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\(\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx\) [90]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [B] (verified)
   Fricas [A] (verification not implemented)
   Sympy [F(-1)]
   Maxima [B] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [F(-1)]

Optimal result

Integrand size = 33, antiderivative size = 209 \[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {a^{3/2} (75 A+88 B) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{64 d}+\frac {a^2 (75 A+88 B) \tan (c+d x)}{64 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (75 A+88 B) \sec (c+d x) \tan (c+d x)}{96 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (9 A+8 B) \sec ^2(c+d x) \tan (c+d x)}{24 d \sqrt {a+a \cos (c+d x)}}+\frac {a A \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{4 d} \]

[Out]

1/64*a^(3/2)*(75*A+88*B)*arctanh(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2))/d+1/64*a^2*(75*A+88*B)*tan(d*x+c)/
d/(a+a*cos(d*x+c))^(1/2)+1/96*a^2*(75*A+88*B)*sec(d*x+c)*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/24*a^2*(9*A+8*B
)*sec(d*x+c)^2*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/4*a*A*sec(d*x+c)^3*(a+a*cos(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3054, 3059, 2851, 2852, 212} \[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {a^{3/2} (75 A+88 B) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{64 d}+\frac {a^2 (75 A+88 B) \tan (c+d x)}{64 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (9 A+8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (75 A+88 B) \tan (c+d x) \sec (c+d x)}{96 d \sqrt {a \cos (c+d x)+a}}+\frac {a A \tan (c+d x) \sec ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d} \]

[In]

Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]

[Out]

(a^(3/2)*(75*A + 88*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B)*T
an[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(75*A + 88*B)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*
Cos[c + d*x]]) + (a^2*(9*A + 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a +
 a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*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]

Rubi steps \begin{align*} \text {integral}& = \frac {a A \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int \sqrt {a+a \cos (c+d x)} \left (\frac {1}{2} a (9 A+8 B)+\frac {1}{2} a (5 A+8 B) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {a^2 (9 A+8 B) \sec ^2(c+d x) \tan (c+d x)}{24 d \sqrt {a+a \cos (c+d x)}}+\frac {a A \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{48} (a (75 A+88 B)) \int \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \, dx \\ & = \frac {a^2 (75 A+88 B) \sec (c+d x) \tan (c+d x)}{96 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (9 A+8 B) \sec ^2(c+d x) \tan (c+d x)}{24 d \sqrt {a+a \cos (c+d x)}}+\frac {a A \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{64} (a (75 A+88 B)) \int \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \, dx \\ & = \frac {a^2 (75 A+88 B) \tan (c+d x)}{64 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (75 A+88 B) \sec (c+d x) \tan (c+d x)}{96 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (9 A+8 B) \sec ^2(c+d x) \tan (c+d x)}{24 d \sqrt {a+a \cos (c+d x)}}+\frac {a A \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{128} (a (75 A+88 B)) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx \\ & = \frac {a^2 (75 A+88 B) \tan (c+d x)}{64 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (75 A+88 B) \sec (c+d x) \tan (c+d x)}{96 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (9 A+8 B) \sec ^2(c+d x) \tan (c+d x)}{24 d \sqrt {a+a \cos (c+d x)}}+\frac {a A \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {\left (a^2 (75 A+88 B)\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 )}{64 d} \\ & = \frac {a^{3/2} (75 A+88 B) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{64 d}+\frac {a^2 (75 A+88 B) \tan (c+d x)}{64 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (75 A+88 B) \sec (c+d x) \tan (c+d x)}{96 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (9 A+8 B) \sec ^2(c+d x) \tan (c+d x)}{24 d \sqrt {a+a \cos (c+d x)}}+\frac {a A \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.72 \[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (6 \sqrt {2} (75 A+88 B) \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4(c+d x)+(492 A+352 B+(1155 A+1048 B) \cos (c+d x)+4 (75 A+88 B) \cos (2 (c+d x))+225 A \cos (3 (c+d x))+264 B \cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{768 d} \]

[In]

Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]

[Out]

(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^4*(6*Sqrt[2]*(75*A + 88*B)*ArcTanh[Sqrt[2]*Sin[(c
+ d*x)/2]]*Cos[c + d*x]^4 + (492*A + 352*B + (1155*A + 1048*B)*Cos[c + d*x] + 4*(75*A + 88*B)*Cos[2*(c + d*x)]
 + 225*A*Cos[3*(c + d*x)] + 264*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1616\) vs. \(2(185)=370\).

Time = 6.36 (sec) , antiderivative size = 1617, normalized size of antiderivative = 7.74

method result size
parts \(\text {Expression too large to display}\) \(1617\)
default \(\text {Expression too large to display}\) \(1651\)

[In]

int((a+cos(d*x+c)*a)^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x,method=_RETURNVERBOSE)

[Out]

1/8*A*a^(1/2)*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(1200*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))+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
)^8-1200*(2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*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*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)^6+200*
(11*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+9*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+9*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+(-1460*2^(
1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-600*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-600*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+362*2^(1/2)*(
a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+75*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+75*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))^4/(2*cos(1/
2*d*x+1/2*c)-2^(1/2))^4/sin(1/2*d*x+1/2*c)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d+1/12*B*a^(1/2)*cos(1/2*d*x+1/2*c)*
(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-264*2^(1/2)*a*(ln(2/(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))+ln(-2/(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)^6+132*(3*2^(1/2)*ln(-2/
(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+3*2^(1/2)*ln(2/(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+4*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))*sin(1/2*d*x+1/2*c)^4-22*(9*2^(1/2)*ln(-2/
(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+9*2^(1/2)*ln(2/(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+32*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))*sin(1/2*d*x+1/2*c)^2+33*2^(1/2)*ln(2/(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+33*2^(1/2)*ln(-2/(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+252*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))/(2*cos(1/2*d*x+1/2*c)+2^(1/2))^3/(2*cos(
1/2*d*x+1/2*c)-2^(1/2))^3/sin(1/2*d*x+1/2*c)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.05 \[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {3 \, {\left ({\left (75 \, A + 88 \, B\right )} a \cos \left (d x + c\right )^{5} + {\left (75 \, A + 88 \, B\right )} a \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (3 \, {\left (75 \, A + 88 \, B\right )} a \cos \left (d x + c\right )^{3} + 2 \, {\left (75 \, A + 88 \, B\right )} a \cos \left (d x + c\right )^{2} + 8 \, {\left (15 \, A + 8 \, B\right )} a \cos \left (d x + c\right ) + 48 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{768 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \]

[In]

integrate((a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x, algorithm="fricas")

[Out]

1/768*(3*((75*A + 88*B)*a*cos(d*x + c)^5 + (75*A + 88*B)*a*cos(d*x + c)^4)*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*(3*(75*A + 88*B)*a*cos(d*x + c)^3 + 2*(75*A + 88*B)*a*cos(d*x + c)^2 + 8*(15*A + 8*B)*a*c
os(d*x + c) + 48*A*a)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c))/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4)

Sympy [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)**5,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10504 vs. \(2 (185) = 370\).

Time = 154.62 (sec) , antiderivative size = 10504, normalized size of antiderivative = 50.26 \[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\text {Too large to display} \]

[In]

integrate((a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x, algorithm="maxima")

[Out]

-1/768*(3*(140*a*cos(8*d*x + 8*c)^2*sin(3/2*d*x + 3/2*c) + 2240*a*cos(6*d*x + 6*c)^2*sin(3/2*d*x + 3/2*c) + 50
40*a*cos(4*d*x + 4*c)^2*sin(3/2*d*x + 3/2*c) + 2240*a*cos(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) + 140*a*sin(8*d*
x + 8*c)^2*sin(3/2*d*x + 3/2*c) + 2240*a*sin(6*d*x + 6*c)^2*sin(3/2*d*x + 3/2*c) + 5040*a*sin(4*d*x + 4*c)^2*s
in(3/2*d*x + 3/2*c) + 2240*a*sin(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) + 4064*a*cos(7/2*d*x + 7/2*c)*sin(2*d*x +
 2*c) + 336*a*cos(5/2*d*x + 5/2*c)*sin(2*d*x + 2*c) - 240*a*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) + 1360*a*cos
(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) - 36*(a*sin(8*d*x + 8*c) + 4*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + 4*
a*sin(2*d*x + 2*c))*cos(21/2*d*x + 21/2*c) + 140*(a*sin(8*d*x + 8*c) + 4*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x +
4*c) + 4*a*sin(2*d*x + 2*c))*cos(19/2*d*x + 19/2*c) + 456*(a*sin(8*d*x + 8*c) + 4*a*sin(6*d*x + 6*c) + 6*a*sin
(4*d*x + 4*c) + 4*a*sin(2*d*x + 2*c))*cos(17/2*d*x + 17/2*c) + 4*(280*a*cos(6*d*x + 6*c)*sin(3/2*d*x + 3/2*c)
+ 420*a*cos(4*d*x + 4*c)*sin(3/2*d*x + 3/2*c) + 280*a*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) - 290*a*sin(15/2*d
*x + 15/2*c) - 596*a*sin(13/2*d*x + 13/2*c) - 780*a*sin(11/2*d*x + 11/2*c) - 750*a*sin(9/2*d*x + 9/2*c) - 254*
a*sin(7/2*d*x + 7/2*c) - 21*a*sin(5/2*d*x + 5/2*c) + 85*a*sin(3/2*d*x + 3/2*c))*cos(8*d*x + 8*c) + 2320*(2*a*s
in(6*d*x + 6*c) + 3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*cos(15/2*d*x + 15/2*c) + 4768*(2*a*sin(6*d*x +
6*c) + 3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*cos(13/2*d*x + 13/2*c) + 16*(420*a*cos(4*d*x + 4*c)*sin(3/
2*d*x + 3/2*c) + 280*a*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) - 780*a*sin(11/2*d*x + 11/2*c) - 750*a*sin(9/2*d*
x + 9/2*c) - 254*a*sin(7/2*d*x + 7/2*c) - 21*a*sin(5/2*d*x + 5/2*c) + 85*a*sin(3/2*d*x + 3/2*c))*cos(6*d*x + 6
*c) + 6240*(3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*cos(11/2*d*x + 11/2*c) + 6000*(3*a*sin(4*d*x + 4*c) +
 2*a*sin(2*d*x + 2*c))*cos(9/2*d*x + 9/2*c) + 24*(280*a*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) - 254*a*sin(7/2*
d*x + 7/2*c) - 21*a*sin(5/2*d*x + 5/2*c) + 85*a*sin(3/2*d*x + 3/2*c))*cos(4*d*x + 4*c) - 75*(sqrt(2)*a*cos(8*d
*x + 8*c)^2 + 16*sqrt(2)*a*cos(6*d*x + 6*c)^2 + 36*sqrt(2)*a*cos(4*d*x + 4*c)^2 + 16*sqrt(2)*a*cos(2*d*x + 2*c
)^2 + sqrt(2)*a*sin(8*d*x + 8*c)^2 + 16*sqrt(2)*a*sin(6*d*x + 6*c)^2 + 36*sqrt(2)*a*sin(4*d*x + 4*c)^2 + 48*sq
rt(2)*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 8*sqrt(2)*a*cos(2*d*x + 2*c) + 2
*(4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(
8*d*x + 8*c) + 8*(6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(6*d*x + 6*c) +
12*(4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(4*d*x + 4*c) + 4*(2*sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a
*sin(4*d*x + 4*c) + 2*sqrt(2)*a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*sqrt(2)*a*sin(4*d*x + 4*c) + 2*sqrt
(2)*a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + sqrt(2)*a)*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(si
n(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*cos(8*d*x + 8*c)^2 + 16*sqrt(2)*a*cos(6*d*x + 6*c)^2 + 36*sqrt(2)*a*cos(4*d*x + 4*c
)^2 + 16*sqrt(2)*a*cos(2*d*x + 2*c)^2 + sqrt(2)*a*sin(8*d*x + 8*c)^2 + 16*sqrt(2)*a*sin(6*d*x + 6*c)^2 + 36*sq
rt(2)*a*sin(4*d*x + 4*c)^2 + 48*sqrt(2)*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*sqrt(2)*a*sin(2*d*x + 2*c)^2
+ 8*sqrt(2)*a*cos(2*d*x + 2*c) + 2*(4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*
cos(2*d*x + 2*c) + sqrt(2)*a)*cos(8*d*x + 8*c) + 8*(6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c
) + sqrt(2)*a)*cos(6*d*x + 6*c) + 12*(4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(4*d*x + 4*c) + 4*(2*sqrt(2
)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 2*sqrt(2)*a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*s
qrt(2)*a*sin(4*d*x + 4*c) + 2*sqrt(2)*a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + sqrt(2)*a)*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*cos(8*d*x + 8*c)^2 + 16*sqrt(2)*a*cos(6*d*x + 6*c
)^2 + 36*sqrt(2)*a*cos(4*d*x + 4*c)^2 + 16*sqrt(2)*a*cos(2*d*x + 2*c)^2 + sqrt(2)*a*sin(8*d*x + 8*c)^2 + 16*sq
rt(2)*a*sin(6*d*x + 6*c)^2 + 36*sqrt(2)*a*sin(4*d*x + 4*c)^2 + 48*sqrt(2)*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c)
+ 16*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 8*sqrt(2)*a*cos(2*d*x + 2*c) + 2*(4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)
*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(8*d*x + 8*c) + 8*(6*sqrt(2)*a*cos(4*d*x +
4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(6*d*x + 6*c) + 12*(4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)
*a)*cos(4*d*x + 4*c) + 4*(2*sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 2*sqrt(2)*a*sin(2*d*x
+ 2*c))*sin(8*d*x + 8*c) + 16*(3*sqrt(2)*a*sin(4*d*x + 4*c) + 2*sqrt(2)*a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) +
 sqrt(2)*a)*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*cos(8*d*x + 8*c
)^2 + 16*sqrt(2)*a*cos(6*d*x + 6*c)^2 + 36*sqrt(2)*a*cos(4*d*x + 4*c)^2 + 16*sqrt(2)*a*cos(2*d*x + 2*c)^2 + sq
rt(2)*a*sin(8*d*x + 8*c)^2 + 16*sqrt(2)*a*sin(6*d*x + 6*c)^2 + 36*sqrt(2)*a*sin(4*d*x + 4*c)^2 + 48*sqrt(2)*a*
sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 8*sqrt(2)*a*cos(2*d*x + 2*c) + 2*(4*sqrt
(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(8*d*x +
8*c) + 8*(6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(6*d*x + 6*c) + 12*(4*sq
rt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(4*d*x + 4*c) + 4*(2*sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d
*x + 4*c) + 2*sqrt(2)*a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*sqrt(2)*a*sin(4*d*x + 4*c) + 2*sqrt(2)*a*si
n(2*d*x + 2*c))*sin(6*d*x + 6*c) + sqrt(2)*a)*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) + 36*(a*cos(8*d*x + 8*c) + 4*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*sin(21/2
*d*x + 21/2*c) - 140*(a*cos(8*d*x + 8*c) + 4*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c)
+ a)*sin(19/2*d*x + 19/2*c) - 456*(a*cos(8*d*x + 8*c) + 4*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + 4*a*cos(
2*d*x + 2*c) + a)*sin(17/2*d*x + 17/2*c) + 4*(280*a*sin(6*d*x + 6*c)*sin(3/2*d*x + 3/2*c) + 420*a*sin(4*d*x +
4*c)*sin(3/2*d*x + 3/2*c) + 280*a*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 290*a*cos(15/2*d*x + 15/2*c) + 596*a
*cos(13/2*d*x + 13/2*c) + 780*a*cos(11/2*d*x + 11/2*c) + 750*a*cos(9/2*d*x + 9/2*c) + 254*a*cos(7/2*d*x + 7/2*
c) + 21*a*cos(5/2*d*x + 5/2*c) - 15*a*cos(3/2*d*x + 3/2*c))*sin(8*d*x + 8*c) - 1160*(4*a*cos(6*d*x + 6*c) + 6*
a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*sin(15/2*d*x + 15/2*c) - 2384*(4*a*cos(6*d*x + 6*c) + 6*a*cos(4
*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*sin(13/2*d*x + 13/2*c) + 16*(420*a*sin(4*d*x + 4*c)*sin(3/2*d*x + 3/2*
c) + 280*a*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 780*a*cos(11/2*d*x + 11/2*c) + 750*a*cos(9/2*d*x + 9/2*c) +
 254*a*cos(7/2*d*x + 7/2*c) + 21*a*cos(5/2*d*x + 5/2*c) - 15*a*cos(3/2*d*x + 3/2*c))*sin(6*d*x + 6*c) - 3120*(
6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*sin(11/2*d*x + 11/2*c) - 3000*(6*a*cos(4*d*x + 4*c) + 4*a*cos
(2*d*x + 2*c) + a)*sin(9/2*d*x + 9/2*c) + 24*(280*a*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 254*a*cos(7/2*d*x
+ 7/2*c) + 21*a*cos(5/2*d*x + 5/2*c) - 15*a*cos(3/2*d*x + 3/2*c))*sin(4*d*x + 4*c) - 1016*(4*a*cos(2*d*x + 2*c
) + a)*sin(7/2*d*x + 7/2*c) - 84*(4*a*cos(2*d*x + 2*c) + a)*sin(5/2*d*x + 5/2*c) + 200*a*sin(3/2*d*x + 3/2*c)
- 36*(a*cos(8*d*x + 8*c)^2 + 16*a*cos(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 16*a*cos(2*d*x + 2*c)^2 + a*s
in(8*d*x + 8*c)^2 + 16*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 48*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c)
 + 16*a*sin(2*d*x + 2*c)^2 + 2*(4*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*cos(8*
d*x + 8*c) + 8*(6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 12*(4*a*cos(2*d*x + 2*c) +
 a)*cos(4*d*x + 4*c) + 8*a*cos(2*d*x + 2*c) + 4*(2*a*sin(6*d*x + 6*c) + 3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x +
 2*c))*sin(8*d*x + 8*c) + 16*(3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*sin(5/3*arcta
n2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 600*(a*cos(8*d*x + 8*c)^2 + 16*a*cos(6*d*x + 6*c)^2 + 36*a*c
os(4*d*x + 4*c)^2 + 16*a*cos(2*d*x + 2*c)^2 + a*sin(8*d*x + 8*c)^2 + 16*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x
+ 4*c)^2 + 48*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a*sin(2*d*x + 2*c)^2 + 2*(4*a*cos(6*d*x + 6*c) + 6*a*co
s(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 8*(6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) +
 a)*cos(6*d*x + 6*c) + 12*(4*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 8*a*cos(2*d*x + 2*c) + 4*(2*a*sin(6*d*
x + 6*c) + 3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a*sin(4*d*x + 4*c) + 2*a*sin(
2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*A*sqrt(a)/(s
qrt(2)*cos(8*d*x + 8*c)^2 + 16*sqrt(2)*cos(6*d*x + 6*c)^2 + 36*sqrt(2)*cos(4*d*x + 4*c)^2 + 16*sqrt(2)*cos(2*d
*x + 2*c)^2 + sqrt(2)*sin(8*d*x + 8*c)^2 + 16*sqrt(2)*sin(6*d*x + 6*c)^2 + 36*sqrt(2)*sin(4*d*x + 4*c)^2 + 48*
sqrt(2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*sqrt(2)*sin(2*d*x + 2*c)^2 + 2*(4*sqrt(2)*cos(6*d*x + 6*c) + 6*
sqrt(2)*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*cos(8*d*x + 8*c) + 8*(6*sqrt(2)*cos(4*d*x + 4
*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*cos(6*d*x + 6*c) + 12*(4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*cos(4
*d*x + 4*c) + 4*(2*sqrt(2)*sin(6*d*x + 6*c) + 3*sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*sin(8*d
*x + 8*c) + 16*(3*sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 8*sqrt(2)*cos(2*d*
x + 2*c) + sqrt(2)) + 8*(774*sqrt(2)*a*cos(7/2*d*x + 7/2*c)*sin(2*d*x + 2*c) + 162*sqrt(2)*a*cos(5/2*d*x + 5/2
*c)*sin(2*d*x + 2*c) + (14*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos
(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*
c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sq
rt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos
(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x +
1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*cos(6*d*x + 6*c)^2 + 9*(14*sq
rt(2)*a*sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1
/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*
d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) +
2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)
*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*
d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*cos(4*d*x + 4*c)^2 + 9*(14*sqrt(2)*a*sin(3/2*d*x + 3/2*c)
+ 90*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)
*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*
x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x +
 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) +
33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(
1/2*d*x + 1/2*c) + 2))*cos(2*d*x + 2*c)^2 + (14*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/
2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(
2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/
2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2
*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)
^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*sin(6*d*
x + 6*c)^2 + 9*(14*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos(1/2*d*x
 + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2)
+ 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*si
n(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x
 + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2
 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*sin(4*d*x + 4*c)^2 + 9*(14*sqrt(2)*a*
sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x +
 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/
2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*
a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2
*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/
2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*sin(2*d*x + 2*c)^2 + 44*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 132*sqrt(
2)*a*sin(1/2*d*x + 1/2*c) + 14*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*
d*x + 2*c))*cos(15/2*d*x + 15/2*c) + 90*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)
*a*sin(2*d*x + 2*c))*cos(13/2*d*x + 13/2*c) - 2*(87*sqrt(2)*a*sin(11/2*d*x + 11/2*c) + 157*sqrt(2)*a*sin(9/2*d
*x + 9/2*c) + 129*sqrt(2)*a*sin(7/2*d*x + 7/2*c) + 27*sqrt(2)*a*sin(5/2*d*x + 5/2*c) - 29*sqrt(2)*a*sin(3/2*d*
x + 3/2*c) - 111*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 3*(14*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*
d*x + 1/2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) +
 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2
)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d
*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x
+ 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*
cos(4*d*x + 4*c) - 3*(14*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos(1
/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c)
 + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt
(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1
/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/
2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*cos(2*d*x + 2*c) + 33*a*log(2*c
os(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/
2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*
sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*c
os(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x
+ 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*cos(6*d*x + 6*c) + 522*(sqr
t(2)*a*sin(4*d*x + 4*c) + sqrt(2)*a*sin(2*d*x + 2*c))*cos(11/2*d*x + 11/2*c) + 942*(sqrt(2)*a*sin(4*d*x + 4*c)
 + sqrt(2)*a*sin(2*d*x + 2*c))*cos(9/2*d*x + 9/2*c) - 6*(129*sqrt(2)*a*sin(7/2*d*x + 7/2*c) + 27*sqrt(2)*a*sin
(5/2*d*x + 5/2*c) - 29*sqrt(2)*a*sin(3/2*d*x + 3/2*c) - 111*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 3*(14*sqrt(2)*a*s
in(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x +
1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2
*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a
*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*
d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2
*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*cos(2*d*x + 2*c) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*
x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x +
 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) +
33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(
1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x +
 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*cos(4*d*x + 4*c) + 6*(29*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 111*s
qrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1
/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/
2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c
)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*l
og(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*
x + 1/2*c) + 2))*cos(2*d*x + 2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*c
os(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x
+ 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1
/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33
*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/
2*d*x + 1/2*c) + 2) - 14*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x +
2*c) + sqrt(2)*a)*sin(15/2*d*x + 15/2*c) - 90*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*s
qrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(13/2*d*x + 13/2*c) + 2*(87*sqrt(2)*a*cos(11/2*d*x + 11/2*c) + 157*s
qrt(2)*a*cos(9/2*d*x + 9/2*c) + 129*sqrt(2)*a*cos(7/2*d*x + 7/2*c) + 27*sqrt(2)*a*cos(5/2*d*x + 5/2*c) - 15*sq
rt(2)*a*cos(3/2*d*x + 3/2*c) - 21*sqrt(2)*a*cos(1/2*d*x + 1/2*c) + 3*(14*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 90*s
qrt(2)*a*sin(1/2*d*x + 1/2*c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1
/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/
2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c
)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*l
og(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*
x + 1/2*c) + 2))*sin(4*d*x + 4*c) + 3*(14*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/2*c) -
 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin
(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x
+ 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2
- 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2
*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*sin(2*d*x + 2*
c))*sin(6*d*x + 6*c) - 174*(3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(11/2*
d*x + 11/2*c) - 314*(3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(9/2*d*x + 9/
2*c) + 18*(43*sqrt(2)*a*cos(7/2*d*x + 7/2*c) + 9*sqrt(2)*a*cos(5/2*d*x + 5/2*c) - 5*sqrt(2)*a*cos(3/2*d*x + 3/
2*c) - 7*sqrt(2)*a*cos(1/2*d*x + 1/2*c) + (14*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 90*sqrt(2)*a*sin(1/2*d*x + 1/2*
c) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)
*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*
d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - 33*a*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c
)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 33*a*log(2*cos(1/2*d*x + 1/2*c)^2
 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*sin(2*d*x
+ 2*c))*sin(4*d*x + 4*c) - 258*(3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(7/2*d*x + 7/2*c) - 54*(3*sqrt(2)
*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(5/2*d*x + 5/2*c) - 18*(5*sqrt(2)*a*cos(3/2*d*x + 3/2*c) + 7*sqrt(2)*a*cos
(1/2*d*x + 1/2*c))*sin(2*d*x + 2*c))*B*sqrt(a)/(2*(3*cos(4*d*x + 4*c) + 3*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*
c) + cos(6*d*x + 6*c)^2 + 6*(3*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + 9*cos(4*d*x + 4*c)^2 + 9*cos(2*d*x + 2
*c)^2 + 6*(sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + sin(6*d*x + 6*c)^2 + 9*sin(4*d*x + 4*c)^2 +
 18*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2*c) + 1))/d

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.44 \[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=-\frac {\sqrt {2} {\left (3 \, \sqrt {2} {\left (75 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 88 \, B a \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 (1800 \, A a \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} + 2112 \, B a \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} - 3300 \, A a \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} - 3872 \, B a \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} + 2190 \, A a \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} + 2416 \, B a \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} - 543 \, A a \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 ) - 504 \, B a \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 )}^{4}}\right )} \sqrt {a}}{768 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x, algorithm="giac")

[Out]

-1/768*sqrt(2)*(3*sqrt(2)*(75*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 88*B*a*sgn(cos(1/2*d*x + 1/2*c)))*log(abs(-2*sqr
t(2) + 4*sin(1/2*d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin(1/2*d*x + 1/2*c))) + 4*(1800*A*a*sgn(cos(1/2*d*x + 1/2*c)
)*sin(1/2*d*x + 1/2*c)^7 + 2112*B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^7 - 3300*A*a*sgn(cos(1/2*d*
x + 1/2*c))*sin(1/2*d*x + 1/2*c)^5 - 3872*B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^5 + 2190*A*a*sgn(
cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^3 + 2416*B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^3 - 543
*A*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c) - 504*B*a*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)^4)*sqrt(a)/d

Mupad [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \]

[In]

int(((A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2))/cos(c + d*x)^5,x)

[Out]

int(((A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2))/cos(c + d*x)^5, x)

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