_RETURNVERBO SE)
Output:
1/96*(((38*A+24*C)*cos(2*d*x+2*c)-2*A*cos(3*d*x+3*c)+3*A*cos(4*d*x+4*c)+(- 82*A-48*C)*cos(d*x+c)-221*A-168*C)*tan(1/2*d*x+1/2*c)+180*d*(A+4/5*C)*x)/d /a
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {9 \, {\left (5 \, A + 4 \, C\right )} d x \cos \left (d x + c\right ) + 9 \, {\left (5 \, A + 4 \, C\right )} d x + {\left (6 \, A \cos \left (d x + c\right )^{4} - 2 \, A \cos \left (d x + c\right )^{3} + {\left (13 \, A + 12 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (19 \, A + 12 \, C\right )} \cos \left (d x + c\right ) - 64 \, A - 48 \, C\right )} \sin \left (d x + c\right )}{24 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \] Input:
integrate(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x, algorithm="f ricas")
Output:
1/24*(9*(5*A + 4*C)*d*x*cos(d*x + c) + 9*(5*A + 4*C)*d*x + (6*A*cos(d*x + c)^4 - 2*A*cos(d*x + c)^3 + (13*A + 12*C)*cos(d*x + c)^2 - (19*A + 12*C)*c os(d*x + c) - 64*A - 48*C)*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)
\[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {A \cos ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(cos(d*x+c)**4*(A+C*sec(d*x+c)**2)/(a+a*sec(d*x+c)),x)
Output:
(Integral(A*cos(c + d*x)**4/(sec(c + d*x) + 1), x) + Integral(C*cos(c + d* x)**4*sec(c + d*x)**2/(sec(c + d*x) + 1), x))/a
Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (148) = 296\).
Time = 0.12 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.25 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=-\frac {A {\left (\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {12 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 12 \, C {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{12 \, d} \] Input:
integrate(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x, algorithm="m axima")
Output:
-1/12*(A*((21*sin(d*x + c)/(cos(d*x + c) + 1) + 109*sin(d*x + c)^3/(cos(d* x + c) + 1)^3 + 115*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 75*sin(d*x + c)^ 7/(cos(d*x + c) + 1)^7)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a *sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) - 45*arctan(sin(d*x + c)/(co s(d*x + c) + 1))/a + 12*sin(d*x + c)/(a*(cos(d*x + c) + 1))) + 12*C*((sin( d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a + 2*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + sin(d*x + c)/(a*(co s(d*x + c) + 1))))/d
Time = 0.31 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\frac {9 \, {\left (d x + c\right )} {\left (5 \, A + 4 \, C\right )}}{a} - \frac {24 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} - \frac {2 \, {\left (75 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 115 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 84 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 109 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \] Input:
integrate(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x, algorithm="g iac")
Output:
1/24*(9*(d*x + c)*(5*A + 4*C)/a - 24*(A*tan(1/2*d*x + 1/2*c) + C*tan(1/2*d *x + 1/2*c))/a - 2*(75*A*tan(1/2*d*x + 1/2*c)^7 + 36*C*tan(1/2*d*x + 1/2*c )^7 + 115*A*tan(1/2*d*x + 1/2*c)^5 + 84*C*tan(1/2*d*x + 1/2*c)^5 + 109*A*t an(1/2*d*x + 1/2*c)^3 + 60*C*tan(1/2*d*x + 1/2*c)^3 + 21*A*tan(1/2*d*x + 1 /2*c) + 12*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a))/d
Time = 13.04 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {15\,A\,x}{8\,a}+\frac {3\,C\,x}{2\,a}-\frac {7\,A\,\sin \left (c+d\,x\right )}{4\,a\,d}-\frac {C\,\sin \left (c+d\,x\right )}{a\,d}+\frac {A\,\sin \left (2\,c+2\,d\,x\right )}{2\,a\,d}-\frac {A\,\sin \left (3\,c+3\,d\,x\right )}{12\,a\,d}+\frac {A\,\sin \left (4\,c+4\,d\,x\right )}{32\,a\,d}-\frac {A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d}+\frac {C\,\sin \left (2\,c+2\,d\,x\right )}{4\,a\,d}-\frac {C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \] Input:
int((cos(c + d*x)^4*(A + C/cos(c + d*x)^2))/(a + a/cos(c + d*x)),x)
Output:
(15*A*x)/(8*a) + (3*C*x)/(2*a) - (7*A*sin(c + d*x))/(4*a*d) - (C*sin(c + d *x))/(a*d) + (A*sin(2*c + 2*d*x))/(2*a*d) - (A*sin(3*c + 3*d*x))/(12*a*d) + (A*sin(4*c + 4*d*x))/(32*a*d) - (A*tan(c/2 + (d*x)/2))/(a*d) + (C*sin(2* c + 2*d*x))/(4*a*d) - (C*tan(c/2 + (d*x)/2))/(a*d)
Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a +27 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a +12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} c +24 \cos \left (d x +c \right ) a +24 \cos \left (d x +c \right ) c +8 \sin \left (d x +c \right )^{4} a -48 \sin \left (d x +c \right )^{2} a -24 \sin \left (d x +c \right )^{2} c +45 \sin \left (d x +c \right ) a d x +36 \sin \left (d x +c \right ) c d x -24 a -24 c}{24 \sin \left (d x +c \right ) a d} \] Input:
int(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x)
Output:
( - 6*cos(c + d*x)*sin(c + d*x)**4*a + 27*cos(c + d*x)*sin(c + d*x)**2*a + 12*cos(c + d*x)*sin(c + d*x)**2*c + 24*cos(c + d*x)*a + 24*cos(c + d*x)*c + 8*sin(c + d*x)**4*a - 48*sin(c + d*x)**2*a - 24*sin(c + d*x)**2*c + 45* sin(c + d*x)*a*d*x + 36*sin(c + d*x)*c*d*x - 24*a - 24*c)/(24*sin(c + d*x) *a*d)