\(\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx\) [31]

Optimal result

Integrand size = 29, antiderivative size = 151 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {1}{8} a^4 (48 A+35 B) x+\frac {a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d} \] Output:

1/8*a^4*(48*A+35*B)*x+a^4*A*arctanh(sin(d*x+c))/d+5/8*a^4*(8*A+7*B)*sin(d* 
x+c)/d+1/4*a*B*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+1/12*(4*A+7*B)*(a^2+a^2*cos 
(d*x+c))^2*sin(d*x+c)/d+1/24*(32*A+35*B)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^4 \left (576 A d x+420 B d x-96 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 (27 A+28 B) \sin (c+d x)+24 (4 A+7 B) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+32 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))\right )}{96 d} \] Input:

Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x],x]
 

Output:

(a^4*(576*A*d*x + 420*B*d*x - 96*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2] 
] + 96*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(27*A + 28*B)*Sin[c 
 + d*x] + 24*(4*A + 7*B)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 32*B*Si 
n[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d)
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3455, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3214, 3042, 4257}

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.

\(\displaystyle \int \sec (c+d x) (a \cos (c+d x)+a)^4 (A+B \cos (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {1}{4} \int (\cos (c+d x) a+a)^3 (4 a A+a (4 A+7 B) \cos (c+d x)) \sec (c+d x)dx+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (4 a A+a (4 A+7 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (\cos (c+d x) a+a)^2 \left (12 A a^2+(32 A+35 B) \cos (c+d x) a^2\right ) \sec (c+d x)dx+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (12 A a^2+(32 A+35 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int 3 (\cos (c+d x) a+a) \left (8 A a^3+5 (8 A+7 B) \cos (c+d x) a^3\right ) \sec (c+d x)dx+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int (\cos (c+d x) a+a) \left (8 A a^3+5 (8 A+7 B) \cos (c+d x) a^3\right ) \sec (c+d x)dx+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (8 A a^3+5 (8 A+7 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3447

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \left (5 (8 A+7 B) \cos ^2(c+d x) a^4+8 A a^4+\left (8 A a^4+5 (8 A+7 B) a^4\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {5 (8 A+7 B) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+8 A a^4+\left (8 A a^4+5 (8 A+7 B) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\int \left (8 A a^4+(48 A+35 B) \cos (c+d x) a^4\right ) \sec (c+d x)dx+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{d}\right )+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\int \frac {8 A a^4+(48 A+35 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{d}\right )+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3214

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^4 A \int \sec (c+d x)dx+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{d}+a^4 x (48 A+35 B)\right )+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^4 A \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{d}+a^4 x (48 A+35 B)\right )+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\frac {8 a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{d}+a^4 x (48 A+35 B)\right )+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\)

Input:

Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x],x]
 

Output:

(a*B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (((4*A + 7*B)*(a^2 + a^2 
*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (((32*A + 35*B)*(a^4 + a^4*Cos[c + 
d*x])*Sin[c + d*x])/(2*d) + (3*(a^4*(48*A + 35*B)*x + (8*a^4*A*ArcTanh[Sin 
[c + d*x]])/d + (5*a^4*(8*A + 7*B)*Sin[c + d*x])/d))/2)/3)/4
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3214

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d   Int[1/(c + d 
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

rule 3447

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a 
 + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), 
 x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
 

rule 3455

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n 
 + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Sin[e + 
f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 
) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + 
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 
 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1 
] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 25.79 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(-\frac {\left (A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-A -\frac {7 B}{4}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {A}{12}-\frac {B}{3}\right ) \sin \left (3 d x +3 c \right )-\frac {\sin \left (4 d x +4 c \right ) B}{32}+\left (-\frac {27 A}{4}-7 B \right ) \sin \left (d x +c \right )-6 \left (A +\frac {35 B}{48}\right ) x d \right ) a^{4}}{d}\)	\(108\)
parts	\(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(181\)
derivativedivides	\(\frac {\frac {a^{4} A \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \sin \left (d x +c \right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \left (d x +c \right )}{d}\)	\(208\)
default	\(\frac {\frac {a^{4} A \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \sin \left (d x +c \right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \left (d x +c \right )}{d}\)	\(208\)
risch	\(6 a^{4} x A +\frac {35 a^{4} B x}{8}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} a^{4} A}{8 d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{8 d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {B \,a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} A}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{3 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4} A}{d}+\frac {7 \sin \left (2 d x +2 c \right ) B \,a^{4}}{4 d}\)	\(224\)
norman	\(\frac {\left (6 a^{4} A +\frac {35}{8} B \,a^{4}\right ) x +\left (6 a^{4} A +\frac {35}{8} B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (30 a^{4} A +\frac {175}{8} B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (30 a^{4} A +\frac {175}{8} B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (60 a^{4} A +\frac {175}{4} B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (60 a^{4} A +\frac {175}{4} B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {5 a^{4} \left (8 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}+\frac {4 a^{4} \left (59 A +56 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}+\frac {a^{4} \left (272 A +245 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {a^{4} \left (368 A +395 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}+\frac {3 a^{4} \left (31 B +24 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {a^{4} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{4} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\)	\(333\)

Input:

int((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c),x,method=_RETURNVERBOSE 
)
 

Output:

-(A*ln(tan(1/2*d*x+1/2*c)-1)-A*ln(tan(1/2*d*x+1/2*c)+1)+(-A-7/4*B)*sin(2*d 
*x+2*c)+(-1/12*A-1/3*B)*sin(3*d*x+3*c)-1/32*sin(4*d*x+4*c)*B+(-27/4*A-7*B) 
*sin(d*x+c)-6*(A+35/48*B)*x*d)*a^4/d
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.78 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {3 \, {\left (48 \, A + 35 \, B\right )} a^{4} d x + 12 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, B a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \, {\left (16 \, A + 27 \, B\right )} a^{4} \cos \left (d x + c\right ) + 160 \, {\left (A + B\right )} a^{4}\right )} \sin \left (d x + c\right )}{24 \, d} \] Input:

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c),x, algorithm="fri 
cas")
 

Output:

1/24*(3*(48*A + 35*B)*a^4*d*x + 12*A*a^4*log(sin(d*x + c) + 1) - 12*A*a^4* 
log(-sin(d*x + c) + 1) + (6*B*a^4*cos(d*x + c)^3 + 8*(A + 4*B)*a^4*cos(d*x 
 + c)^2 + 3*(16*A + 27*B)*a^4*cos(d*x + c) + 160*(A + B)*a^4)*sin(d*x + c) 
)/d
 

Sympy [F]

\[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=a^{4} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \] Input:

integrate((a+a*cos(d*x+c))**4*(A+B*cos(d*x+c))*sec(d*x+c),x)
 

Output:

a**4*(Integral(A*sec(c + d*x), x) + Integral(4*A*cos(c + d*x)*sec(c + d*x) 
, x) + Integral(6*A*cos(c + d*x)**2*sec(c + d*x), x) + Integral(4*A*cos(c 
+ d*x)**3*sec(c + d*x), x) + Integral(A*cos(c + d*x)**4*sec(c + d*x), x) + 
 Integral(B*cos(c + d*x)*sec(c + d*x), x) + Integral(4*B*cos(c + d*x)**2*s 
ec(c + d*x), x) + Integral(6*B*cos(c + d*x)**3*sec(c + d*x), x) + Integral 
(4*B*cos(c + d*x)**4*sec(c + d*x), x) + Integral(B*cos(c + d*x)**5*sec(c + 
 d*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.31 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 384 \, {\left (d x + c\right )} A a^{4} + 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 96 \, {\left (d x + c\right )} B a^{4} - 96 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 576 \, A a^{4} \sin \left (d x + c\right ) - 384 \, B a^{4} \sin \left (d x + c\right )}{96 \, d} \] Input:

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c),x, algorithm="max 
ima")
 

Output:

-1/96*(32*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 96*(2*d*x + 2*c + sin( 
2*d*x + 2*c))*A*a^4 - 384*(d*x + c)*A*a^4 + 128*(sin(d*x + c)^3 - 3*sin(d* 
x + c))*B*a^4 - 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))* 
B*a^4 - 144*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 96*(d*x + c)*B*a^4 - 
96*A*a^4*log(sec(d*x + c) + tan(d*x + c)) - 576*A*a^4*sin(d*x + c) - 384*B 
*a^4*sin(d*x + c))/d
 

Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.42 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {24 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (48 \, A a^{4} + 35 \, B a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (120 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 424 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 520 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 216 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 279 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \] Input:

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c),x, algorithm="gia 
c")
 

Output:

1/24*(24*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 24*A*a^4*log(abs(tan(1 
/2*d*x + 1/2*c) - 1)) + 3*(48*A*a^4 + 35*B*a^4)*(d*x + c) + 2*(120*A*a^4*t 
an(1/2*d*x + 1/2*c)^7 + 105*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 424*A*a^4*tan(1 
/2*d*x + 1/2*c)^5 + 385*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 520*A*a^4*tan(1/2*d 
*x + 1/2*c)^3 + 511*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 216*A*a^4*tan(1/2*d*x + 
 1/2*c) + 279*B*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/ 
d
 

Mupad [B] (verification not implemented)

Time = 41.86 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.25 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {144\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+24\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+105\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+12\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+A\,a^4\,\sin \left (3\,c+3\,d\,x\right )+21\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )+4\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {3\,B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+81\,A\,a^4\,\sin \left (c+d\,x\right )+84\,B\,a^4\,\sin \left (c+d\,x\right )}{12\,d} \] Input:

int(((A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4)/cos(c + d*x),x)
                                                                                    
                                                                                    
 

Output:

(144*A*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + 24*A*a^4*atanh(si 
n(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + 105*B*a^4*atan(sin(c/2 + (d*x)/2)/c 
os(c/2 + (d*x)/2)) + 12*A*a^4*sin(2*c + 2*d*x) + A*a^4*sin(3*c + 3*d*x) + 
21*B*a^4*sin(2*c + 2*d*x) + 4*B*a^4*sin(3*c + 3*d*x) + (3*B*a^4*sin(4*c + 
4*d*x))/8 + 81*A*a^4*sin(c + d*x) + 84*B*a^4*sin(c + d*x))/(12*d)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.90 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^{4} \left (-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +48 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +87 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a +24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a -8 \sin \left (d x +c \right )^{3} a -32 \sin \left (d x +c \right )^{3} b +168 \sin \left (d x +c \right ) a +192 \sin \left (d x +c \right ) b +144 a d x +105 b d x \right )}{24 d} \] Input:

int((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c),x)
 

Output:

(a**4*( - 6*cos(c + d*x)*sin(c + d*x)**3*b + 48*cos(c + d*x)*sin(c + d*x)* 
a + 87*cos(c + d*x)*sin(c + d*x)*b - 24*log(tan((c + d*x)/2) - 1)*a + 24*l 
og(tan((c + d*x)/2) + 1)*a - 8*sin(c + d*x)**3*a - 32*sin(c + d*x)**3*b + 
168*sin(c + d*x)*a + 192*sin(c + d*x)*b + 144*a*d*x + 105*b*d*x))/(24*d)