\(\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx\) [124]
Optimal result
Integrand size = 23, antiderivative size = 104 \[
\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}-\frac {4 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d}
\]
[Out]
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*sin(d*x+c)/d/(a+a*cos(d*x
+c))^(1/2)+2/3*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/a/d
Rubi [A] (verified)
Time = 0.16 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2838, 2830, 2728, 212}
\[
\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 a d}-\frac {4 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}
\]
[In]
Int[Cos[c + d*x]^2/Sqrt[a + a*Cos[c + d*x]],x]
[Out]
(Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*Sin[c + d*x])/(3
*d*Sqrt[a + a*Cos[c + d*x]]) + (2*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 2838
Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-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*(b*(m + 1) -
a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Rubi steps \begin{align*}
\text {integral}& = \frac {2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d}+\frac {2 \int \frac {\frac {a}{2}-a \cos (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{3 a} \\ & = -\frac {4 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d}+\int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx \\ & = -\frac {4 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d}-\frac {2 \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} \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 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 a d} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.81
\[
\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\left (\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\cos (c+d x)}}{\sqrt {2}}\right )-\frac {2}{3} (1-\cos (c+d x))^{3/2}\right ) \sin (c+d x)}{d \sqrt {1-\cos (c+d x)} \sqrt {a (1+\cos (c+d x))}}
\]
[In]
Integrate[Cos[c + d*x]^2/Sqrt[a + a*Cos[c + d*x]],x]
[Out]
((Sqrt[2]*ArcTanh[Sqrt[1 - Cos[c + d*x]]/Sqrt[2]] - (2*(1 - Cos[c + d*x])^(3/2))/3)*Sin[c + d*x])/(d*Sqrt[1 -
Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])])
Maple [A] (verified)
Time = 1.41 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.27
| | |
| method | result | size |
| | |
| default |
\(\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-4 \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \right )}{3 a^{\frac {3}{2}} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) |
\(132\) |
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[In]
int(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*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*sin(1
/2*d*x+1/2*c)^2+3*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 = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.26
\[
\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {4 \, \sqrt {a \cos \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + \frac {3 \, \sqrt {2} {\left (a \cos \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - \frac {2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt {a}}}{6 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}}
\]
[In]
integrate(cos(d*x+c)^2/(a+a*cos(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
1/6*(4*sqrt(a*cos(d*x + c) + a)*(cos(d*x + c) - 1)*sin(d*x + c) + 3*sqrt(2)*(a*cos(d*x + 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*co
s(d*x + c) + 1))/sqrt(a))/(a*d*cos(d*x + c) + a*d)
Sympy [F]
\[
\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )}}{\sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}}\, dx
\]
[In]
integrate(cos(d*x+c)**2/(a+a*cos(d*x+c))**(1/2),x)
[Out]
Integral(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. 19437 vs. \(2 (87) = 174\).
Time = 0.66 (sec) , antiderivative size = 19437, normalized size of antiderivative = 186.89
\[
\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)^2/(a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
1/60*(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*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 + 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))*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*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) + 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))*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*(c
os(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*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) + sin(5/2*d*x + 5/2*c)^2*sin(d*x + c) + 2*si
n(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(si
n(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 + s
in(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 + (l
og(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 + 3
0*(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*(l
og(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*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 + 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*si
n(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))*s
in(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)*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*(l
og(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*(c
os(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)*co
s(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*co
s(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*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(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 + 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) + 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*si
n(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))*c
os(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*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 + 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(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) - log(cos(1/2*d*x + 1/2*c)^2 + si
n(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)*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*si
n(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))*c
os(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) - 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) + (c
os(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*s
in(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))) + 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) + 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))) + 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*c
os(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*co
s(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(co
s(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*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)*co
s(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)*si
n(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*c
os(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(1/3*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*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*(co
s(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 + s
in(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), cos
(d*x + c)))^2 + 2*sin(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)*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)*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)*s
in(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*arc
tan2(sin(d*x + c), cos(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 + 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) - 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) + (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))) - 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)*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))))*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)*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 + ((sqrt(2)*a*cos(d*x + c)^2 + s
qrt(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 + (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)*s
in(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)*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 + (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)*si
n(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) + sqr
t(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*s
qrt(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*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*s
in(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*c
os(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 + (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)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*((s
qrt(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 + (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 + sq
rt(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 = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.99
\[
\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=-\frac {\frac {8 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{\sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3 \, \sqrt {2} \log \left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {3 \, \sqrt {2} \log \left (-\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{6 \, d}
\]
[In]
integrate(cos(d*x+c)^2/(a+a*cos(d*x+c))^(1/2),x, algorithm="giac")
[Out]
-1/6*(8*sqrt(2)*sin(1/2*d*x + 1/2*c)^3/(sqrt(a)*sgn(cos(1/2*d*x + 1/2*c))) - 3*sqrt(2)*log(sin(1/2*d*x + 1/2*c
) + 1)/(sqrt(a)*sgn(cos(1/2*d*x + 1/2*c))) + 3*sqrt(2)*log(-sin(1/2*d*x + 1/2*c) + 1)/(sqrt(a)*sgn(cos(1/2*d*x
+ 1/2*c))))/d
Mupad [B] (verification not implemented)
Time = 14.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93
\[
\int \frac {\cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {2\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{3\,a\,d}-\frac {2\,\left (4\,a^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |1\right )-3\,a^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |1\right )\right )\,\sqrt {\frac {a+a\,\cos \left (c+d\,x\right )}{2\,a}}}{3\,a^2\,d\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}
\]
[In]
int(cos(c + d*x)^2/(a + a*cos(c + d*x))^(1/2),x)
[Out]
(2*sin(c + d*x)*(a + a*cos(c + d*x))^(1/2))/(3*a*d) - (2*(4*a^2*ellipticE(c/2 + (d*x)/2, 1) - 3*a^2*ellipticF(
c/2 + (d*x)/2, 1))*((a + a*cos(c + d*x))/(2*a))^(1/2))/(3*a^2*d*(a + a*cos(c + d*x))^(1/2))