\(\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx\) [141]
Optimal result
Integrand size = 23, antiderivative size = 222 \[
\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx=-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^3 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {2 a^4 \tan ^7(c+d x)}{d (a+a \sec (c+d x))^{7/2}}+\frac {10 a^5 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {2 a^6 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}
\]
[Out]
-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))*a^(1/2)/d+2*a*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-2/3*a^2
*tan(d*x+c)^3/d/(a+a*sec(d*x+c))^(3/2)+2/5*a^3*tan(d*x+c)^5/d/(a+a*sec(d*x+c))^(5/2)+2*a^4*tan(d*x+c)^7/d/(a+a
*sec(d*x+c))^(7/2)+10/9*a^5*tan(d*x+c)^9/d/(a+a*sec(d*x+c))^(9/2)+2/11*a^6*tan(d*x+c)^11/d/(a+a*sec(d*x+c))^(1
1/2)
Rubi [A] (verified)
Time = 0.14 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3972, 472, 209}
\[
\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx=\frac {2 a^6 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac {10 a^5 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac {2 a^4 \tan ^7(c+d x)}{d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}}
\]
[In]
Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^6,x]
[Out]
(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[
c + d*x]]) - (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[
c + d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(d*(a + a*Sec[c + d*x])^(7/2)) + (10*a^5*Tan[c + d*x]^9)/(9*d*(a + a
*Sec[c + d*x])^(9/2)) + (2*a^6*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2))
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 472
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Rule 3972
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[-2*(a^(m/2 +
n + 1/2)/d), Subst[Int[x^m*((2 + a*x^2)^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]
Rubi steps \begin{align*}
\text {integral}& = -\frac {\left (2 a^4\right ) \text {Subst}\left (\int \frac {x^6 \left (2+a x^2\right )^3}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {\left (2 a^4\right ) \text {Subst}\left (\int \left (\frac {1}{a^3}-\frac {x^2}{a^2}+\frac {x^4}{a}+7 x^6+5 a x^8+a^2 x^{10}-\frac {1}{a^3 \left (1+a x^2\right )}\right ) \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = \frac {2 a \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^3 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {2 a^4 \tan ^7(c+d x)}{d (a+a \sec (c+d x))^{7/2}}+\frac {10 a^5 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {2 a^6 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^3 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {2 a^4 \tan ^7(c+d x)}{d (a+a \sec (c+d x))^{7/2}}+\frac {10 a^5 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {2 a^6 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}} \\
\end{align*}
Mathematica [A] (verified)
Time = 7.81 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.60
\[
\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx=-\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \sec ^5(c+d x) \sqrt {a (1+\sec (c+d x))} \left (3960 \sqrt {2} \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^{\frac {11}{2}}(c+d x)+792 \sin \left (\frac {1}{2} (c+d x)\right )-1386 \sin \left (\frac {3}{2} (c+d x)\right )+495 \sin \left (\frac {5}{2} (c+d x)\right )-616 \sin \left (\frac {7}{2} (c+d x)\right )-247 \sin \left (\frac {11}{2} (c+d x)\right )\right )}{3960 d}
\]
[In]
Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^6,x]
[Out]
-1/3960*(Sec[(c + d*x)/2]*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*(3960*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)
/2]]*Cos[c + d*x]^(11/2) + 792*Sin[(c + d*x)/2] - 1386*Sin[(3*(c + d*x))/2] + 495*Sin[(5*(c + d*x))/2] - 616*S
in[(7*(c + d*x))/2] - 247*Sin[(11*(c + d*x))/2]))/d
Maple [A] (verified)
Time = 5.35 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.05
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| method | result | size |
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| default |
\(\frac {2 \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (-495 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-495 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+494 \sin \left (d x +c \right )+247 \tan \left (d x +c \right )-186 \sec \left (d x +c \right ) \tan \left (d x +c \right )-155 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}+50 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}+45 \sec \left (d x +c \right )^{4} \tan \left (d x +c \right )\right )}{495 d \left (\cos \left (d x +c \right )+1\right )}\) |
\(233\) |
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[In]
int((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^6,x,method=_RETURNVERBOSE)
[Out]
2/495/d*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)*(-495*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(c
os(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)-495*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin
(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+494*sin(d*x+c)+247*tan(d*x+c)-186*sec(d*x+c)*tan(d*
x+c)-155*tan(d*x+c)*sec(d*x+c)^2+50*tan(d*x+c)*sec(d*x+c)^3+45*sec(d*x+c)^4*tan(d*x+c))
Fricas [A] (verification not implemented)
none
Time = 0.34 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.67
\[
\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx=\left [\frac {495 \, {\left (\cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (494 \, \cos \left (d x + c\right )^{5} + 247 \, \cos \left (d x + c\right )^{4} - 186 \, \cos \left (d x + c\right )^{3} - 155 \, \cos \left (d x + c\right )^{2} + 50 \, \cos \left (d x + c\right ) + 45\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{495 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}, \frac {2 \, {\left (495 \, {\left (\cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (494 \, \cos \left (d x + c\right )^{5} + 247 \, \cos \left (d x + c\right )^{4} - 186 \, \cos \left (d x + c\right )^{3} - 155 \, \cos \left (d x + c\right )^{2} + 50 \, \cos \left (d x + c\right ) + 45\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{495 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}\right ]
\]
[In]
integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^6,x, algorithm="fricas")
[Out]
[1/495*(495*(cos(d*x + c)^6 + cos(d*x + c)^5)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x +
c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(494*cos(d*x + c
)^5 + 247*cos(d*x + c)^4 - 186*cos(d*x + c)^3 - 155*cos(d*x + c)^2 + 50*cos(d*x + c) + 45)*sqrt((a*cos(d*x + c
) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5), 2/495*(495*(cos(d*x + c)^6 + cos(d*x
+ c)^5)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (494*co
s(d*x + c)^5 + 247*cos(d*x + c)^4 - 186*cos(d*x + c)^3 - 155*cos(d*x + c)^2 + 50*cos(d*x + c) + 45)*sqrt((a*co
s(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)]
Sympy [F]
\[
\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \tan ^{6}{\left (c + d x \right )}\, dx
\]
[In]
integrate((a+a*sec(d*x+c))**(1/2)*tan(d*x+c)**6,x)
[Out]
Integral(sqrt(a*(sec(c + d*x) + 1))*tan(c + d*x)**6, x)
Maxima [F]
\[
\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \tan \left (d x + c\right )^{6} \,d x }
\]
[In]
integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^6,x, algorithm="maxima")
[Out]
1/990*(495*((cos(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x + 2*c)^2 + 2*cos(2*
d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*arctan2((cos(2*d*x + 2*c)^
2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)
), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c) + 1)) + 1) - (cos(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x +
2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*arctan2(
(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos
(2*d*x + 2*c) + 1)), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(
sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - 1) - 2*(d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x
+ 2*c)^3 + 6*d*cos(2*d*x + 2*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 +
4*d*cos(2*d*x + 2*c) + d)*integrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x +
2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos
(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c)
+ 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*
d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*
cos(13/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2
*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*co
s(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) -
6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x
+ 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(13/2*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*s
in(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*
d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos
(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2
*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d
*x + 2*c))*cos(13/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*co
s(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2
*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14
*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c)
+ 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x
+ 2*c) + sin(2*d*x + 2*c)^2)*sin(13/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c) + 1)))/(((2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*
cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*c
os(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*
d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + c
os(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c)
+ cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos
(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*
d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6
*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) +
20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin
(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin
(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*
sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*
sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/
2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + (2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*c
os(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x
+ 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(
2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6
*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c)
+ 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) +
cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(
2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15
*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*
sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12
*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) +
sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c)
+ sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*si
n(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2
*d*x + 2*c)^2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2
*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)), x) + 12*(d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x +
2*c)^3 + 6*d*cos(2*d*x + 2*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 4*
d*cos(2*d*x + 2*c) + d)*integrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*
c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2
*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) +
6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*
x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*co
s(11/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c
)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(
2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*
cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x +
2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(11/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin
(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*
x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(1
4*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c
) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x
+ 2*c))*cos(11/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(
12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c
) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d
*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) +
20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x +
2*c) + sin(2*d*x + 2*c)^2)*sin(11/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c) + 1)))/(((2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*co
s(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos
(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*
x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos
(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) +
cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6
*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*
x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*s
in(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20
*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(1
2*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(1
0*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*si
n(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*si
n(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + (2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos
(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x +
14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*
d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*c
os(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) +
6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + co
s(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*
d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*s
in(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*si
n(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d
*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + si
n(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) +
sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(
6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d
*x + 2*c)^2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c
)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)), x) - 30*(d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x + 2
*c)^3 + 6*d*cos(2*d*x + 2*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 4*d*
cos(2*d*x + 2*c) + d)*integrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c)
+ 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d
*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6
*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x
+ 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(
9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*s
in(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d
*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos
(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*
c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(9/2*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(14*
d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x +
2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*
x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) -
20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2
*c))*cos(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*
x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 1
5*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x +
14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*s
in(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c)
+ sin(2*d*x + 2*c)^2)*sin(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c) + 1)))/(((2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*
x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*
x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12
*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x
+ 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*
d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x +
6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*
c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d
*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8
*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x
+ 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x
+ 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*
x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*
x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*arctan
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + (2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x
+ 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)
^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x +
2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d
*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(
4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*
x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x +
2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d
*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d
*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 1
2*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*
x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2
*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x
+ 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2
*c)^2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 +
2*cos(2*d*x + 2*c) + 1)^(1/4)), x) + 40*(d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x + 2*c)^3
+ 6*d*cos(2*d*x + 2*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 4*d*cos(2*
d*x + 2*c) + d)*integrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*
cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2
*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(1
2*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c)
+ 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(7/2*ar
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*
d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2
*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*
x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 1
5*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(7/2*arctan2(sin(2*d*x + 2*c), c
os(2*d*x + 2*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(14*d*x +
14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*s
in(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14
*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos
(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*c
os(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12
*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(
6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*
sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d
*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(
2*d*x + 2*c)^2)*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*
d*x + 2*c) + 1)))/(((2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*
c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10
*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) +
36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c
))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x +
2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c)
+ 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 +
2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4
*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x +
8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c
)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c
) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*
c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*
c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*arctan2(sin(
2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + (2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c
) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 1
2*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*
cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4
*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x
+ 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*
c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) +
cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6
*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 1
0*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) +
36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*
c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x +
2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c)
+ 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)
*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(
2*d*x + 2*c) + 1)^(1/4)), x) - 30*(d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x + 2*c)^3 + 6*d*
cos(2*d*x + 2*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 4*d*cos(2*d*x +
2*c) + d)*integrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10
*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) +
6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x
+ 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*
sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(5/2*arctan2(
sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x +
12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*si
n(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12
*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(
6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d
*x + 2*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(14*d*x + 14*c)
+ 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d
*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*si
n(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x
+ 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(5/2
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*co
s(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x
+ 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*
d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8
*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x
+ 2*c)^2)*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c) + 1)))/(((2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6
*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) +
20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos
(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos
(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*
cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*
cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*
sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) +
sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c)
+ 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 +
30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 22
5*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 4
00*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 +
36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c) + 1))^2 + (2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15
*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*
cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12
*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) +
cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c)
+ cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*co
s(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2
*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) +
6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) +
20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*si
n(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*si
n(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))
*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225
*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(1
/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x
+ 2*c) + 1)^(1/4)), x) + 12*(d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x + 2*c)^3 + 6*d*cos(2*
d*x + 2*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 4*d*cos(2*d*x + 2*c) +
d)*integrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x +
10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(
4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c
)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*
d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(3/2*arctan2(sin(2*
d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c)
+ 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*
x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*si
n(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x
+ 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2
*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*co
s(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8
*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*
x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c
)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(3/2*arcta
n2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*
x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)
*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x +
2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*si
n(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)
^2)*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) +
1)))/(((2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4
*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos
(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*
x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*
x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*
d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*
d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12
*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*
d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*s
in(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20
*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(
10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin
(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*si
n(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*arctan2(sin(2*d*x + 2*c)
, cos(2*d*x + 2*c) + 1))^2 + (2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6
*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10
*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x +
12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*
d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos
(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*
x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x +
2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(
4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*si
n(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d
*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d
*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8
*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6
*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(1/2*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)
+ 1)^(1/4)), x) - 2*(d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x + 2*c)^3 + 6*d*cos(2*d*x + 2
*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 4*d*cos(2*d*x + 2*c) + d)*int
egrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*
cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x +
4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2
*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6
*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*co
s(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c
) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x
+ 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*
sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*
cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x
+ 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 1
5*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c
) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2
*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(1/2*arctan2(sin(
2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c
) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*
d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) +
6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x
+ 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin
(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(
((2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x +
4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x
+ 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*
c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*
c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8
*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6
*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x +
12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2
*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*
x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*
d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x
+ 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x
+ 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x
+ 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2
*d*x + 2*c) + 1))^2 + (2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x +
6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x +
10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c)
+ 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2
*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x
+ 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c
) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2
+ 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x +
4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x
+ 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12
*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10
*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x +
8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x +
6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(1/2*arctan2(si
n(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(
1/4)), x))*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(3/4)*sqrt(a) - 8*(11*(45*sin(8*
d*x + 8*c) + 72*sin(6*d*x + 6*c) + 126*sin(4*d*x + 4*c) + 56*sin(2*d*x + 2*c))*cos(11/2*arctan2(sin(2*d*x + 2*
c), cos(2*d*x + 2*c) + 1)) - (495*cos(8*d*x + 8*c) + 792*cos(6*d*x + 6*c) + 1386*cos(4*d*x + 4*c) + 616*cos(2*
d*x + 2*c) + 247)*sin(11/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))*sqrt(a))/((d*cos(2*d*x + 2*c)^4 +
d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x + 2*c)^3 + 6*d*cos(2*d*x + 2*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*
d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 4*d*cos(2*d*x + 2*c) + d)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*co
s(2*d*x + 2*c) + 1)^(3/4))
Giac [F]
\[
\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \tan \left (d x + c\right )^{6} \,d x }
\]
[In]
integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^6,x, algorithm="giac")
[Out]
sage0*x
Mupad [F(-1)]
Timed out. \[
\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^6\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x
\]
[In]
int(tan(c + d*x)^6*(a + a/cos(c + d*x))^(1/2),x)
[Out]
int(tan(c + d*x)^6*(a + a/cos(c + d*x))^(1/2), x)