\(\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx\) [135]
Optimal result
Integrand size = 14, antiderivative size = 77 \[
\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}
\]
[Out]
1/2*sin(d*x+c)/d/(a+a*cos(d*x+c))^(3/2)+1/4*arctanh(1/2*sin(d*x+c)*a^(1/2)*2^(1/2)/(a+a*cos(d*x+c))^(1/2))/a^(
3/2)/d*2^(1/2)
Rubi [A] (verified)
Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of
steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2729, 2728, 212}
\[
\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}
\]
[In]
Int[(a + a*Cos[c + d*x])^(-3/2),x]
[Out]
ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*(
a + a*Cos[c + d*x])^(3/2))
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 2729
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d
*(2*n + 1))), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Rubi steps \begin{align*}
\text {integral}& = \frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{4 a} \\ & = \frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {\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 )}{2 a d} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.82
\[
\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (\text {arctanh}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos \left (\frac {1}{2} (c+d x)\right )+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d (a (1+\cos (c+d x)))^{3/2}}
\]
[In]
Integrate[(a + a*Cos[c + d*x])^(-3/2),x]
[Out]
(Cos[(c + d*x)/2]^2*(ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + Tan[(c + d*x)/2]))/(d*(a*(1 + Cos[c + d*x]))
^(3/2))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(62)=124\).
Time = 1.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.79
| | |
| method | result | size |
| | |
| default |
\(\frac {\sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\right )}{4 a^{\frac {5}{2}} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) |
\(138\) |
| | |
|
|
|
|
[In]
int(1/(a+cos(d*x+c)*a)^(3/2),x,method=_RETURNVERBOSE)
[Out]
1/4/a^(5/2)/cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2^(1/2)*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2
)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*a*cos(1/2*d*x+1/2*c)^2+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2))/sin(1/
2*d*x+1/2*c)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (62) = 124\).
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.99
\[
\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}
\]
[In]
integrate(1/(a+a*cos(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
1/8*(sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*cos(d*x +
c) + a)*sqrt(a)*sin(d*x + c) - 2*a*cos(d*x + c) - 3*a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 4*sqrt(a*cos(
d*x + c) + a)*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)
Sympy [F]
\[
\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (a \cos {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx
\]
[In]
integrate(1/(a+a*cos(d*x+c))**(3/2),x)
[Out]
Integral((a*cos(c + d*x) + a)**(-3/2), x)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 15721 vs. \(2 (62) = 124\).
Time = 1.14 (sec) , antiderivative size = 15721, normalized size of antiderivative = 204.17
\[
\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+a*cos(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
1/4*(32*(cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + cos(d*x + c)*sin(3/2*
d*x + 3/2*c) + cos(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(3*d*x + 3*c)^2 + 96*(cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*
c) + 3*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) - (3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - cos(3*d*x + 3*c)*si
n(3/2*d*x + 3/2*c) - 3*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 3*cos(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(4/3*ar
ctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 96*(cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) + 3*cos(3/2*d
*x + 3/2*c)*sin(2*d*x + 2*c) - (3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*
c) - 3*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 3*cos(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(2/3*arctan2(sin(3/2*d*
x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 - 32*(cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c)*sin(3/2*d*
x + 3/2*c) + cos(d*x + c)*sin(3/2*d*x + 3/2*c) + cos(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(3*d*x + 3*c)^2 + 32*(6
*cos(d*x + c) + 1)*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 96*cos(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) + 96*sin
(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) + 96*(cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) + 3*cos(3/2*d*x + 3/2*c)*sin(
2*d*x + 2*c) - (3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - 3*cos(2*d*x
+ 2*c)*sin(3/2*d*x + 3/2*c) + 3*cos(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(
3/2*d*x + 3/2*c)))^2 + 96*(cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) + 3*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) - (
3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - 3*cos(2*d*x + 2*c)*sin(3/2*
d*x + 3/2*c) + 3*cos(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)
))^2 + 32*(2*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 3*cos(2*d*x + 2*c)^2*sin(3/2*d*x + 3
/2*c) + 3*sin(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) + 2*(3*sin(3/2*d*x + 3/2*c)*sin(d*x + c) + cos(3/2*d*x + 3/2
*c))*sin(2*d*x + 2*c) + (3*cos(d*x + c)^2 + 3*sin(d*x + c)^2 + 2*cos(d*x + c))*sin(3/2*d*x + 3/2*c) + 2*cos(3/
2*d*x + 3/2*c)*sin(d*x + c))*cos(3*d*x + 3*c) - 4*(6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c)^2 + si
n(3*d*x + 3*c)^3 + (2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*c
os(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*
d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(3*d*x + 3*c) + 3*(2*(3*cos(2*d*x + 2*c) +
3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2
*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 +
9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(4/3*arct
an2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x +
3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2
+ 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*
d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2
*d*x + 3/2*c))))*cos(5/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 4*(8*cos(3*d*x + 3*c)^2*sin(3/
2*d*x + 3/2*c) - 72*cos(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) - 144*cos(2*d*x + 2*c)*cos(d*x + c)*sin(3/2*d*x +
3/2*c) - 8*sin(3*d*x + 3*c)^2*sin(3/2*d*x + 3/2*c) - 72*sin(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) - 16*(3*cos(3/
2*d*x + 3/2*c)*sin(2*d*x + 2*c) + 3*cos(3/2*d*x + 3/2*c)*sin(d*x + c) - sin(3/2*d*x + 3/2*c))*cos(3*d*x + 3*c)
- 48*(cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) + 3*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) - (3*cos(d*x + c) + 1)*
sin(3/2*d*x + 3/2*c) - cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - 3*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 3*cos
(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 16*(cos(3*d*x +
3*c)*cos(3/2*d*x + 3/2*c) + 3*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 3*sin(3/2*d*x + 3/2*c)*sin(d*x + c) + c
os(3/2*d*x + 3/2*c))*sin(3*d*x + 3*c) - 48*(3*sin(3/2*d*x + 3/2*c)*sin(d*x + c) + cos(3/2*d*x + 3/2*c))*sin(2*
d*x + 2*c) - 8*(9*cos(d*x + c)^2 + 9*sin(d*x + c)^2 - 1)*sin(3/2*d*x + 3/2*c) - 48*cos(3/2*d*x + 3/2*c)*sin(d*
x + c) + 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x +
c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(
3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2
+ 6*cos(d*x + c) + 1)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*cos(4/3*arctan2(sin(3/2*d
*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 4*(8*cos(3*d*x + 3*c)^2*sin(3/2*d*x + 3/2*c) - 72*cos(2*d*x + 2*c)^2*sin
(3/2*d*x + 3/2*c) - 144*cos(2*d*x + 2*c)*cos(d*x + c)*sin(3/2*d*x + 3/2*c) - 8*sin(3*d*x + 3*c)^2*sin(3/2*d*x
+ 3/2*c) - 72*sin(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) - 16*(3*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) + 3*cos(3/
2*d*x + 3/2*c)*sin(d*x + c) - sin(3/2*d*x + 3/2*c))*cos(3*d*x + 3*c) - 16*(cos(3*d*x + 3*c)*cos(3/2*d*x + 3/2*
c) + 3*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 3*sin(3/2*d*x + 3/2*c)*sin(d*x + c) + cos(3/2*d*x + 3/2*c))*sin
(3*d*x + 3*c) - 48*(3*sin(3/2*d*x + 3/2*c)*sin(d*x + c) + cos(3/2*d*x + 3/2*c))*sin(2*d*x + 2*c) - 8*(9*cos(d*
x + c)^2 + 9*sin(d*x + c)^2 - 1)*sin(3/2*d*x + 3/2*c) - 48*cos(3/2*d*x + 3/2*c)*sin(d*x + c) + 3*(2*(3*cos(2*d
*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c
) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x
+ 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*s
in(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x
+ 3/2*c))) + 4*(6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c)^2 + sin(3*d*x + 3*c)^3 + (2*(3*cos(2*d*x
+ 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c)
+ 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*
x + c)^2 + 6*cos(d*x + c) + 1)*sin(3*d*x + 3*c))*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))
+ (2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 2)*cos(3*d*x + 3*c)^3 + cos(3*d*x + 3*c)^4 + 6*(sin(2*d*x + 2*c) +
sin(d*x + c))*sin(3*d*x + 3*c)^3 + sin(3*d*x + 3*c)^4 + 3*(6*(cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 3*cos(2*d*
x + 2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(2*d*x + 2*c)^2 + 6*sin(2*d*x + 2*c)*sin(d*x + c) + 3*sin(d*x + c)^2 + 6*
cos(d*x + c) + 2)*cos(3*d*x + 3*c)^2 + 9*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3
*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d
*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*s
in(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c
)))^2 + 9*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x +
c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3
*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2
+ 6*cos(d*x + c) + 1)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + (2*(3*cos(2*d*x + 2*c)
+ 3*cos(d*x + c) + 2)*cos(3*d*x + 3*c) + 2*cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*co
s(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)
^2 + 6*cos(d*x + c) + 2)*sin(3*d*x + 3*c)^2 + 9*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c)
+ cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(
sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x +
2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x
+ 3/2*c)))^2 + 9*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos
(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c)
)*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x
+ c)^2 + 6*cos(d*x + c) + 1)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*(9*(2*cos(d*x
+ c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2
*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 9*cos(d*x + c) + 2)*cos(3*d*x + 3*c) + 6*(3*cos(d*x + c) + 1)*cos(2*d*x
+ 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(3*(2*cos(2*d*x + 2*c) + 2*cos(d*x + c) + 1)*cos(3*d*x +
3*c)^2 + cos(3*d*x + 3*c)^3 + (cos(3*d*x + 3*c) + 1)*sin(3*d*x + 3*c)^2 + 3*(2*(3*cos(d*x + c) + 2)*cos(2*d*x
+ 2*c) + 3*cos(2*d*x + 2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(2*d*x + 2*c)^2 + 6*sin(2*d*x + 2*c)*sin(d*x + c) + 3*
sin(d*x + c)^2 + 4*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x
+ 2*c)^2 + 9*cos(d*x + c)^2 + 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*
c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c)
+ sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x +
c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(
(sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c) + sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + 9*sin
(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(4/3*arctan2(si
n(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(3*(2*cos(2*d*x + 2*c) + 2*cos(d*x + c) + 1)*cos(3*d*x + 3*c)^2
+ cos(3*d*x + 3*c)^3 + (cos(3*d*x + 3*c) + 1)*sin(3*d*x + 3*c)^2 + 3*(2*(3*cos(d*x + c) + 2)*cos(2*d*x + 2*c)
+ 3*cos(2*d*x + 2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(2*d*x + 2*c)^2 + 6*sin(2*d*x + 2*c)*sin(d*x + c) + 3*sin(d*
x + c)^2 + 4*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)
^2 + 9*cos(d*x + c)^2 + 6*((sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c) + sin(2*d*x + 2*c) + sin(d*x + c
))*sin(3*d*x + 3*c) + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x +
c) + 1)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*((sin(2*d*x + 2*c) + sin(d*x + c))*c
os(3*d*x + 3*c)^2 + 2*(sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c) + sin(2*d*x + 2*c) + sin(d*x + c))*si
n(3*d*x + 3*c) + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*(6*(sin(2*d*x
+ 2*c) + sin(d*x + c))*sin(3*d*x + 3*c)^2 + sin(3*d*x + 3*c)^3 + (2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*
cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos
(d*x + c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)
*sin(3*d*x + 3*c) + 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(
3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x
+ c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin
(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*sin(4/3*arctan
2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c)^2 + s
in(3*d*x + 3*c)^3 + (2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*
cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2
*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(3*d*x + 3*c))*sin(2/3*arctan2(sin(3/2*d*
x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*cos(d*x + c) + 1)*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) - (2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 2)*cos(3*d*x + 3*c)^3 + cos
(3*d*x + 3*c)^4 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c)^3 + sin(3*d*x + 3*c)^4 + 3*(6*(cos(d*x
+ c) + 1)*cos(2*d*x + 2*c) + 3*cos(2*d*x + 2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(2*d*x + 2*c)^2 + 6*sin(2*d*x + 2*
c)*sin(d*x + c) + 3*sin(d*x + c)^2 + 6*cos(d*x + c) + 2)*cos(3*d*x + 3*c)^2 + 9*(2*(3*cos(2*d*x + 2*c) + 3*cos
(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x +
2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin
(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(4/3*arctan2(si
n(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 9*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c)
+ cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*
(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x
+ 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x
+ 3/2*c)))^2 + (2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 2)*cos(3*d*x + 3*c) + 2*cos(3*d*x + 3*c)^2 + 6*(3*co
s(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d
*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 2)*sin(3*d*x + 3*c)^2 + 9*(2*(3*cos(2*d*x + 2*c)
+ 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(
2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2
+ 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(4/3*arc
tan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 9*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x
+ 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)
^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin
(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(
3/2*d*x + 3/2*c)))^2 + 2*(9*(2*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 +
9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 9*cos(d*x + c) + 2)*cos(3*d*x + 3
*c) + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(3*(2*cos(2*d*x +
2*c) + 2*cos(d*x + c) + 1)*cos(3*d*x + 3*c)^2 + cos(3*d*x + 3*c)^3 + (cos(3*d*x + 3*c) + 1)*sin(3*d*x + 3*c)^2
+ 3*(2*(3*cos(d*x + c) + 2)*cos(2*d*x + 2*c) + 3*cos(2*d*x + 2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(2*d*x + 2*c)^2
+ 6*sin(2*d*x + 2*c)*sin(d*x + c) + 3*sin(d*x + c)^2 + 4*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + 6*(3*cos(d*x +
c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c)
+ 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 +
9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x +
2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(2/3*arctan2(sin(3/2*d*x
+ 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*((sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c) + sin(2*d*x + 2*c) +
sin(d*x + c))*sin(3*d*x + 3*c) + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 +
6*cos(d*x + c) + 1)*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(3*(2*cos(2*d*x + 2*c) +
2*cos(d*x + c) + 1)*cos(3*d*x + 3*c)^2 + cos(3*d*x + 3*c)^3 + (cos(3*d*x + 3*c) + 1)*sin(3*d*x + 3*c)^2 + 3*(
2*(3*cos(d*x + c) + 2)*cos(2*d*x + 2*c) + 3*cos(2*d*x + 2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(2*d*x + 2*c)^2 + 6*s
in(2*d*x + 2*c)*sin(d*x + c) + 3*sin(d*x + c)^2 + 4*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + 6*(3*cos(d*x + c) + 1
)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*((sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x
+ 3*c) + sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*
x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) +
6*((sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c)^2 + 2*(sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c
) + sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c
) + 9*sin(d*x + c)^2 + 6*(6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c)^2 + sin(3*d*x + 3*c)^3 + (2*(3*
cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*
x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) +
9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(3*d*x + 3*c) + 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(
3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x
+ c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 1
8*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c),
cos(3/2*d*x + 3/2*c))))*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(6*(sin(2*d*x + 2*c)
+ sin(d*x + c))*sin(3*d*x + 3*c)^2 + sin(3*d*x + 3*c)^3 + (2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*
d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x +
c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(3
*d*x + 3*c))*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*cos(d*x + c) + 1)*log(cos(1/3*ar
ctan2(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) - 32*(2*(3*cos(d*x + c) + 1)*cos
(2*d*x + 2*c)*cos(3/2*d*x + 3/2*c) + 3*cos(2*d*x + 2*c)^2*cos(3/2*d*x + 3/2*c) + 3*cos(3/2*d*x + 3/2*c)*sin(2*
d*x + 2*c)^2 + 2*(cos(2*d*x + 2*c)*cos(3/2*d*x + 3/2*c) + cos(3/2*d*x + 3/2*c)*cos(d*x + c) - sin(2*d*x + 2*c)
*sin(3/2*d*x + 3/2*c) - sin(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(3*d*x + 3*c) + (3*cos(d*x + c)^2 + 3*sin(d*x +
c)^2 + 2*cos(d*x + c))*cos(3/2*d*x + 3/2*c) + 2*(3*cos(3/2*d*x + 3/2*c)*sin(d*x + c) - sin(3/2*d*x + 3/2*c))*s
in(2*d*x + 2*c) - 2*sin(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(3*d*x + 3*c) + 32*(6*sin(3/2*d*x + 3/2*c)*sin(d*x +
c) + cos(3/2*d*x + 3/2*c))*sin(2*d*x + 2*c) + 32*(3*cos(d*x + c)^2 + 3*sin(d*x + c)^2 + cos(d*x + c))*sin(3/2
*d*x + 3/2*c) + 32*cos(3/2*d*x + 3/2*c)*sin(d*x + c) + 4*(3*(2*cos(2*d*x + 2*c) + 2*cos(d*x + c) + 1)*cos(3*d*
x + 3*c)^2 + cos(3*d*x + 3*c)^3 + (cos(3*d*x + 3*c) + 1)*sin(3*d*x + 3*c)^2 + 3*(2*(3*cos(d*x + c) + 2)*cos(2*
d*x + 2*c) + 3*cos(2*d*x + 2*c)^2 + 3*cos(d*x + c)^2 + 3*sin(2*d*x + 2*c)^2 + 6*sin(2*d*x + 2*c)*sin(d*x + c)
+ 3*sin(d*x + c)^2 + 4*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*
d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x
+ 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x +
2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*
x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) +
3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)
*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x +
3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos
(d*x + c) + 1)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*((sin(2*d*x + 2*c) + sin(d*x +
c))*cos(3*d*x + 3*c) + sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + 9*sin(2*d*x + 2*c)^2 + 18*sin(2*d*
x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(5/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d
*x + 3/2*c))) - 4*(8*cos(3*d*x + 3*c)^2*cos(3/2*d*x + 3/2*c) + 48*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c)*cos(3/
2*d*x + 3/2*c) + 72*cos(2*d*x + 2*c)^2*cos(3/2*d*x + 3/2*c) - 8*cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c)^2 + 72*c
os(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c)^2 + 144*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c)*sin(d*x + c) + 16*((3*cos(d
*x + c) + 1)*cos(3/2*d*x + 3/2*c) + 3*cos(2*d*x + 2*c)*cos(3/2*d*x + 3/2*c))*cos(3*d*x + 3*c) + 8*(9*cos(d*x +
c)^2 + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c) - 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c
) + 1)*cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2
+ 9*cos(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x +
2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(1/3*arctan2(sin(3/2*d*
x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 16*((3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) + cos(3*d*x + 3*c)*sin(3/2*
d*x + 3/2*c) + 3*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c))*sin(3*d*x + 3*c) - 48*(cos(3/2*d*x + 3/2*c)*sin(3*d*x
+ 3*c) + 3*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) - (3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - cos(3*d*x + 3*c
)*sin(3/2*d*x + 3/2*c) - 3*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 3*cos(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(2/
3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/
2*c))) - 4*(8*cos(3*d*x + 3*c)^2*cos(3/2*d*x + 3/2*c) + 48*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c)*cos(3/2*d*x +
3/2*c) + 72*cos(2*d*x + 2*c)^2*cos(3/2*d*x + 3/2*c) - 8*cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c)^2 + 72*cos(3/2*
d*x + 3/2*c)*sin(2*d*x + 2*c)^2 + 144*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c)*sin(d*x + c) + 16*((3*cos(d*x + c)
+ 1)*cos(3/2*d*x + 3/2*c) + 3*cos(2*d*x + 2*c)*cos(3/2*d*x + 3/2*c))*cos(3*d*x + 3*c) + 8*(9*cos(d*x + c)^2 +
9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c) - 3*(2*(3*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*
cos(3*d*x + 3*c) + cos(3*d*x + 3*c)^2 + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos
(d*x + c)^2 + 6*(sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + sin(3*d*x + 3*c)^2 + 9*sin(2*d*x + 2*c)^2
+ 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*cos(1/3*arctan2(sin(3/2*d*x + 3/2
*c), cos(3/2*d*x + 3/2*c))) + 16*((3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) + cos(3*d*x + 3*c)*sin(3/2*d*x + 3
/2*c) + 3*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c))*sin(3*d*x + 3*c))*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3
/2*d*x + 3/2*c))) - 4*(3*(2*cos(2*d*x + 2*c) + 2*cos(d*x + c) + 1)*cos(3*d*x + 3*c)^2 + cos(3*d*x + 3*c)^3 + (
cos(3*d*x + 3*c) + 1)*sin(3*d*x + 3*c)^2 + 3*(2*(3*cos(d*x + c) + 2)*cos(2*d*x + 2*c) + 3*cos(2*d*x + 2*c)^2 +
3*cos(d*x + c)^2 + 3*sin(2*d*x + 2*c)^2 + 6*sin(2*d*x + 2*c)*sin(d*x + c) + 3*sin(d*x + c)^2 + 4*cos(d*x + c)
+ 1)*cos(3*d*x + 3*c) + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c) + 9*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 6
*((sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c) + sin(2*d*x + 2*c) + sin(d*x + c))*sin(3*d*x + 3*c) + 9*s
in(2*d*x + 2*c)^2 + 18*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(1/3*arctan2(
sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))/((sqrt(2)*a*cos(3*d*x + 3*c)^4 + sqrt(2)*a*sin(3*d*x + 3*c)^4 +
2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) + 2*sqrt(2)*a)*cos(3*d*x + 3*c)^3 + 6*(sqrt(2)*a*si
n(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c))*sin(3*d*x + 3*c)^3 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos
(d*x + c)^2 + 9*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*
x + c)^2 + 3*(3*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 3*sqrt(2)*a*cos(d*x + c)^2 + 3*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 6
*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 3*sqrt(2)*a*sin(d*x + c)^2 + 6*sqrt(2)*a*cos(d*x + c) + 6*(sqrt(2)*
a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 2*sqrt(2)*a)*cos(3*d*x + 3*c)^2 + 9*(sqrt(2)*a*cos(3*d*x + 3*c)
^2 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(3*d*x + 3*c)^2 + 9*sqrt(2)*a*
sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x + c)^2 + 6*sqrt(2)*a*cos
(d*x + c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sq
rt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 6*(sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c))*s
in(3*d*x + 3*c) + sqrt(2)*a)*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 9*(sqrt(2)*a*cos
(3*d*x + 3*c)^2 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(3*d*x + 3*c)^2 +
9*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x + c)^2 + 6*
sqrt(2)*a*cos(d*x + c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3*d*x + 3
*c) + 6*(3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 6*(sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*si
n(d*x + c))*sin(3*d*x + 3*c) + sqrt(2)*a)*cos(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(3*d*x + 3*c)^2 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + 9*sqrt(2)*a*sin(2
*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x + c)^2 + 6*sqrt(2)*a*cos(d*x
+ c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) + 2*sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sqrt(
2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 2*sqrt(2)*a)*sin(3*d*x + 3*c)^2 + 9*(sqrt(2)*a*cos(3*d*x + 3
*c)^2 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(3*d*x + 3*c)^2 + 9*sqrt(2)
*a*sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x + c)^2 + 6*sqrt(2)*a*
cos(d*x + c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3
*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 6*(sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c)
)*sin(3*d*x + 3*c) + sqrt(2)*a)*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 9*(sqrt(2)*a*
cos(3*d*x + 3*c)^2 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(3*d*x + 3*c)^
2 + 9*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x + c)^2 +
6*sqrt(2)*a*cos(d*x + c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3*d*x
+ 3*c) + 6*(3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 6*(sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a
*sin(d*x + c))*sin(3*d*x + 3*c) + sqrt(2)*a)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 +
6*sqrt(2)*a*cos(d*x + c) + 2*(9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + 9*sqrt(2)*a*sin(2*
d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x + c)^2 + 9*sqrt(2)*a*cos(d*x +
c) + 9*(2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 2*sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sqrt(2)
*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 6*(sqrt(2)*a*cos(3*d*x + 3*c)^3 + 9*sqrt(2)*a*cos(2*d*x + 2*c)
^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + 9*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c)
+ 9*sqrt(2)*a*sin(d*x + c)^2 + 3*(2*sqrt(2)*a*cos(2*d*x + 2*c) + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3*d
*x + 3*c)^2 + (sqrt(2)*a*cos(3*d*x + 3*c) + sqrt(2)*a)*sin(3*d*x + 3*c)^2 + 6*sqrt(2)*a*cos(d*x + c) + 3*(3*sq
rt(2)*a*cos(2*d*x + 2*c)^2 + 3*sqrt(2)*a*cos(d*x + c)^2 + 3*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 6*sqrt(2)*a*sin(2*d
*x + 2*c)*sin(d*x + c) + 3*sqrt(2)*a*sin(d*x + c)^2 + 4*sqrt(2)*a*cos(d*x + c) + 2*(3*sqrt(2)*a*cos(d*x + c) +
2*sqrt(2)*a)*cos(2*d*x + 2*c) + sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*
d*x + 2*c) + 3*(sqrt(2)*a*cos(3*d*x + 3*c)^2 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + s
qrt(2)*a*sin(3*d*x + 3*c)^2 + 9*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*
sqrt(2)*a*sin(d*x + c)^2 + 6*sqrt(2)*a*cos(d*x + c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x +
c) + sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 6*(sqrt(2)*a*si
n(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c))*sin(3*d*x + 3*c) + sqrt(2)*a)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), c
os(3/2*d*x + 3/2*c))) + 6*(sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c) + (sqrt(2)*a*sin(2*d*x + 2*c) +
sqrt(2)*a*sin(d*x + c))*cos(3*d*x + 3*c))*sin(3*d*x + 3*c) + sqrt(2)*a)*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c),
cos(3/2*d*x + 3/2*c))) + 6*(sqrt(2)*a*cos(3*d*x + 3*c)^3 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d
*x + c)^2 + 9*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x
+ c)^2 + 3*(2*sqrt(2)*a*cos(2*d*x + 2*c) + 2*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3*d*x + 3*c)^2 + (sqrt(2)
*a*cos(3*d*x + 3*c) + sqrt(2)*a)*sin(3*d*x + 3*c)^2 + 6*sqrt(2)*a*cos(d*x + c) + 3*(3*sqrt(2)*a*cos(2*d*x + 2*
c)^2 + 3*sqrt(2)*a*cos(d*x + c)^2 + 3*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 6*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c)
+ 3*sqrt(2)*a*sin(d*x + c)^2 + 4*sqrt(2)*a*cos(d*x + c) + 2*(3*sqrt(2)*a*cos(d*x + c) + 2*sqrt(2)*a)*cos(2*d*
x + 2*c) + sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 6*(sqrt(2
)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c) + (sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c))*cos(3*d*
x + 3*c))*sin(3*d*x + 3*c) + sqrt(2)*a)*cos(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*((sqr
t(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c))*cos(3*d*x + 3*c)^2 + sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*
sin(d*x + c) + 2*(sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c))*cos(3*d*x + 3*c))*sin(3*d*x + 3*c) + 6*
(sqrt(2)*a*sin(3*d*x + 3*c)^3 + 6*(sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c))*sin(3*d*x + 3*c)^2 + (
sqrt(2)*a*cos(3*d*x + 3*c)^2 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + 9*sqrt(2)*a*sin(2
*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x + c)^2 + 6*sqrt(2)*a*cos(d*x
+ c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sqrt(2)
*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(3*d*x + 3*c) + 3*(sqrt(2)*a*cos(3*d*x + 3*c)^2
+ 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*sqrt(2)*a*cos(d*x + c)^2 + sqrt(2)*a*sin(3*d*x + 3*c)^2 + 9*sqrt(2)*a*sin
(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt(2)*a*sin(d*x + c)^2 + 6*sqrt(2)*a*cos(d*
x + c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sqrt(
2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + 6*(sqrt(2)*a*sin(2*d*x + 2*c) + sqrt(2)*a*sin(d*x + c))*sin(
3*d*x + 3*c) + sqrt(2)*a)*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*sin(4/3*arctan2(sin(3/
2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(sqrt(2)*a*sin(3*d*x + 3*c)^3 + 6*(sqrt(2)*a*sin(2*d*x + 2*c) + sqr
t(2)*a*sin(d*x + c))*sin(3*d*x + 3*c)^2 + (sqrt(2)*a*cos(3*d*x + 3*c)^2 + 9*sqrt(2)*a*cos(2*d*x + 2*c)^2 + 9*s
qrt(2)*a*cos(d*x + c)^2 + 9*sqrt(2)*a*sin(2*d*x + 2*c)^2 + 18*sqrt(2)*a*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sqrt
(2)*a*sin(d*x + c)^2 + 6*sqrt(2)*a*cos(d*x + c) + 2*(3*sqrt(2)*a*cos(2*d*x + 2*c) + 3*sqrt(2)*a*cos(d*x + c) +
sqrt(2)*a)*cos(3*d*x + 3*c) + 6*(3*sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(3*d*
x + 3*c))*sin(2/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + sqrt(2)*a)*sqrt(a)*d)
Giac [A] (verification not implemented)
none
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.45
\[
\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (\frac {\log \left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {\log \left (-\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{8 \, \sqrt {a} d}
\]
[In]
integrate(1/(a+a*cos(d*x+c))^(3/2),x, algorithm="giac")
[Out]
1/8*sqrt(2)*(log(sin(1/2*d*x + 1/2*c) + 1)/(a*sgn(cos(1/2*d*x + 1/2*c))) - log(-sin(1/2*d*x + 1/2*c) + 1)/(a*s
gn(cos(1/2*d*x + 1/2*c))) - 2*sin(1/2*d*x + 1/2*c)/((sin(1/2*d*x + 1/2*c)^2 - 1)*a*sgn(cos(1/2*d*x + 1/2*c))))
/(sqrt(a)*d)
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{(a+a \cos (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x
\]
[In]
int(1/(a + a*cos(c + d*x))^(3/2),x)
[Out]
int(1/(a + a*cos(c + d*x))^(3/2), x)