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\(\int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx\) [105]

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

Optimal result

Integrand size = 27, antiderivative size = 109 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\sqrt {2} (A+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}-\frac {4 C \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d} \]

[Out]

(A+C)*arctanh(1/2*sin(d*x+c)*a^(1/2)*2^(1/2)/(a+a*cos(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)-4/3*C*sin(d*x+c)/d/(a+a
*cos(d*x+c))^(1/2)+2/3*C*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/a/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3103, 2830, 2728, 212} \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\sqrt {2} (A+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {2 C \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 a d}-\frac {4 C \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}} \]

[In]

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

[Out]

(Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*C*Sin[c
+ d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*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 2728

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

Rule 2830

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

Rule 3103

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d}+\frac {2 \int \frac {\frac {1}{2} a (3 A+C)-a C \cos (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{3 a} \\ & = -\frac {4 C \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d}+(A+C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx \\ & = -\frac {4 C \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d}-\frac {(2 (A+C)) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d} \\ & = \frac {\sqrt {2} (A+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}-\frac {4 C \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.58 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (3 (A+C) \text {arctanh}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 C \sin ^3\left (\frac {1}{2} (c+d x)\right )\right )}{3 d \sqrt {a (1+\cos (c+d x))}} \]

[In]

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

[Out]

(2*Cos[(c + d*x)/2]*(3*(A + C)*ArcTanh[Sin[(c + d*x)/2]] - 4*C*Sin[(c + d*x)/2]^3))/(3*d*Sqrt[a*(1 + Cos[c + d
*x])])

Maple [A] (verified)

Time = 5.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.59

method result size
default \(\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-4 C \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 A \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a +3 C \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \right )}{3 a^{\frac {3}{2}} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) \(173\)
parts \(\frac {A \sqrt {2}\, \operatorname {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}| 1\right )}{d \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \operatorname {csgn}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-4 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \right )}{3 a^{\frac {3}{2}} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) \(193\)

[In]

int((A+C*cos(d*x+c)^2)/(a+cos(d*x+c)*a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*cos(1/2*d*x+1/2*c)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-4*C*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*sin
(1/2*d*x+1/2*c)^2+3*A*ln(4*(a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+a)/cos(1/2*d*x+1/2*c))*a+3*C*ln(4*(a^(1/2)*
(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+a)/cos(1/2*d*x+1/2*c))*a)/a^(3/2)/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 = 0.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.30 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {4 \, {\left (C \cos \left (d x + c\right ) - C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + \frac {3 \, \sqrt {2} {\left ({\left (A + C\right )} a \cos \left (d x + c\right ) + {\left (A + C\right )} a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - \frac {2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt {a}}}{6 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]

[In]

integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/6*(4*(C*cos(d*x + c) - C)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c) + 3*sqrt(2)*((A + C)*a*cos(d*x + c) + (A + C
)*a)*log(-(cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(a) - 2*cos(d*x + c) - 3)/(cos
(d*x + c)^2 + 2*cos(d*x + c) + 1))/sqrt(a))/(a*d*cos(d*x + c) + a*d)

Sympy [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {A + C \cos ^{2}{\left (c + d x \right )}}{\sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}}\, dx \]

[In]

integrate((A+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**(1/2),x)

[Out]

Integral((A + C*cos(c + d*x)**2)/sqrt(a*(cos(c + d*x) + 1)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19528 vs. \(2 (92) = 184\).

Time = 0.88 (sec) , antiderivative size = 19528, normalized size of antiderivative = 179.16 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\text {Too large to display} \]

[In]

integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

1/60*(30*(sqrt(2)*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - sqrt(2)*
log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*A/sqrt(a) + (20*(cos(d*x +
c) + 1)*sin(5/2*d*x + 5/2*c)^3 + 8*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)
^3 - 20*cos(5/2*d*x + 5/2*c)^3*sin(d*x + c) + 2*(15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*s
in(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*
cos(d*x + c)^2 + 15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(c
os(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 30*(log(cos(1/2
*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*
d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + 4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x +
c) + 1)*sin(3/2*d*x + 3/2*c) - 20*cos(3/2*d*x + 3/2*c)*sin(d*x + c) + 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*
d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(
1/2*d*x + 1/2*c) + 1))*cos(5/2*d*x + 5/2*c)^2 + 30*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*s
in(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*
cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(
1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x
 + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x
+ 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 +
 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) +
1))*cos(3/2*d*x + 3/2*c)^2 + 2*(10*(cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^3 + 4*(cos(d*x + c)^2 + sin(d*x + c
)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^3 - 10*cos(5/2*d*x + 5/2*c)^3*sin(d*x + c) + (15*(log(cos(1/2*d
*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*
x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + 15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/
2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1
/2*c) + 1))*sin(d*x + c)^2 + 30*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)
+ 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + 4*(co
s(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - 20*cos(3/2*d*x + 3/2*c)*sin(d*x + c
) + 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - 15*log(cos(1/2*d*x
+ 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(5/2*d*x + 5/2*c)^2 + 15*((log(cos(1/2*d
*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*
x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c
)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*
c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1)
 - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1
/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/
2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(3/2*d*x + 3/2*c)^2 + (15*(log(cos(1/2*d*x + 1/2*c)^2 + sin
(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*si
n(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + 15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*
d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x
 + c)^2 + 30*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*
d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + 4*(cos(d*x + c)^2 + sin(
d*x + c)^2 + 7*cos(d*x + c) + 6)*sin(3/2*d*x + 3/2*c) - 10*cos(5/2*d*x + 5/2*c)*sin(d*x + c) + 15*log(cos(1/2*
d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/
2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(5/2*d*x + 5/2*c)^2 + 15*((log(cos(1/2*d*x + 1/2*c)^2 + sin
(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*si
n(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x
 + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x +
c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x
+ 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 +
 sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 -
2*sin(1/2*d*x + 1/2*c) + 1))*sin(3/2*d*x + 3/2*c)^2 + 2*(4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) +
 1)*cos(3/2*d*x + 3/2*c)*sin(3/2*d*x + 3/2*c) - 5*cos(3/2*d*x + 3/2*c)^2*sin(d*x + c) - 5*sin(3/2*d*x + 3/2*c)
^2*sin(d*x + c) + 15*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log
(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*
d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d
*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/
2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1
/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) -
 log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(3/2*d*x + 3/2*c))*cos(
5/2*d*x + 5/2*c) + 10*(cos(5/2*d*x + 5/2*c)^2*sin(d*x + c) + 2*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c)*sin(d
*x + c) + cos(3/2*d*x + 3/2*c)^2*sin(d*x + c) + sin(5/2*d*x + 5/2*c)^2*sin(d*x + c) + 2*sin(5/2*d*x + 5/2*c)*s
in(3/2*d*x + 3/2*c)*sin(d*x + c) + sin(3/2*d*x + 3/2*c)^2*sin(d*x + c))*cos(1/2*arctan2(sin(d*x + c), cos(d*x
+ c))) + 2*(5*(cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 5*(cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (4*cos
(d*x + c)^2 + 4*sin(d*x + c)^2 + 13*cos(d*x + c) + 9)*sin(3/2*d*x + 3/2*c)^2 + 10*((cos(d*x + c) + 1)*cos(3/2*
d*x + 3/2*c) - sin(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(5/2*d*x + 5/2*c) + 3*cos(d*x + c)^2 + 15*((log(cos(1/2*d
*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*
x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c
)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*
c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1)
 - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1
/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/
2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(3/2*d*x + 3/2*c) + 3*sin(d*x + c)^2 + 6*cos(d*x + c) + 3)*
sin(5/2*d*x + 5/2*c) + 2*(2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + 3*
cos(d*x + c)^2 + 3*sin(d*x + c)^2 + 6*cos(d*x + c) + 3)*sin(3/2*d*x + 3/2*c) - 24*((cos(d*x + c)^2 + sin(d*x +
 c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*
cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x
 + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^
2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x
 + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*
c))) - 10*((2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(5/2*d*x + 5/2*c)^2 + 2*(2*cos(d*x +
c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (2*cos(d*x + c)^2 +
2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(3/2*d*x + 3/2*c)^2 + (2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d
*x + c) + 3)*sin(5/2*d*x + 5/2*c)^2 + 2*(2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(5/2*d*x
 + 5/2*c)*sin(3/2*d*x + 3/2*c) + (2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(3/2*d*x + 3/2*
c)^2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))))*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)
))^2 + 2*(15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*
d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + 15*(log(cos(1/2*d*x +
1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1
/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 30*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^
2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c)
 + 1))*cos(d*x + c) + 4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 7*cos(d*x + c) + 6)*sin(3/2*d*x + 3/2*c) - 10*cos(5
/2*d*x + 5/2*c)*sin(d*x + c) + 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)
 + 1) - 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(5/2*d*x + 5/
2*c)^2 + 30*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*
d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2
*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*
c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 +
2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1
))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(
1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(3/2*d*x + 3/2*c)^2 + 12*((cos(d
*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 +
 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x
+ c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)
^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos
(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c
), cos(3/2*d*x + 3/2*c)))*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2*(10*(cos(d*x + c) +
 1)*sin(5/2*d*x + 5/2*c)^3 + 4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^3 -
 10*cos(5/2*d*x + 5/2*c)^3*sin(d*x + c) + (15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2
*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*
x + c)^2 + 15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2
*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 30*(log(cos(1/2*d*x +
 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x +
1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + 4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1
)*sin(3/2*d*x + 3/2*c) - 20*cos(3/2*d*x + 3/2*c)*sin(d*x + c) + 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x +
1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*
x + 1/2*c) + 1))*cos(5/2*d*x + 5/2*c)^2 + 15*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2
*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*
x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*
x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2
*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*
c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin
(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*co
s(3/2*d*x + 3/2*c)^2 + (15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1)
- log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + 15*(log(
cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + s
in(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 30*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2
*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/
2*d*x + 1/2*c) + 1))*cos(d*x + c) + 4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 7*cos(d*x + c) + 6)*sin(3/2*d*x + 3/2
*c) - 10*cos(5/2*d*x + 5/2*c)*sin(d*x + c) + 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/
2*d*x + 1/2*c) + 1) - 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*si
n(5/2*d*x + 5/2*c)^2 + 15*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1)
- log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos
(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(
1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x
 + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*
x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) +
 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(3/2*d*x + 3/2*c)^
2 + 2*(4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)*sin(3/2*d*x + 3/2*c) - 5*
cos(3/2*d*x + 3/2*c)^2*sin(d*x + c) - 5*sin(3/2*d*x + 3/2*c)^2*sin(d*x + c) + 15*((log(cos(1/2*d*x + 1/2*c)^2
+ sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 -
 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/
2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d
*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2
*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c
)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)
^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(3/2*d*x + 3/2*c))*cos(5/2*d*x + 5/2*c) + 10*(cos(5/2*d*x + 5/2*c)^2*sin(
d*x + c) + 2*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c)*sin(d*x + c) + cos(3/2*d*x + 3/2*c)^2*sin(d*x + c) + si
n(5/2*d*x + 5/2*c)^2*sin(d*x + c) + 2*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c)*sin(d*x + c) + sin(3/2*d*x + 3
/2*c)^2*sin(d*x + c))*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2*(5*(cos(d*x + c) + 1)*cos(5/2*d*x + 5/2
*c)^2 + 5*(cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 13*cos(d*x + c) +
 9)*sin(3/2*d*x + 3/2*c)^2 + 10*((cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c) - sin(3/2*d*x + 3/2*c)*sin(d*x + c))*
cos(5/2*d*x + 5/2*c) + 3*cos(d*x + c)^2 + 15*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2
*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*
x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*
x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2
*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*
c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin
(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*si
n(3/2*d*x + 3/2*c) + 3*sin(d*x + c)^2 + 6*cos(d*x + c) + 3)*sin(5/2*d*x + 5/2*c) + 2*(2*(cos(d*x + c)^2 + sin(
d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(d*x + c)^2 + 6*cos(d*x + c)
 + 3)*sin(3/2*d*x + 3/2*c) - 24*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2
 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d
*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2
*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d
*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)
^2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 10*((2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 +
5*cos(d*x + c) + 3)*cos(5/2*d*x + 5/2*c)^2 + 2*(2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(
5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(3/2*d*x
 + 3/2*c)^2 + (2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(5/2*d*x + 5/2*c)^2 + 2*(2*cos(d*x
 + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (2*cos(d*x + c)^2
 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(3/2*d*x + 3/2*c)^2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))
)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 4*(4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*c
os(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)*sin(3/2*d*x + 3/2*c) - 5*cos(3/2*d*x + 3/2*c)^2*sin(d*x + c) - 5*sin(3/2
*d*x + 3/2*c)^2*sin(d*x + c) + 15*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*
c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 +
(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^
2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin
(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*si
n(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x +
1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(3/2*d*x +
 3/2*c))*cos(5/2*d*x + 5/2*c) + 4*(10*(cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^3 + 4*(cos(d*x + c)^2 + sin(d*x
+ c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^3 - 10*cos(5/2*d*x + 5/2*c)^3*sin(d*x + c) + (15*(log(cos(1/
2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2
*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + 15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x +
 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x
+ 1/2*c) + 1))*sin(d*x + c)^2 + 30*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*
c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + 4*
(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - 20*cos(3/2*d*x + 3/2*c)*sin(d*x
+ c) + 15*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - 15*log(cos(1/2*d
*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(5/2*d*x + 5/2*c)^2 + 15*((log(cos(1/
2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2
*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/
2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1
/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) +
 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(co
s(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin
(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(3/2*d*x + 3/2*c)^2 + (15*(log(cos(1/2*d*x + 1/2*c)^2 +
sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2
*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + 15*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1
/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(
d*x + c)^2 + 30*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1
/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + 4*(cos(d*x + c)^2 + s
in(d*x + c)^2 + 7*cos(d*x + c) + 6)*sin(3/2*d*x + 3/2*c) - 10*cos(5/2*d*x + 5/2*c)*sin(d*x + c) + 15*log(cos(1
/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - 15*log(cos(1/2*d*x + 1/2*c)^2 + sin
(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(5/2*d*x + 5/2*c)^2 + 15*((log(cos(1/2*d*x + 1/2*c)^2 +
sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2
*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*
d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x
 + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d
*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^
2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2
 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(3/2*d*x + 3/2*c)^2 + 2*(4*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c
) + 1)*cos(3/2*d*x + 3/2*c)*sin(3/2*d*x + 3/2*c) - 5*cos(3/2*d*x + 3/2*c)^2*sin(d*x + c) - 5*sin(3/2*d*x + 3/2
*c)^2*sin(d*x + c) + 15*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) -
log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1
/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/
2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x +
 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x
+ 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1
) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(3/2*d*x + 3/2*c))*c
os(5/2*d*x + 5/2*c) + 10*(cos(5/2*d*x + 5/2*c)^2*sin(d*x + c) + 2*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c)*si
n(d*x + c) + cos(3/2*d*x + 3/2*c)^2*sin(d*x + c) + sin(5/2*d*x + 5/2*c)^2*sin(d*x + c) + 2*sin(5/2*d*x + 5/2*c
)*sin(3/2*d*x + 3/2*c)*sin(d*x + c) + sin(3/2*d*x + 3/2*c)^2*sin(d*x + c))*cos(1/2*arctan2(sin(d*x + c), cos(d
*x + c))) + 2*(5*(cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 5*(cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (4*
cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 13*cos(d*x + c) + 9)*sin(3/2*d*x + 3/2*c)^2 + 10*((cos(d*x + c) + 1)*cos(3
/2*d*x + 3/2*c) - sin(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(5/2*d*x + 5/2*c) + 3*cos(d*x + c)^2 + 15*((log(cos(1/
2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2
*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/
2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1
/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) +
 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(co
s(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin
(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(3/2*d*x + 3/2*c) + 3*sin(d*x + c)^2 + 6*cos(d*x + c) +
3)*sin(5/2*d*x + 5/2*c) + 2*(2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 +
 3*cos(d*x + c)^2 + 3*sin(d*x + c)^2 + 6*cos(d*x + c) + 3)*sin(3/2*d*x + 3/2*c) - 27*((cos(d*x + c)^2 + sin(d*
x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) +
1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*
d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x +
c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(
d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3
/2*c))) - 10*((2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(5/2*d*x + 5/2*c)^2 + 2*(2*cos(d*x
 + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (2*cos(d*x + c)^2
 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(3/2*d*x + 3/2*c)^2 + (2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*co
s(d*x + c) + 3)*sin(5/2*d*x + 5/2*c)^2 + 2*(2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(5/2*
d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(3/2*d*x + 3
/2*c)^2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))))*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2
*c))) + 20*(cos(5/2*d*x + 5/2*c)^2*sin(d*x + c) + 2*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c)*sin(d*x + c) + c
os(3/2*d*x + 3/2*c)^2*sin(d*x + c) + sin(5/2*d*x + 5/2*c)^2*sin(d*x + c) + 2*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x
+ 3/2*c)*sin(d*x + c) + sin(3/2*d*x + 3/2*c)^2*sin(d*x + c))*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 15
*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x
+ c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*
cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + ((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x
 + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*
c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x
 + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1
)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d
*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + (cos(d*x + c)^2 + sin(d*x + c)
^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin
(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x +
3/2*c)^2 + ((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2
+ sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x +
 c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*si
n(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d
*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*sin(2/3*arctan2(s
in(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(
5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x
 + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 +
sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x +
 c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*si
n(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*log(cos(1/3*arctan2(sin(3/
2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + sin(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))) + 1) - 15*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*c
os(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x
 + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2
 + ((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*
x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 +
2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*
x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2
*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d
*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5
/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) +
 (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2 + ((cos(d*x + c)^2 + sin(d*x +
c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*c
os(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x
+ 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2
 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x
+ c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c
)))^2 + 2*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 +
 sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x +
c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin
(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*
x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(si
n(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*log(cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2
 + sin(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), co
s(3/2*d*x + 3/2*c))) + 1) + 15*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2
+ 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*
x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + ((cos(d*x + c)^2 + sin(d*x + c)^2 + 2
*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d
*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)
^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d
*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 +
 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 +
(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x +
c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*co
s(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2 + ((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x +
 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c)
 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x +
 c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*
sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x
 + 3/2*c)^2)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*((cos(d*x + c)^2 + sin(d*x + c
)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*co
s(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x +
 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2
+ sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x +
 c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)
)))*log(cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*si
n(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 1) - 15*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*c
os(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*
d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + ((cos(d*x + c)^
2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d
*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1
)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(
cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c
)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3
/2*d*x + 3/2*c)))^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d
*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 +
 sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2 + ((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x +
 c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c
)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(
d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2
+ 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x
 + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*((cos(d*
x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 +
2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x +
 c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^
2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(
d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c)
, cos(3/2*d*x + 3/2*c))))*log(cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + sin(1/2*arctan2(sin(d*x + c), c
os(d*x + c)))^2 - 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 1) + 4*(5*(cos(d*x + c) + 1)*cos(5/2*d*x +
5/2*c)^2 + 5*(cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 13*cos(d*x + c
) + 9)*sin(3/2*d*x + 3/2*c)^2 + 10*((cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c) - sin(3/2*d*x + 3/2*c)*sin(d*x + c
))*cos(5/2*d*x + 5/2*c) + 3*cos(d*x + c)^2 + 15*((log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(
1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos
(d*x + c)^2 + (log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2
*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*sin(d*x + c)^2 + 2*(log(cos(1/2*d*x +
1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1
/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))*cos(d*x + c) + log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*
sin(1/2*d*x + 1/2*c) + 1) - log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1))
*sin(3/2*d*x + 3/2*c) + 3*sin(d*x + c)^2 + 6*cos(d*x + c) + 3)*sin(5/2*d*x + 5/2*c) + 4*(2*(cos(d*x + c)^2 + s
in(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(d*x + c)^2 + 6*cos(d*x +
 c) + 3)*sin(3/2*d*x + 3/2*c) - 60*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c
)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (co
s(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2
+ 2*cos(d*x + c) + 1)*sin(5/2*d*x + 5/2*c)^2 + 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(5/
2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2
*c)^2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 20*((2*cos(d*x + c)^2 + 2*sin(d*x + c)^2
 + 5*cos(d*x + c) + 3)*cos(5/2*d*x + 5/2*c)^2 + 2*(2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*c
os(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*cos(3/2*
d*x + 3/2*c)^2 + (2*cos(d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(5/2*d*x + 5/2*c)^2 + 2*(2*cos(
d*x + c)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (2*cos(d*x + c
)^2 + 2*sin(d*x + c)^2 + 5*cos(d*x + c) + 3)*sin(3/2*d*x + 3/2*c)^2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c
))))*C*sqrt(a)/((sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*c
os(5/2*d*x + 5/2*c)^2 + 2*(sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sq
rt(2)*a)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*
sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3/2*d*x + 3/2*c)^2 + ((sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x +
c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(5/2*d*x + 5/2*c)^2 + 2*(sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*
sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (sqrt(2)*a*
cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3/2*d*x + 3/2*c)^2 + (sq
rt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sin(5/2*d*x + 5/2*c)
^2 + 2*(sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sin(5/2*d*
x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x +
 c) + sqrt(2)*a)*sin(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + (sqr
t(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sin(5/2*d*x + 5/2*c)^
2 + 2*(sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sin(5/2*d*x
 + 5/2*c)*sin(3/2*d*x + 3/2*c) + (sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x +
c) + sqrt(2)*a)*sin(3/2*d*x + 3/2*c)^2 + ((sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*c
os(d*x + c) + sqrt(2)*a)*cos(5/2*d*x + 5/2*c)^2 + 2*(sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*s
qrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(5/2*d*x + 5/2*c)*cos(3/2*d*x + 3/2*c) + (sqrt(2)*a*cos(d*x + c)^2 + sqr
t(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3/2*d*x + 3/2*c)^2 + (sqrt(2)*a*cos(d*x + c)
^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sin(5/2*d*x + 5/2*c)^2 + 2*(sqrt(2)*a*co
s(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sin(5/2*d*x + 5/2*c)*sin(3/2*d
*x + 3/2*c) + (sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sin
(3/2*d*x + 3/2*c)^2)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*((sqrt(2)*a*cos(d*x +
c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(5/2*d*x + 5/2*c)^2 + 2*(sqrt(2)*a*
cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(5/2*d*x + 5/2*c)*cos(3/2
*d*x + 3/2*c) + (sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*c
os(3/2*d*x + 3/2*c)^2 + (sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt
(2)*a)*sin(5/2*d*x + 5/2*c)^2 + 2*(sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c)^2 + 2*sqrt(2)*a*cos(d*x +
 c) + sqrt(2)*a)*sin(5/2*d*x + 5/2*c)*sin(3/2*d*x + 3/2*c) + (sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(d*x + c
)^2 + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sin(3/2*d*x + 3/2*c)^2)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(
3/2*d*x + 3/2*c)))))/d

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.16 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=-\frac {\frac {8 \, \sqrt {2} C \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{\sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3 \, \sqrt {2} {\left (A \sqrt {a} + C \sqrt {a}\right )} \log \left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {3 \, \sqrt {2} {\left (A \sqrt {a} + C \sqrt {a}\right )} \log \left (-\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{6 \, d} \]

[In]

integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-1/6*(8*sqrt(2)*C*sin(1/2*d*x + 1/2*c)^3/(sqrt(a)*sgn(cos(1/2*d*x + 1/2*c))) - 3*sqrt(2)*(A*sqrt(a) + C*sqrt(a
))*log(sin(1/2*d*x + 1/2*c) + 1)/(a*sgn(cos(1/2*d*x + 1/2*c))) + 3*sqrt(2)*(A*sqrt(a) + C*sqrt(a))*log(-sin(1/
2*d*x + 1/2*c) + 1)/(a*sgn(cos(1/2*d*x + 1/2*c))))/d

Mupad [B] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.28 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {2\,C\,\sin \left (c+d\,x\right )\,\left (a+a\,\cos \left (c+d\,x\right )\right )+3\,\sqrt {2}\,A\,a\,\sqrt {\frac {a+a\,\cos \left (c+d\,x\right )}{a}}\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |1\right )-4\,\sqrt {2}\,C\,a\,\sqrt {\frac {a+a\,\cos \left (c+d\,x\right )}{a}}\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |1\right )+3\,\sqrt {2}\,C\,a\,\sqrt {\frac {a+a\,\cos \left (c+d\,x\right )}{a}}\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |1\right )}{3\,a\,d\,\sqrt {a+a\,\cos \left (c+d\,x\right )}} \]

[In]

int((A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2),x)

[Out]

(2*C*sin(c + d*x)*(a + a*cos(c + d*x)) + 3*2^(1/2)*A*a*((a + a*cos(c + d*x))/a)^(1/2)*ellipticF(c/2 + (d*x)/2,
 1) - 4*2^(1/2)*C*a*((a + a*cos(c + d*x))/a)^(1/2)*ellipticE(c/2 + (d*x)/2, 1) + 3*2^(1/2)*C*a*((a + a*cos(c +
 d*x))/a)^(1/2)*ellipticF(c/2 + (d*x)/2, 1))/(3*a*d*(a + a*cos(c + d*x))^(1/2))

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