\(\int \cos (c+d x) (a+b \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [666]

Optimal result

Integrand size = 31, antiderivative size = 229 \[ \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=4 a^3 A b x+\frac {\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]

[Out]

4*a^3*A*b*x+1/8*(8*a^4*C+24*a^2*b^2*(2*A+C)+b^4*(4*A+3*C))*arctanh(sin(d*x+c))/d+A*(a+b*sec(d*x+c))^4*sin(d*x+
c)/d-1/6*a*b*(a^2*(12*A-19*C)-8*b^2*(3*A+2*C))*tan(d*x+c)/d-1/24*b^2*(a^2*(24*A-26*C)-3*b^2*(4*A+3*C))*sec(d*x
+c)*tan(d*x+c)/d-1/12*a*b*(12*A-7*C)*(a+b*sec(d*x+c))^2*tan(d*x+c)/d-1/4*b*(4*A-C)*(a+b*sec(d*x+c))^3*tan(d*x+
c)/d

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4180, 4141, 4133, 3855, 3852, 8} \[ \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=4 a^3 A b x-\frac {a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}-\frac {a b (12 A-7 C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^4}{d} \]

[In]

Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]

[Out]

4*a^3*A*b*x + ((8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + b*Sec
[c + d*x])^4*Sin[c + d*x])/d - (a*b*(a^2*(12*A - 19*C) - 8*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) - (b^2*(a^2*(2
4*A - 26*C) - 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (a*b*(12*A - 7*C)*(a + b*Sec[c + d*x])^2*
Tan[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3855

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

Rule 4133

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*(Cot[e + f*x]/(2*f)), x] + Dist[1/2, Int[Simp[2*A*a + (2*B*a + b
*(2*A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x]

Rule 4141

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_.), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), I
nt[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)*Csc[e + f*x] + (b*B*(m + 1) +
a*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]

Rule 4180

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*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Dis
t[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e +
f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2,
0] && GtQ[m, 0] && LeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\int (a+b \sec (c+d x))^3 \left (4 A b+a C \sec (c+d x)-b (4 A-C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \sec (c+d x))^2 \left (16 a A b+\left (4 A b^2+4 a^2 C+3 b^2 C\right ) \sec (c+d x)-a b (12 A-7 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \sec (c+d x)) \left (48 a^2 A b+a \left (36 A b^2+12 a^2 C+23 b^2 C\right ) \sec (c+d x)-b \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{24} \int \left (96 a^3 A b+3 \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \sec (c+d x)-4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = 4 a^3 A b x+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {1}{6} \left (a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \int \sec (c+d x) \, dx \\ & = 4 a^3 A b x+\frac {\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\left (a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d} \\ & = 4 a^3 A b x+\frac {\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1357\) vs. \(2(229)=458\).

Time = 11.14 (sec) , antiderivative size = 1357, normalized size of antiderivative = 5.93 \[ \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^3 A b (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right )}{d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x))}+\frac {\left (-48 a^2 A b^2-4 A b^4-8 a^4 C-24 a^2 b^2 C-3 b^4 C\right ) \cos ^6(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right )}{4 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x))}+\frac {\left (48 a^2 A b^2+4 A b^4+8 a^4 C+24 a^2 b^2 C+3 b^4 C\right ) \cos ^6(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right )}{4 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x))}+\frac {b^4 C \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right )}{8 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {\left (12 A b^4+72 a^2 b^2 C+16 a b^3 C+9 b^4 C\right ) \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right )}{24 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 a b^3 C \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{3 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {b^4 C \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right )}{8 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {4 a b^3 C \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{3 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {\left (-12 A b^4-72 a^2 b^2 C-16 a b^3 C-9 b^4 C\right ) \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right )}{24 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {8 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \left (3 a A b^3 \sin \left (\frac {1}{2} (c+d x)\right )+3 a^3 b C \sin \left (\frac {1}{2} (c+d x)\right )+2 a b^3 C \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {8 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \left (3 a A b^3 \sin \left (\frac {1}{2} (c+d x)\right )+3 a^3 b C \sin \left (\frac {1}{2} (c+d x)\right )+2 a b^3 C \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 a^4 A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \sin (c+d x)}{d (b+a \cos (c+d x))^4 (A+2 C+A \cos (2 c+2 d x))} \]

[In]

Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]

[Out]

(8*a^3*A*b*(c + d*x)*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(d*(b + a*Cos[c + d*x])^4*(
A + 2*C + A*Cos[2*c + 2*d*x])) + ((-48*a^2*A*b^2 - 4*A*b^4 - 8*a^4*C - 24*a^2*b^2*C - 3*b^4*C)*Cos[c + d*x]^6*
Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*
x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((48*a^2*A*b^2 + 4*A*b^4 + 8*a^4*C + 24*a^2*b^2*C + 3*b^4*C)*Cos[c + d
*x]^6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[
c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])) + (b^4*C*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]
^2))/(8*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((1
2*A*b^4 + 72*a^2*b^2*C + 16*a*b^3*C + 9*b^4*C)*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(
24*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*a*b^3
*C*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(3*d*(b + a*Cos[c + d*x])^4*
(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (b^4*C*Cos[c + d*x]^6*(a + b*Sec[c +
d*x])^4*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2])^4) + (4*a*b^3*C*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)
/2])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((-
12*A*b^4 - 72*a^2*b^2*C - 16*a*b^3*C - 9*b^4*C)*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/
(24*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (8*Cos[
c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(3*a*A*b^3*Sin[(c + d*x)/2] + 3*a^3*b*C*Sin[(c + d*x)
/2] + 2*a*b^3*C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2
] - Sin[(c + d*x)/2])) + (8*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(3*a*A*b^3*Sin[(c + d
*x)/2] + 3*a^3*b*C*Sin[(c + d*x)/2] + 2*a*b^3*C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Co
s[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (2*a^4*A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C
*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x]))

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {a^{4} A \sin \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{3} b \left (d x +c \right )+4 C \tan \left (d x +c \right ) a^{3} b +6 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 C \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A \tan \left (d x +c \right ) a \,b^{3}-4 C a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+A \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(247\)
default	\(\frac {a^{4} A \sin \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{3} b \left (d x +c \right )+4 C \tan \left (d x +c \right ) a^{3} b +6 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 C \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A \tan \left (d x +c \right ) a \,b^{3}-4 C a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+A \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(247\)
parallelrisch	\(\frac {-576 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (\left (\frac {A}{12}+\frac {C}{16}\right ) b^{4}+\left (A +\frac {C}{2}\right ) a^{2} b^{2}+\frac {a^{4} C}{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+576 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (\left (\frac {A}{12}+\frac {C}{16}\right ) b^{4}+\left (A +\frac {C}{2}\right ) a^{2} b^{2}+\frac {a^{4} C}{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+384 a^{3} A b x d \cos \left (2 d x +2 c \right )+96 a^{3} A b x d \cos \left (4 d x +4 c \right )+\left (\left (24 A +18 C \right ) b^{4}+144 C \,a^{2} b^{2}+36 a^{4} A \right ) \sin \left (3 d x +3 c \right )+192 a \left (b^{2} \left (A +\frac {4 C}{3}\right )+C \,a^{2}\right ) b \sin \left (2 d x +2 c \right )+96 a b \left (C \,a^{2}+b^{2} \left (A +\frac {2 C}{3}\right )\right ) \sin \left (4 d x +4 c \right )+12 a^{4} A \sin \left (5 d x +5 c \right )+\left (\left (24 A +66 C \right ) b^{4}+144 C \,a^{2} b^{2}+24 a^{4} A \right ) \sin \left (d x +c \right )+288 a^{3} A b x d}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\)	\(350\)
risch	\(4 a^{3} A b x -\frac {i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i b \left (12 A \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+72 C \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+9 C \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-96 A a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-96 C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 A \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+72 C \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+33 C \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-288 A a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-288 C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-192 C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 A \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-72 C \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-33 C \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-288 A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-288 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-256 C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 A \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-72 C \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-9 C \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-96 a A \,b^{2}-96 a^{3} C -64 C a \,b^{2}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,a^{2} b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{2 d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,a^{2} b^{2}}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{4}}{8 d}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,a^{2} b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{2 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,a^{2} b^{2}}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{4}}{8 d}\)	\(629\)
norman	\(\frac {\frac {\left (8 a^{4} A -32 a A \,b^{3}+4 A \,b^{4}-32 a^{3} b C +24 C \,a^{2} b^{2}-32 C a \,b^{3}+5 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}-\frac {\left (8 a^{4} A +32 a A \,b^{3}+4 A \,b^{4}+32 a^{3} b C +24 C \,a^{2} b^{2}+32 C a \,b^{3}+5 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (120 a^{4} A -288 a A \,b^{3}+12 A \,b^{4}-288 a^{3} b C +72 C \,a^{2} b^{2}-160 C a \,b^{3}-9 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}+\frac {\left (120 a^{4} A -96 a A \,b^{3}-12 A \,b^{4}-96 a^{3} b C -72 C \,a^{2} b^{2}-32 C a \,b^{3}-3 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}-\frac {\left (120 a^{4} A +96 a A \,b^{3}-12 A \,b^{4}+96 a^{3} b C -72 C \,a^{2} b^{2}+32 C a \,b^{3}-3 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 d}+\frac {\left (120 a^{4} A +288 a A \,b^{3}+12 A \,b^{4}+288 a^{3} b C +72 C \,a^{2} b^{2}+160 C a \,b^{3}-9 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}-4 a^{3} A b x +16 a^{3} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-20 a^{3} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+20 a^{3} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-16 a^{3} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+4 a^{3} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {\left (48 A \,a^{2} b^{2}+4 A \,b^{4}+8 a^{4} C +24 C \,a^{2} b^{2}+3 C \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (48 A \,a^{2} b^{2}+4 A \,b^{4}+8 a^{4} C +24 C \,a^{2} b^{2}+3 C \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\)	\(633\)

[In]

int(cos(d*x+c)*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(a^4*A*sin(d*x+c)+a^4*C*ln(sec(d*x+c)+tan(d*x+c))+4*A*a^3*b*(d*x+c)+4*C*tan(d*x+c)*a^3*b+6*A*a^2*b^2*ln(se
c(d*x+c)+tan(d*x+c))+6*C*a^2*b^2*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+4*A*tan(d*x+c)*a*b^
3-4*C*a*b^3*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+A*b^4*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))
+C*b^4*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c))))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.03 \[ \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {192 \, A a^{3} b d x \cos \left (d x + c\right )^{4} + 3 \, {\left (8 \, C a^{4} + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, C a^{4} + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \, C a b^{3} \cos \left (d x + c\right ) + 6 \, C b^{4} + 32 \, {\left (3 \, C a^{3} b + {\left (3 \, A + 2 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (24 \, C a^{2} b^{2} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]

[In]

integrate(cos(d*x+c)*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/48*(192*A*a^3*b*d*x*cos(d*x + c)^4 + 3*(8*C*a^4 + 24*(2*A + C)*a^2*b^2 + (4*A + 3*C)*b^4)*cos(d*x + c)^4*log
(sin(d*x + c) + 1) - 3*(8*C*a^4 + 24*(2*A + C)*a^2*b^2 + (4*A + 3*C)*b^4)*cos(d*x + c)^4*log(-sin(d*x + c) + 1
) + 2*(24*A*a^4*cos(d*x + c)^4 + 32*C*a*b^3*cos(d*x + c) + 6*C*b^4 + 32*(3*C*a^3*b + (3*A + 2*C)*a*b^3)*cos(d*
x + c)^3 + 3*(24*C*a^2*b^2 + (4*A + 3*C)*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^4)

Sympy [F]

\[ \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4} \cos {\left (c + d x \right )}\, dx \]

[In]

integrate(cos(d*x+c)*(a+b*sec(d*x+c))**4*(A+C*sec(d*x+c)**2),x)

[Out]

Integral((A + C*sec(c + d*x)**2)*(a + b*sec(c + d*x))**4*cos(c + d*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.34 \[ \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {192 \, {\left (d x + c\right )} A a^{3} b + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{3} - 3 \, C b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, C a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, A a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \]

[In]

integrate(cos(d*x+c)*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/48*(192*(d*x + c)*A*a^3*b + 64*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a*b^3 - 3*C*b^4*(2*(3*sin(d*x + c)^3 - 5*
sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 7
2*C*a^2*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 12*A*b^4*(
2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 24*C*a^4*(log(sin(d*x +
c) + 1) - log(sin(d*x + c) - 1)) + 144*A*a^2*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*A*a^4*s
in(d*x + c) + 192*C*a^3*b*tan(d*x + c) + 192*A*a*b^3*tan(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (220) = 440\).

Time = 0.38 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.58 \[ \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {96 \, {\left (d x + c\right )} A a^{3} b + \frac {48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (8 \, C a^{4} + 48 \, A a^{2} b^{2} + 24 \, C a^{2} b^{2} + 4 \, A b^{4} + 3 \, C b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, C a^{4} + 48 \, A a^{2} b^{2} + 24 \, C a^{2} b^{2} + 4 \, A b^{4} + 3 \, C b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, C b^{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} \]

[In]

integrate(cos(d*x+c)*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/24*(96*(d*x + c)*A*a^3*b + 48*A*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 3*(8*C*a^4 + 48*A*a^
2*b^2 + 24*C*a^2*b^2 + 4*A*b^4 + 3*C*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(8*C*a^4 + 48*A*a^2*b^2 + 24*
C*a^2*b^2 + 4*A*b^4 + 3*C*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(96*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 72*
C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 96*C*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 12*
A*b^4*tan(1/2*d*x + 1/2*c)^7 - 15*C*b^4*tan(1/2*d*x + 1/2*c)^7 - 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 72*C*a^2
*b^2*tan(1/2*d*x + 1/2*c)^5 - 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 160*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 12*A*b
^4*tan(1/2*d*x + 1/2*c)^5 - 9*C*b^4*tan(1/2*d*x + 1/2*c)^5 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 72*C*a^2*b^2
*tan(1/2*d*x + 1/2*c)^3 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 160*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 12*A*b^4*t
an(1/2*d*x + 1/2*c)^3 - 9*C*b^4*tan(1/2*d*x + 1/2*c)^3 - 96*C*a^3*b*tan(1/2*d*x + 1/2*c) - 72*C*a^2*b^2*tan(1/
2*d*x + 1/2*c) - 96*A*a*b^3*tan(1/2*d*x + 1/2*c) - 96*C*a*b^3*tan(1/2*d*x + 1/2*c) - 12*A*b^4*tan(1/2*d*x + 1/
2*c) - 15*C*b^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 = 19.09 (sec) , antiderivative size = 1988, normalized size of antiderivative = 8.68 \[ \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)*(A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^4,x)

[Out]

((9*A*b^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + 9*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2
)) + (27*C*b^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/8 + (9*A*a^4*sin(3*c + 3*d*x))/4 + (3*A*a^4*sin(5
*c + 5*d*x))/4 + (3*A*b^4*sin(3*c + 3*d*x))/2 + (9*C*b^4*sin(3*c + 3*d*x))/8 + (3*A*a^4*sin(c + d*x))/2 + (3*A
*b^4*sin(c + d*x))/2 + (33*C*b^4*sin(c + d*x))/8 + 12*A*a*b^3*sin(2*c + 2*d*x) + 6*A*a*b^3*sin(4*c + 4*d*x) +
16*C*a*b^3*sin(2*c + 2*d*x) + 12*C*a^3*b*sin(2*c + 2*d*x) + 4*C*a*b^3*sin(4*c + 4*d*x) + 6*C*a^3*b*sin(4*c + 4
*d*x) + 9*C*a^2*b^2*sin(c + d*x) + 36*A*a^3*b*atan((16*A^2*b^8*sin(c/2 + (d*x)/2) + 64*C^2*a^8*sin(c/2 + (d*x)
/2) + 9*C^2*b^8*sin(c/2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 2304*A^2*a^4*b^4*sin(c/2 + (d*x)/2)
+ 1024*A^2*a^6*b^2*sin(c/2 + (d*x)/2) + 144*C^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624*C^2*a^4*b^4*sin(c/2 + (d*x)/2
) + 384*C^2*a^6*b^2*sin(c/2 + (d*x)/2) + 24*A*C*b^8*sin(c/2 + (d*x)/2) + 480*A*C*a^2*b^6*sin(c/2 + (d*x)/2) +
2368*A*C*a^4*b^4*sin(c/2 + (d*x)/2) + 768*A*C*a^6*b^2*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(16*A^2*b^8 + 64
*C^2*a^8 + 9*C^2*b^8 + 384*A^2*a^2*b^6 + 2304*A^2*a^4*b^4 + 1024*A^2*a^6*b^2 + 144*C^2*a^2*b^6 + 624*C^2*a^4*b
^4 + 384*C^2*a^6*b^2 + 24*A*C*b^8 + 480*A*C*a^2*b^6 + 2368*A*C*a^4*b^4 + 768*A*C*a^6*b^2))) + 54*A*a^2*b^2*ata
nh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + 6*A*b^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d
*x) + (3*A*b^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*d*x))/2 + 12*C*a^4*atanh(sin(c/2 + (d*
x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x) + 3*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*
d*x) + 27*C*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + (9*C*b^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 +
(d*x)/2))*cos(2*c + 2*d*x))/2 + (9*C*b^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*d*x))/8 + 9
*C*a^2*b^2*sin(3*c + 3*d*x) + 48*A*a^3*b*cos(2*c + 2*d*x)*atan((16*A^2*b^8*sin(c/2 + (d*x)/2) + 64*C^2*a^8*sin
(c/2 + (d*x)/2) + 9*C^2*b^8*sin(c/2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 2304*A^2*a^4*b^4*sin(c/2
+ (d*x)/2) + 1024*A^2*a^6*b^2*sin(c/2 + (d*x)/2) + 144*C^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624*C^2*a^4*b^4*sin(c
/2 + (d*x)/2) + 384*C^2*a^6*b^2*sin(c/2 + (d*x)/2) + 24*A*C*b^8*sin(c/2 + (d*x)/2) + 480*A*C*a^2*b^6*sin(c/2 +
(d*x)/2) + 2368*A*C*a^4*b^4*sin(c/2 + (d*x)/2) + 768*A*C*a^6*b^2*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(16*
A^2*b^8 + 64*C^2*a^8 + 9*C^2*b^8 + 384*A^2*a^2*b^6 + 2304*A^2*a^4*b^4 + 1024*A^2*a^6*b^2 + 144*C^2*a^2*b^6 + 6
24*C^2*a^4*b^4 + 384*C^2*a^6*b^2 + 24*A*C*b^8 + 480*A*C*a^2*b^6 + 2368*A*C*a^4*b^4 + 768*A*C*a^6*b^2))) + 12*A
*a^3*b*cos(4*c + 4*d*x)*atan((16*A^2*b^8*sin(c/2 + (d*x)/2) + 64*C^2*a^8*sin(c/2 + (d*x)/2) + 9*C^2*b^8*sin(c/
2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 2304*A^2*a^4*b^4*sin(c/2 + (d*x)/2) + 1024*A^2*a^6*b^2*sin
(c/2 + (d*x)/2) + 144*C^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624*C^2*a^4*b^4*sin(c/2 + (d*x)/2) + 384*C^2*a^6*b^2*si
n(c/2 + (d*x)/2) + 24*A*C*b^8*sin(c/2 + (d*x)/2) + 480*A*C*a^2*b^6*sin(c/2 + (d*x)/2) + 2368*A*C*a^4*b^4*sin(c
/2 + (d*x)/2) + 768*A*C*a^6*b^2*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(16*A^2*b^8 + 64*C^2*a^8 + 9*C^2*b^8 +
384*A^2*a^2*b^6 + 2304*A^2*a^4*b^4 + 1024*A^2*a^6*b^2 + 144*C^2*a^2*b^6 + 624*C^2*a^4*b^4 + 384*C^2*a^6*b^2 +
24*A*C*b^8 + 480*A*C*a^2*b^6 + 2368*A*C*a^4*b^4 + 768*A*C*a^6*b^2))) + 72*A*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/
cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x) + 18*A*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*d
*x) + 36*C*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x) + 9*C*a^2*b^2*atanh(sin(c/2 +
(d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*d*x))/(12*d*(cos(2*c + 2*d*x)/2 + cos(4*c + 4*d*x)/8 + 3/8))