\(\int (a+a \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x)) \sec ^6(c+d x) \, dx\) [100]

Optimal result

Integrand size = 35, antiderivative size = 245 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a^{5/2} (283 A+400 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d}+\frac {a^3 (283 A+400 C) \tan (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^3 (787 A+1040 C) \sec (c+d x) \tan (c+d x)}{960 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (79 A+80 C) \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d} \]

[Out]

1/128*a^(5/2)*(283*A+400*C)*arctanh(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2))/d+1/8*a*A*(a+a*cos(d*x+c))^(3/2
)*sec(d*x+c)^3*tan(d*x+c)/d+1/5*A*(a+a*cos(d*x+c))^(5/2)*sec(d*x+c)^4*tan(d*x+c)/d+1/128*a^3*(283*A+400*C)*tan
(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/960*a^3*(787*A+1040*C)*sec(d*x+c)*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/240
*a^2*(79*A+80*C)*sec(d*x+c)^2*(a+a*cos(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3123, 3054, 3059, 2851, 2852, 212} \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a^{5/2} (283 A+400 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{128 d}+\frac {a^3 (283 A+400 C) \tan (c+d x)}{128 d \sqrt {a \cos (c+d x)+a}}+\frac {a^3 (787 A+1040 C) \tan (c+d x) \sec (c+d x)}{960 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (79 A+80 C) \tan (c+d x) \sec ^2(c+d x) \sqrt {a \cos (c+d x)+a}}{240 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d} \]

[In]

Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]

[Out]

(a^(5/2)*(283*A + 400*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(283*A + 400
*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(787*A + 1040*C)*Sec[c + d*x]*Tan[c + d*x])/(960*d*S
qrt[a + a*Cos[c + d*x]]) + (a^2*(79*A + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) +
(a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x
]^4*Tan[c + d*x])/(5*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2851

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x]
+ Dist[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rule 2852

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[-2*(
b/f), Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3054

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d
*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x
])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n
+ 1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, 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 3059

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n +
1)*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rule 3123

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

Rubi steps \begin{align*} \text {integral}& = \frac {A (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^{5/2} \left (\frac {5 a A}{2}+\frac {1}{2} a (3 A+10 C) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{5 a} \\ & = \frac {a A (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^{3/2} \left (\frac {1}{4} a^2 (79 A+80 C)+\frac {1}{4} a^2 (39 A+80 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{20 a} \\ & = \frac {a^2 (79 A+80 C) \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int \sqrt {a+a \cos (c+d x)} \left (\frac {1}{8} a^3 (787 A+1040 C)+\frac {3}{8} a^3 (157 A+240 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{60 a} \\ & = \frac {a^3 (787 A+1040 C) \sec (c+d x) \tan (c+d x)}{960 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (79 A+80 C) \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{128} \left (a^2 (283 A+400 C)\right ) \int \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \, dx \\ & = \frac {a^3 (283 A+400 C) \tan (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^3 (787 A+1040 C) \sec (c+d x) \tan (c+d x)}{960 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (79 A+80 C) \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{256} \left (a^2 (283 A+400 C)\right ) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx \\ & = \frac {a^3 (283 A+400 C) \tan (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^3 (787 A+1040 C) \sec (c+d x) \tan (c+d x)}{960 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (79 A+80 C) \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {\left (a^3 (283 A+400 C)\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d} \\ & = \frac {a^{5/2} (283 A+400 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d}+\frac {a^3 (283 A+400 C) \tan (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^3 (787 A+1040 C) \sec (c+d x) \tan (c+d x)}{960 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (79 A+80 C) \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.36 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.72 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (60 \sqrt {2} (283 A+400 C) \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^5(c+d x)+(24863 A+20560 C+12 (2343 A+1360 C) \cos (c+d x)+4 (6509 A+6640 C) \cos (2 (c+d x))+5660 A \cos (3 (c+d x))+5440 C \cos (3 (c+d x))+4245 A \cos (4 (c+d x))+6000 C \cos (4 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{15360 d} \]

[In]

Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^5*(60*Sqrt[2]*(283*A + 400*C)*ArcTanh[Sqrt[2]*Si
n[(c + d*x)/2]]*Cos[c + d*x]^5 + (24863*A + 20560*C + 12*(2343*A + 1360*C)*Cos[c + d*x] + 4*(6509*A + 6640*C)*
Cos[2*(c + d*x)] + 5660*A*Cos[3*(c + d*x)] + 5440*C*Cos[3*(c + d*x)] + 4245*A*Cos[4*(c + d*x)] + 6000*C*Cos[4*
(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1974\) vs. \(2(217)=434\).

Time = 4.45 (sec) , antiderivative size = 1975, normalized size of antiderivative = 8.06

\[\text {Expression too large to display}\]

[In]

int((a+cos(d*x+c)*a)^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^6,x)

[Out]

1/120*a^(3/2)*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-480*a*(283*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1
/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))+283*A*ln(-4/(2*cos(1/2
*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))+400*C*
ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/
2)+2*a))+400*C*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c
)^2)^(1/2)*a^(1/2)-2*a)))*sin(1/2*d*x+1/2*c)^10+240*(566*A*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+800*
C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+1415*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2
*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+1415*A*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))
*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a+2000*C*ln(4/(2*cos(1/2*d
*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+2000*C
*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(
1/2)-2*a))*a)*sin(1/2*d*x+1/2*c)^8-80*(3962*A*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+5344*C*2^(1/2)*(a
*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+4245*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+
2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+4245*A*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*
cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a+6000*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^
(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+6000*C*ln(-4/(2*co
s(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a
)*sin(1/2*d*x+1/2*c)^6+8*(36224*A*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+44800*C*2^(1/2)*(a*sin(1/2*d*
x+1/2*c)^2)^(1/2)*a^(1/2)+21225*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a
*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+21225*A*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d
*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a+30000*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(
2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+30000*C*ln(-4/(2*cos(1/2*d
*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)*sin(1
/2*d*x+1/2*c)^4-10*(12556*A*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+13376*C*2^(1/2)*(a*sin(1/2*d*x+1/2*
c)^2)^(1/2)*a^(1/2)+4245*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/
2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+4245*A*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c
)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a+6000*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a
*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+6000*C*ln(-4/(2*cos(1/2*d*x+1/2*c)-
2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)*sin(1/2*d*x+1/2
*c)^2+22230*A*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+4245*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/
2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+4245*A*ln(-4/(2*cos(1/2*d*x+1/2
*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a+18720*C*2^(1
/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+6000*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1
/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+6000*C*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1
/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)/(2*cos(1/2*d*x+1/2*c)+2^(1/2)
)^5/(2*cos(1/2*d*x+1/2*c)-2^(1/2))^5/sin(1/2*d*x+1/2*c)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {15 \, {\left ({\left (283 \, A + 400 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + {\left (283 \, A + 400 \, C\right )} a^{2} \cos \left (d x + c\right )^{5}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (15 \, {\left (283 \, A + 400 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (283 \, A + 272 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (283 \, A + 80 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 1392 \, A a^{2} \cos \left (d x + c\right ) + 384 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{7680 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \]

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="fricas")

[Out]

1/7680*(15*((283*A + 400*C)*a^2*cos(d*x + c)^6 + (283*A + 400*C)*a^2*cos(d*x + c)^5)*sqrt(a)*log((a*cos(d*x +
c)^3 - 7*a*cos(d*x + c)^2 - 4*sqrt(a*cos(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - 2)*sin(d*x + c) + 8*a)/(cos(d*x
+ c)^3 + cos(d*x + c)^2)) + 4*(15*(283*A + 400*C)*a^2*cos(d*x + c)^4 + 10*(283*A + 272*C)*a^2*cos(d*x + c)^3
+ 8*(283*A + 80*C)*a^2*cos(d*x + c)^2 + 1392*A*a^2*cos(d*x + c) + 384*A*a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x
+ c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)

Sympy [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))**(5/2)*(A+C*cos(d*x+c)**2)*sec(d*x+c)**6,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 13407 vs. \(2 (217) = 434\).

Time = 4.16 (sec) , antiderivative size = 13407, normalized size of antiderivative = 54.72 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Too large to display} \]

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="maxima")

[Out]

-1/7680*((1015000*a^2*cos(4*d*x + 4*c)^2*sin(3/2*d*x + 3/2*c) + 253750*a^2*cos(2*d*x + 2*c)^2*sin(3/2*d*x + 3/
2*c) + 1015000*a^2*sin(4*d*x + 4*c)^2*sin(3/2*d*x + 3/2*c) + 253750*a^2*sin(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c
) + 162300*a^2*cos(7/2*d*x + 7/2*c)*sin(2*d*x + 2*c) + 11550*a^2*cos(5/2*d*x + 5/2*c)*sin(2*d*x + 2*c) - 5850*
a^2*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) + 107350*a^2*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 10*(39*a^2*sin(
9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*cos(10*d*x + 10*c)^2 + 250*(39*a^2*sin(9/2*d*x + 9/2*c) + 10
15*a^2*sin(3/2*d*x + 3/2*c))*cos(8*d*x + 8*c)^2 + 1000*(39*a^2*sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3
/2*c))*cos(6*d*x + 6*c)^2 + 10*(39*a^2*sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*sin(10*d*x + 10*c
)^2 + 250*(39*a^2*sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*sin(8*d*x + 8*c)^2 + 1000*(39*a^2*sin(
9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*sin(6*d*x + 6*c)^2 + 11320*a^2*sin(3/2*d*x + 3/2*c) + 390*(a
^2*sin(10*d*x + 10*c) + 5*a^2*sin(8*d*x + 8*c) + 10*a^2*sin(6*d*x + 6*c) + 10*a^2*sin(4*d*x + 4*c) + 5*a^2*sin
(2*d*x + 2*c))*cos(29/2*d*x + 29/2*c) - 1650*(a^2*sin(10*d*x + 10*c) + 5*a^2*sin(8*d*x + 8*c) + 10*a^2*sin(6*d
*x + 6*c) + 10*a^2*sin(4*d*x + 4*c) + 5*a^2*sin(2*d*x + 2*c))*cos(27/2*d*x + 27/2*c) - 2532*(a^2*sin(10*d*x +
10*c) + 5*a^2*sin(8*d*x + 8*c) + 10*a^2*sin(6*d*x + 6*c) + 10*a^2*sin(4*d*x + 4*c) + 5*a^2*sin(2*d*x + 2*c))*c
os(25/2*d*x + 25/2*c) + 1900*(a^2*sin(10*d*x + 10*c) + 5*a^2*sin(8*d*x + 8*c) + 10*a^2*sin(6*d*x + 6*c) + 10*a
^2*sin(4*d*x + 4*c) + 5*a^2*sin(2*d*x + 2*c))*cos(23/2*d*x + 23/2*c) + 15450*(a^2*sin(10*d*x + 10*c) + 5*a^2*s
in(8*d*x + 8*c) + 10*a^2*sin(6*d*x + 6*c) + 10*a^2*sin(4*d*x + 4*c) + 5*a^2*sin(2*d*x + 2*c))*cos(21/2*d*x + 2
1/2*c) + 2*(101500*a^2*cos(4*d*x + 4*c)*sin(3/2*d*x + 3/2*c) + 50750*a^2*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c)
- 34105*a^2*sin(19/2*d*x + 19/2*c) - 70100*a^2*sin(17/2*d*x + 17/2*c) - 124004*a^2*sin(15/2*d*x + 15/2*c) - 1
56685*a^2*sin(13/2*d*x + 13/2*c) - 133065*a^2*sin(11/2*d*x + 11/2*c) - 16230*a^2*sin(7/2*d*x + 7/2*c) - 1155*a
^2*sin(5/2*d*x + 5/2*c) + 10735*a^2*sin(3/2*d*x + 3/2*c) + 50*(39*a^2*sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*
d*x + 3/2*c))*cos(8*d*x + 8*c) + 100*(39*a^2*sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*cos(6*d*x +
6*c) + 10*(390*a^2*cos(4*d*x + 4*c) + 195*a^2*cos(2*d*x + 2*c) - 7196*a^2)*sin(9/2*d*x + 9/2*c))*cos(10*d*x +
10*c) + 341050*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c)
)*cos(19/2*d*x + 19/2*c) + 701000*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^
2*sin(2*d*x + 2*c))*cos(17/2*d*x + 17/2*c) + 10*(101500*a^2*cos(4*d*x + 4*c)*sin(3/2*d*x + 3/2*c) + 50750*a^2*
cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) - 124004*a^2*sin(15/2*d*x + 15/2*c) - 156685*a^2*sin(13/2*d*x + 13/2*c)
- 133065*a^2*sin(11/2*d*x + 11/2*c) - 16230*a^2*sin(7/2*d*x + 7/2*c) - 1155*a^2*sin(5/2*d*x + 5/2*c) + 10735*a
^2*sin(3/2*d*x + 3/2*c) + 100*(39*a^2*sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*cos(6*d*x + 6*c) +
10*(390*a^2*cos(4*d*x + 4*c) + 195*a^2*cos(2*d*x + 2*c) - 7196*a^2)*sin(9/2*d*x + 9/2*c))*cos(8*d*x + 8*c) +
1240040*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*cos(15/2*d*x + 15/2*c) + 1566
850*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*cos(13/2*d*x + 13/2*c) + 100*(203
00*a^2*cos(4*d*x + 4*c)*sin(3/2*d*x + 3/2*c) + 10150*a^2*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) - 26613*a^2*sin
(11/2*d*x + 11/2*c) - 3246*a^2*sin(7/2*d*x + 7/2*c) - 231*a^2*sin(5/2*d*x + 5/2*c) + 2147*a^2*sin(3/2*d*x + 3/
2*c) + 2*(390*a^2*cos(4*d*x + 4*c) + 195*a^2*cos(2*d*x + 2*c) - 7196*a^2)*sin(9/2*d*x + 9/2*c))*cos(6*d*x + 6*
c) + 1330650*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*cos(11/2*d*x + 11/2*c) + 723500*(2*a^2*sin(4*d*x
+ 4*c) + a^2*sin(2*d*x + 2*c))*cos(9/2*d*x + 9/2*c) + 100*(10150*a^2*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) - 3
246*a^2*sin(7/2*d*x + 7/2*c) - 231*a^2*sin(5/2*d*x + 5/2*c) + 2147*a^2*sin(3/2*d*x + 3/2*c))*cos(4*d*x + 4*c)
- 4245*(sqrt(2)*a^2*cos(10*d*x + 10*c)^2 + 25*sqrt(2)*a^2*cos(8*d*x + 8*c)^2 + 100*sqrt(2)*a^2*cos(6*d*x + 6*c
)^2 + 100*sqrt(2)*a^2*cos(4*d*x + 4*c)^2 + 25*sqrt(2)*a^2*cos(2*d*x + 2*c)^2 + sqrt(2)*a^2*sin(10*d*x + 10*c)^
2 + 25*sqrt(2)*a^2*sin(8*d*x + 8*c)^2 + 100*sqrt(2)*a^2*sin(6*d*x + 6*c)^2 + 100*sqrt(2)*a^2*sin(4*d*x + 4*c)^
2 + 100*sqrt(2)*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*sqrt(2)*a^2*sin(2*d*x + 2*c)^2 + 10*sqrt(2)*a^2*cos
(2*d*x + 2*c) + sqrt(2)*a^2 + 2*(5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)
*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(10*d*x + 10*c) + 10*(10*sqrt(2)*a^2*
cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(8*d*x +
8*c) + 20*(10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(6*d*x + 6*c) +
20*(5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(4*d*x + 4*c) + 10*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 2*sqr
t(2)*a^2*sin(6*d*x + 6*c) + 2*sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c)
+ 50*(2*sqrt(2)*a^2*sin(6*d*x + 6*c) + 2*sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(8*d*
x + 8*c) + 100*(2*sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/3
*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x
+ 3/2*c)))^2 + 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2*sqrt(2)*sin(1/3*arct
an2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) + 4245*(sqrt(2)*a^2*cos(10*d*x + 10*c)^2 + 25*sqrt(2)*a^
2*cos(8*d*x + 8*c)^2 + 100*sqrt(2)*a^2*cos(6*d*x + 6*c)^2 + 100*sqrt(2)*a^2*cos(4*d*x + 4*c)^2 + 25*sqrt(2)*a^
2*cos(2*d*x + 2*c)^2 + sqrt(2)*a^2*sin(10*d*x + 10*c)^2 + 25*sqrt(2)*a^2*sin(8*d*x + 8*c)^2 + 100*sqrt(2)*a^2*
sin(6*d*x + 6*c)^2 + 100*sqrt(2)*a^2*sin(4*d*x + 4*c)^2 + 100*sqrt(2)*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) +
25*sqrt(2)*a^2*sin(2*d*x + 2*c)^2 + 10*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2 + 2*(5*sqrt(2)*a^2*cos(8*d*x
+ 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) +
sqrt(2)*a^2)*cos(10*d*x + 10*c) + 10*(10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*s
qrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(8*d*x + 8*c) + 20*(10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*
a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(6*d*x + 6*c) + 20*(5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(4
*d*x + 4*c) + 10*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 2*sqrt(2)*a^2*sin(6*d*x + 6*c) + 2*sqrt(2)*a^2*sin(4*d*x + 4*
c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*sqrt(2)*a^2*sin(6*d*x + 6*c) + 2*sqrt(2)*a^2*sin
(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)
*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2
+ 2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3
/2*c), cos(3/2*d*x + 3/2*c))) - 2*sqrt(2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) -
4245*(sqrt(2)*a^2*cos(10*d*x + 10*c)^2 + 25*sqrt(2)*a^2*cos(8*d*x + 8*c)^2 + 100*sqrt(2)*a^2*cos(6*d*x + 6*c)^
2 + 100*sqrt(2)*a^2*cos(4*d*x + 4*c)^2 + 25*sqrt(2)*a^2*cos(2*d*x + 2*c)^2 + sqrt(2)*a^2*sin(10*d*x + 10*c)^2
+ 25*sqrt(2)*a^2*sin(8*d*x + 8*c)^2 + 100*sqrt(2)*a^2*sin(6*d*x + 6*c)^2 + 100*sqrt(2)*a^2*sin(4*d*x + 4*c)^2
+ 100*sqrt(2)*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*sqrt(2)*a^2*sin(2*d*x + 2*c)^2 + 10*sqrt(2)*a^2*cos(2
*d*x + 2*c) + sqrt(2)*a^2 + 2*(5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a
^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(10*d*x + 10*c) + 10*(10*sqrt(2)*a^2*co
s(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(8*d*x + 8
*c) + 20*(10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(6*d*x + 6*c) + 2
0*(5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(4*d*x + 4*c) + 10*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 2*sqrt(
2)*a^2*sin(6*d*x + 6*c) + 2*sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) +
50*(2*sqrt(2)*a^2*sin(6*d*x + 6*c) + 2*sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(8*d*x
+ 8*c) + 100*(2*sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/3*a
rctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x +
3/2*c)))^2 - 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2*sqrt(2)*sin(1/3*arctan
2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) + 4245*(sqrt(2)*a^2*cos(10*d*x + 10*c)^2 + 25*sqrt(2)*a^2*
cos(8*d*x + 8*c)^2 + 100*sqrt(2)*a^2*cos(6*d*x + 6*c)^2 + 100*sqrt(2)*a^2*cos(4*d*x + 4*c)^2 + 25*sqrt(2)*a^2*
cos(2*d*x + 2*c)^2 + sqrt(2)*a^2*sin(10*d*x + 10*c)^2 + 25*sqrt(2)*a^2*sin(8*d*x + 8*c)^2 + 100*sqrt(2)*a^2*si
n(6*d*x + 6*c)^2 + 100*sqrt(2)*a^2*sin(4*d*x + 4*c)^2 + 100*sqrt(2)*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25
*sqrt(2)*a^2*sin(2*d*x + 2*c)^2 + 10*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2 + 2*(5*sqrt(2)*a^2*cos(8*d*x +
8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + s
qrt(2)*a^2)*cos(10*d*x + 10*c) + 10*(10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqr
t(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(8*d*x + 8*c) + 20*(10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^
2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(6*d*x + 6*c) + 20*(5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(4*d
*x + 4*c) + 10*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 2*sqrt(2)*a^2*sin(6*d*x + 6*c) + 2*sqrt(2)*a^2*sin(4*d*x + 4*c)
+ sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*sqrt(2)*a^2*sin(6*d*x + 6*c) + 2*sqrt(2)*a^2*sin(4
*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a
^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 +
2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 - 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2
*c), cos(3/2*d*x + 3/2*c))) - 2*sqrt(2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) - 39
0*(a^2*cos(10*d*x + 10*c) + 5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2
*cos(2*d*x + 2*c) + a^2)*sin(29/2*d*x + 29/2*c) + 1650*(a^2*cos(10*d*x + 10*c) + 5*a^2*cos(8*d*x + 8*c) + 10*a
^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*sin(27/2*d*x + 27/2*c) + 2532*(a
^2*cos(10*d*x + 10*c) + 5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos
(2*d*x + 2*c) + a^2)*sin(25/2*d*x + 25/2*c) - 1900*(a^2*cos(10*d*x + 10*c) + 5*a^2*cos(8*d*x + 8*c) + 10*a^2*c
os(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*sin(23/2*d*x + 23/2*c) - 15450*(a^2*
cos(10*d*x + 10*c) + 5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*
d*x + 2*c) + a^2)*sin(21/2*d*x + 21/2*c) + 2*(101500*a^2*sin(4*d*x + 4*c)*sin(3/2*d*x + 3/2*c) + 50750*a^2*sin
(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 34105*a^2*cos(19/2*d*x + 19/2*c) + 70100*a^2*cos(17/2*d*x + 17/2*c) + 124
004*a^2*cos(15/2*d*x + 15/2*c) + 156685*a^2*cos(13/2*d*x + 13/2*c) + 133065*a^2*cos(11/2*d*x + 11/2*c) + 72350
*a^2*cos(9/2*d*x + 9/2*c) + 16230*a^2*cos(7/2*d*x + 7/2*c) + 1155*a^2*cos(5/2*d*x + 5/2*c) - 585*a^2*cos(3/2*d
*x + 3/2*c) + 50*(39*a^2*sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*sin(8*d*x + 8*c) + 100*(39*a^2*
sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*sin(6*d*x + 6*c) + 1950*(2*a^2*sin(4*d*x + 4*c) + a^2*si
n(2*d*x + 2*c))*sin(9/2*d*x + 9/2*c))*sin(10*d*x + 10*c) - 68210*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x +
6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*sin(19/2*d*x + 19/2*c) - 140200*(5*a^2*cos(8*d*
x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*sin(17/2*d*x + 17
/2*c) + 10*(101500*a^2*sin(4*d*x + 4*c)*sin(3/2*d*x + 3/2*c) + 50750*a^2*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c)
+ 124004*a^2*cos(15/2*d*x + 15/2*c) + 156685*a^2*cos(13/2*d*x + 13/2*c) + 133065*a^2*cos(11/2*d*x + 11/2*c) +
72350*a^2*cos(9/2*d*x + 9/2*c) + 16230*a^2*cos(7/2*d*x + 7/2*c) + 1155*a^2*cos(5/2*d*x + 5/2*c) - 585*a^2*cos
(3/2*d*x + 3/2*c) + 100*(39*a^2*sin(9/2*d*x + 9/2*c) + 1015*a^2*sin(3/2*d*x + 3/2*c))*sin(6*d*x + 6*c) + 1950*
(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(9/2*d*x + 9/2*c))*sin(8*d*x + 8*c) - 248008*(10*a^2*cos(6*
d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*sin(15/2*d*x + 15/2*c) - 313370*(10*a^2*c
os(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*sin(13/2*d*x + 13/2*c) + 100*(20300*
a^2*sin(4*d*x + 4*c)*sin(3/2*d*x + 3/2*c) + 10150*a^2*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 26613*a^2*cos(11
/2*d*x + 11/2*c) + 14470*a^2*cos(9/2*d*x + 9/2*c) + 3246*a^2*cos(7/2*d*x + 7/2*c) + 231*a^2*cos(5/2*d*x + 5/2*
c) - 117*a^2*cos(3/2*d*x + 3/2*c) + 390*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(9/2*d*x + 9/2*c))*
sin(6*d*x + 6*c) - 266130*(10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*sin(11/2*d*x + 11/2*c) + 10
*(3900*a^2*cos(4*d*x + 4*c)^2 + 975*a^2*cos(2*d*x + 2*c)^2 + 3900*a^2*sin(4*d*x + 4*c)^2 + 3900*a^2*sin(4*d*x
+ 4*c)*sin(2*d*x + 2*c) + 975*a^2*sin(2*d*x + 2*c)^2 - 71960*a^2*cos(2*d*x + 2*c) - 14431*a^2 + 20*(195*a^2*co
s(2*d*x + 2*c) - 7196*a^2)*cos(4*d*x + 4*c))*sin(9/2*d*x + 9/2*c) + 100*(10150*a^2*sin(2*d*x + 2*c)*sin(3/2*d*
x + 3/2*c) + 3246*a^2*cos(7/2*d*x + 7/2*c) + 231*a^2*cos(5/2*d*x + 5/2*c) - 117*a^2*cos(3/2*d*x + 3/2*c))*sin(
4*d*x + 4*c) - 32460*(5*a^2*cos(2*d*x + 2*c) + a^2)*sin(7/2*d*x + 7/2*c) - 2310*(5*a^2*cos(2*d*x + 2*c) + a^2)
*sin(5/2*d*x + 5/2*c) - 1650*(a^2*cos(10*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*cos(6*d*x + 6*c)^
2 + 100*a^2*cos(4*d*x + 4*c)^2 + 25*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*sin(8*d*x + 8*c
)^2 + 100*a^2*sin(6*d*x + 6*c)^2 + 100*a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25
*a^2*sin(2*d*x + 2*c)^2 + 10*a^2*cos(2*d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c)
+ 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6*d*x + 6*c) + 1
0*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x + 4*c) + 5*a^2*
cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 10*(a^2*sin(8*
d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(
2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a^2*sin(4*d*
x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*sin(7/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))
) - 4482*(a^2*cos(10*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*cos(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*
x + 4*c)^2 + 25*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*
d*x + 6*c)^2 + 100*a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c
)^2 + 10*a^2*cos(2*d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x +
4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*
c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a
^2)*cos(6*d*x + 6*c) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 10*(a^2*sin(8*d*x + 8*c) + 2*a^2*s
in(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a^2*sin(6*d*x + 6*
c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2
*d*x + 2*c))*sin(6*d*x + 6*c))*sin(5/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 33960*(a^2*cos(1
0*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*cos(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2 + 25*a^2
*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^2 + 100*
a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^2*cos(2*
d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2
*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*
x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c
) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2
*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*
x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*
d*x + 6*c))*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*A*sqrt(a)/(sqrt(2)*cos(10*d*x + 10*c
)^2 + 25*sqrt(2)*cos(8*d*x + 8*c)^2 + 100*sqrt(2)*cos(6*d*x + 6*c)^2 + 100*sqrt(2)*cos(4*d*x + 4*c)^2 + 25*sqr
t(2)*cos(2*d*x + 2*c)^2 + sqrt(2)*sin(10*d*x + 10*c)^2 + 25*sqrt(2)*sin(8*d*x + 8*c)^2 + 100*sqrt(2)*sin(6*d*x
+ 6*c)^2 + 100*sqrt(2)*sin(4*d*x + 4*c)^2 + 100*sqrt(2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*sqrt(2)*sin(2*
d*x + 2*c)^2 + 2*(5*sqrt(2)*cos(8*d*x + 8*c) + 10*sqrt(2)*cos(6*d*x + 6*c) + 10*sqrt(2)*cos(4*d*x + 4*c) + 5*s
qrt(2)*cos(2*d*x + 2*c) + sqrt(2))*cos(10*d*x + 10*c) + 10*(10*sqrt(2)*cos(6*d*x + 6*c) + 10*sqrt(2)*cos(4*d*x
+ 4*c) + 5*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*cos(8*d*x + 8*c) + 20*(10*sqrt(2)*cos(4*d*x + 4*c) + 5*sqrt(2)
*cos(2*d*x + 2*c) + sqrt(2))*cos(6*d*x + 6*c) + 20*(5*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*cos(4*d*x + 4*c) + 1
0*(sqrt(2)*sin(8*d*x + 8*c) + 2*sqrt(2)*sin(6*d*x + 6*c) + 2*sqrt(2)*sin(4*d*x + 4*c) + sqrt(2)*sin(2*d*x + 2*
c))*sin(10*d*x + 10*c) + 50*(2*sqrt(2)*sin(6*d*x + 6*c) + 2*sqrt(2)*sin(4*d*x + 4*c) + sqrt(2)*sin(2*d*x + 2*c
))*sin(8*d*x + 8*c) + 100*(2*sqrt(2)*sin(4*d*x + 4*c) + sqrt(2)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 10*sqrt(2
)*cos(2*d*x + 2*c) + sqrt(2)) + 80*(1530*a^2*cos(4*d*x + 4*c)^2*sin(3/2*d*x + 3/2*c) + 1530*a^2*cos(2*d*x + 2*
c)^2*sin(3/2*d*x + 3/2*c) + 1530*a^2*sin(4*d*x + 4*c)^2*sin(3/2*d*x + 3/2*c) + 1530*a^2*sin(2*d*x + 2*c)^2*sin
(3/2*d*x + 3/2*c) + 4176*a^2*cos(7/2*d*x + 7/2*c)*sin(2*d*x + 2*c) + 2430*a^2*cos(5/2*d*x + 5/2*c)*sin(2*d*x +
2*c) + 678*a^2*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) + 342*a^2*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 10*(a^
2*sin(9/2*d*x + 9/2*c) + 17*a^2*sin(3/2*d*x + 3/2*c))*cos(6*d*x + 6*c)^2 + 10*(a^2*sin(9/2*d*x + 9/2*c) + 17*a
^2*sin(3/2*d*x + 3/2*c))*sin(6*d*x + 6*c)^2 - 56*a^2*sin(3/2*d*x + 3/2*c) + 10*(a^2*sin(6*d*x + 6*c) + 3*a^2*s
in(4*d*x + 4*c) + 3*a^2*sin(2*d*x + 2*c))*cos(21/2*d*x + 21/2*c) - 30*(a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x
+ 4*c) + 3*a^2*sin(2*d*x + 2*c))*cos(19/2*d*x + 19/2*c) - 48*(a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) +
3*a^2*sin(2*d*x + 2*c))*cos(17/2*d*x + 17/2*c) + 80*(a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 3*a^2*sin
(2*d*x + 2*c))*cos(15/2*d*x + 15/2*c) + 396*(a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 3*a^2*sin(2*d*x +
2*c))*cos(13/2*d*x + 13/2*c) + 6*(170*a^2*cos(4*d*x + 4*c)*sin(3/2*d*x + 3/2*c) + 170*a^2*cos(2*d*x + 2*c)*si
n(3/2*d*x + 3/2*c) - 170*a^2*sin(11/2*d*x + 11/2*c) - 232*a^2*sin(7/2*d*x + 7/2*c) - 135*a^2*sin(5/2*d*x + 5/2
*c) + 19*a^2*sin(3/2*d*x + 3/2*c) + 10*(a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c) - 25*a^2)*sin(9/2*d*x + 9/
2*c))*cos(6*d*x + 6*c) + 3060*(a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*cos(11/2*d*x + 11/2*c) + 4560*(a^2
*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*cos(9/2*d*x + 9/2*c) + 18*(170*a^2*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/
2*c) - 232*a^2*sin(7/2*d*x + 7/2*c) - 135*a^2*sin(5/2*d*x + 5/2*c) + 19*a^2*sin(3/2*d*x + 3/2*c))*cos(4*d*x +
4*c) - 75*(sqrt(2)*a^2*cos(6*d*x + 6*c)^2 + 9*sqrt(2)*a^2*cos(4*d*x + 4*c)^2 + 9*sqrt(2)*a^2*cos(2*d*x + 2*c)^
2 + sqrt(2)*a^2*sin(6*d*x + 6*c)^2 + 9*sqrt(2)*a^2*sin(4*d*x + 4*c)^2 + 18*sqrt(2)*a^2*sin(4*d*x + 4*c)*sin(2*
d*x + 2*c) + 9*sqrt(2)*a^2*sin(2*d*x + 2*c)^2 + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2 + 2*(3*sqrt(2)*a^
2*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(6*d*x + 6*c) + 6*(3*sqrt(2)*a^2*cos(2*d
*x + 2*c) + sqrt(2)*a^2)*cos(4*d*x + 4*c) + 6*(sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*si
n(6*d*x + 6*c))*log(2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sin(1/3*arctan2(sin(3
/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*
c))) + 2*sqrt(2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) + 75*(sqrt(2)*a^2*cos(6*d*x
+ 6*c)^2 + 9*sqrt(2)*a^2*cos(4*d*x + 4*c)^2 + 9*sqrt(2)*a^2*cos(2*d*x + 2*c)^2 + sqrt(2)*a^2*sin(6*d*x + 6*c)
^2 + 9*sqrt(2)*a^2*sin(4*d*x + 4*c)^2 + 18*sqrt(2)*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*sqrt(2)*a^2*sin(2
*d*x + 2*c)^2 + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2 + 2*(3*sqrt(2)*a^2*cos(4*d*x + 4*c) + 3*sqrt(2)*a
^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(6*d*x + 6*c) + 6*(3*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(4*d
*x + 4*c) + 6*(sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/3*ar
ctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3
/2*c)))^2 + 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 2*sqrt(2)*sin(1/3*arctan2
(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) - 75*(sqrt(2)*a^2*cos(6*d*x + 6*c)^2 + 9*sqrt(2)*a^2*cos(4*
d*x + 4*c)^2 + 9*sqrt(2)*a^2*cos(2*d*x + 2*c)^2 + sqrt(2)*a^2*sin(6*d*x + 6*c)^2 + 9*sqrt(2)*a^2*sin(4*d*x + 4
*c)^2 + 18*sqrt(2)*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*sqrt(2)*a^2*sin(2*d*x + 2*c)^2 + 6*sqrt(2)*a^2*co
s(2*d*x + 2*c) + sqrt(2)*a^2 + 2*(3*sqrt(2)*a^2*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^
2)*cos(6*d*x + 6*c) + 6*(3*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(4*d*x + 4*c) + 6*(sqrt(2)*a^2*sin(4
*d*x + 4*c) + sqrt(2)*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(
3/2*d*x + 3/2*c)))^2 + 2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 - 2*sqrt(2)*cos(1/3*ar
ctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2*sqrt(2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d
*x + 3/2*c))) + 2) + 75*(sqrt(2)*a^2*cos(6*d*x + 6*c)^2 + 9*sqrt(2)*a^2*cos(4*d*x + 4*c)^2 + 9*sqrt(2)*a^2*cos
(2*d*x + 2*c)^2 + sqrt(2)*a^2*sin(6*d*x + 6*c)^2 + 9*sqrt(2)*a^2*sin(4*d*x + 4*c)^2 + 18*sqrt(2)*a^2*sin(4*d*x
+ 4*c)*sin(2*d*x + 2*c) + 9*sqrt(2)*a^2*sin(2*d*x + 2*c)^2 + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2 + 2
*(3*sqrt(2)*a^2*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(6*d*x + 6*c) + 6*(3*sqrt(
2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*cos(4*d*x + 4*c) + 6*(sqrt(2)*a^2*sin(4*d*x + 4*c) + sqrt(2)*a^2*sin(2*
d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sin(1/3
*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 - 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(
3/2*d*x + 3/2*c))) - 2*sqrt(2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) - 10*(a^2*cos
(6*d*x + 6*c) + 3*a^2*cos(4*d*x + 4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*sin(21/2*d*x + 21/2*c) + 30*(a^2*cos(6*
d*x + 6*c) + 3*a^2*cos(4*d*x + 4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*sin(19/2*d*x + 19/2*c) + 48*(a^2*cos(6*d*x
+ 6*c) + 3*a^2*cos(4*d*x + 4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*sin(17/2*d*x + 17/2*c) - 80*(a^2*cos(6*d*x +
6*c) + 3*a^2*cos(4*d*x + 4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*sin(15/2*d*x + 15/2*c) - 396*(a^2*cos(6*d*x + 6*
c) + 3*a^2*cos(4*d*x + 4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*sin(13/2*d*x + 13/2*c) + 2*(510*a^2*sin(4*d*x + 4*
c)*sin(3/2*d*x + 3/2*c) + 510*a^2*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 510*a^2*cos(11/2*d*x + 11/2*c) + 760
*a^2*cos(9/2*d*x + 9/2*c) + 696*a^2*cos(7/2*d*x + 7/2*c) + 405*a^2*cos(5/2*d*x + 5/2*c) + 113*a^2*cos(3/2*d*x
+ 3/2*c) + 30*(a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(9/2*d*x + 9/2*c))*sin(6*d*x + 6*c) - 1020*(3*a
^2*cos(4*d*x + 4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*sin(11/2*d*x + 11/2*c) + 10*(9*a^2*cos(4*d*x + 4*c)^2 + 9*
a^2*cos(2*d*x + 2*c)^2 + 9*a^2*sin(4*d*x + 4*c)^2 + 18*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a^2*sin(2*d*x
+ 2*c)^2 - 450*a^2*cos(2*d*x + 2*c) - 151*a^2 + 18*(a^2*cos(2*d*x + 2*c) - 25*a^2)*cos(4*d*x + 4*c))*sin(9/2*
d*x + 9/2*c) + 6*(510*a^2*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 696*a^2*cos(7/2*d*x + 7/2*c) + 405*a^2*cos(5
/2*d*x + 5/2*c) + 113*a^2*cos(3/2*d*x + 3/2*c))*sin(4*d*x + 4*c) - 1392*(3*a^2*cos(2*d*x + 2*c) + a^2)*sin(7/2
*d*x + 7/2*c) - 810*(3*a^2*cos(2*d*x + 2*c) + a^2)*sin(5/2*d*x + 5/2*c) - 30*(a^2*cos(6*d*x + 6*c)^2 + 9*a^2*c
os(4*d*x + 4*c)^2 + 9*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(6*d*x + 6*c)^2 + 9*a^2*sin(4*d*x + 4*c)^2 + 18*a^2*sin(
4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a^2*sin(2*d*x + 2*c)^2 + 6*a^2*cos(2*d*x + 2*c) + a^2 + 2*(3*a^2*cos(4*d*x +
4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 6*(3*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 6
*(a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*sin(7/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2
*d*x + 3/2*c))) - 78*(a^2*cos(6*d*x + 6*c)^2 + 9*a^2*cos(4*d*x + 4*c)^2 + 9*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(6
*d*x + 6*c)^2 + 9*a^2*sin(4*d*x + 4*c)^2 + 18*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a^2*sin(2*d*x + 2*c)^2
+ 6*a^2*cos(2*d*x + 2*c) + a^2 + 2*(3*a^2*cos(4*d*x + 4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) +
6*(3*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 6*(a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x
+ 6*c))*sin(5/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 600*(a^2*cos(6*d*x + 6*c)^2 + 9*a^2*co
s(4*d*x + 4*c)^2 + 9*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(6*d*x + 6*c)^2 + 9*a^2*sin(4*d*x + 4*c)^2 + 18*a^2*sin(4
*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a^2*sin(2*d*x + 2*c)^2 + 6*a^2*cos(2*d*x + 2*c) + a^2 + 2*(3*a^2*cos(4*d*x +
4*c) + 3*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 6*(3*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 6*
(a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*
d*x + 3/2*c))))*C*sqrt(a)/(sqrt(2)*cos(6*d*x + 6*c)^2 + 9*sqrt(2)*cos(4*d*x + 4*c)^2 + 9*sqrt(2)*cos(2*d*x + 2
*c)^2 + sqrt(2)*sin(6*d*x + 6*c)^2 + 9*sqrt(2)*sin(4*d*x + 4*c)^2 + 18*sqrt(2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*
c) + 9*sqrt(2)*sin(2*d*x + 2*c)^2 + 2*(3*sqrt(2)*cos(4*d*x + 4*c) + 3*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*cos(
6*d*x + 6*c) + 6*(3*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*cos(4*d*x + 4*c) + 6*(sqrt(2)*sin(4*d*x + 4*c) + sqrt(
2)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2)))/d

Giac [A] (verification not implemented)

none

Time = 1.46 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.53 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=-\frac {\sqrt {2} {\left (15 \, \sqrt {2} {\left (283 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 400 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) + \frac {4 \, {\left (67920 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 96000 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 158480 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 213760 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 144896 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 179200 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 62780 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 66880 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11115 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9360 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}\right )} \sqrt {a}}{7680 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="giac")

[Out]

-1/7680*sqrt(2)*(15*sqrt(2)*(283*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 400*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*log(ab
s(-2*sqrt(2) + 4*sin(1/2*d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin(1/2*d*x + 1/2*c))) + 4*(67920*A*a^2*sgn(cos(1/2*d
*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^9 + 96000*C*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^9 - 158480*A*
a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^7 - 213760*C*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/
2*c)^7 + 144896*A*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^5 + 179200*C*a^2*sgn(cos(1/2*d*x + 1/2*c)
)*sin(1/2*d*x + 1/2*c)^5 - 62780*A*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^3 - 66880*C*a^2*sgn(cos(
1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^3 + 11115*A*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c) + 9360*C
*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c))/(2*sin(1/2*d*x + 1/2*c)^2 - 1)^5)*sqrt(a)/d

Mupad [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^6} \,d x \]

[In]

int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2))/cos(c + d*x)^6,x)

[Out]

int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2))/cos(c + d*x)^6, x)