\(\int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx\) [201]

Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac {2 \tan (c+d x)}{a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {2 \tan ^3(c+d x)}{3 a d (a+a \sec (c+d x))^{3/2}}+\frac {2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}} \]

[Out]

-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(5/2)/d+2*tan(d*x+c)/a^2/d/(a+a*sec(d*x+c))^(1/2)-2/3*t
an(d*x+c)^3/a/d/(a+a*sec(d*x+c))^(3/2)+2/5*tan(d*x+c)^5/d/(a+a*sec(d*x+c))^(5/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 127, 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, 308, 209} \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{5/2} d}+\frac {2 \tan (c+d x)}{a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 \tan ^3(c+d x)}{3 a d (a \sec (c+d x)+a)^{3/2}} \]

[In]

Int[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]

[Out]

(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + (2*Tan[c + d*x])/(a^2*d*Sqrt[a + a*
Sec[c + d*x]]) - (2*Tan[c + d*x]^3)/(3*a*d*(a + a*Sec[c + d*x])^(3/2)) + (2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c
+ d*x])^(5/2))

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 {(2 a) \text {Subst}\left (\int \frac {x^6}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {(2 a) \text {Subst}\left (\int \left (\frac {1}{a^3}-\frac {x^2}{a^2}+\frac {x^4}{a}-\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 \tan (c+d x)}{a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {2 \tan ^3(c+d x)}{3 a d (a+a \sec (c+d x))^{3/2}}+\frac {2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {2 \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 )}{a^2 d} \\ & = -\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac {2 \tan (c+d x)}{a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {2 \tan ^3(c+d x)}{3 a d (a+a \sec (c+d x))^{3/2}}+\frac {2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(447\) vs. \(2(127)=254\).

Time = 6.09 (sec) , antiderivative size = 447, normalized size of antiderivative = 3.52 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\sqrt {2} \cot ^8\left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{9/2} \sqrt {1+\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (-1+\frac {2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^3 \left (\frac {\sqrt {2} \arcsin \left (\frac {\sqrt {2} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}}\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {1+\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {1-\frac {2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}}}+\frac {8 \tan ^6\left (\frac {1}{2} (c+d x)\right )}{5 \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-1+\frac {2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^3}+\frac {4 \tan ^4\left (\frac {1}{2} (c+d x)\right )}{3 \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-1+\frac {2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^2}+\frac {2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (-1+\frac {2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )}\right ) \tan ^7(c+d x)}{d (a (1+\sec (c+d x)))^{5/2} \left (1-\frac {2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^{5/2}} \]

[In]

Integrate[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]

[Out]

(Sqrt[2]*Cot[(c + d*x)/2]^8*((1 + Sec[c + d*x])^(-1))^(9/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(-1 + (2*Tan[(c + d*x
)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^3*((Sqrt[2]*ArcSin[(Sqrt[2]*Tan[(c + d*x)/2])/Sqrt[1 + Tan[(c + d*x)/2]^2]]*
Tan[(c + d*x)/2])/(Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 - (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (
8*Tan[(c + d*x)/2]^6)/(5*(1 + Tan[(c + d*x)/2]^2)^3*(-1 + (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^3)
+ (4*Tan[(c + d*x)/2]^4)/(3*(1 + Tan[(c + d*x)/2]^2)^2*(-1 + (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^
2) + (2*Tan[(c + d*x)/2]^2)/((1 + Tan[(c + d*x)/2]^2)*(-1 + (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))))
*Tan[c + d*x]^7)/(d*(a*(1 + Sec[c + d*x]))^(5/2)*(1 - (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^(5/2))

Maple [A] (warning: unable to verify)

Time = 3.70 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.66

method	result	size
default	\(-\frac {\left (15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {5}{2}}-74 \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}+80 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-30 \csc \left (d x +c \right )+30 \cot \left (d x +c \right )\right ) \sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}{15 d \,a^{3} \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+1\right )^{2} \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )-1\right )^{2}}\)	\(211\)

[In]

int(tan(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/15/d/a^3*(15*2^(1/2)*arctanh(2^(1/2)/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2)*(-cot(d*x+c)+csc(d*x+c)))*((1-
cos(d*x+c))^2*csc(d*x+c)^2-1)^(5/2)-74*(1-cos(d*x+c))^5*csc(d*x+c)^5+80*(1-cos(d*x+c))^3*csc(d*x+c)^3-30*csc(d
*x+c)+30*cot(d*x+c))*(-2*a/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)/(csc(d*x+c)-cot(d*x+c)+1)^2/(-cot(d*x+c)+c
sc(d*x+c)-1)^2

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.54 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\left [-\frac {15 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) - 2 \, {\left (23 \, \cos \left (d x + c\right )^{2} - 11 \, \cos \left (d x + c\right ) + 3\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}}, \frac {2 \, {\left (15 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (23 \, \cos \left (d x + c\right )^{2} - 11 \, \cos \left (d x + c\right ) + 3\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}}\right ] \]

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[-1/15*(15*(cos(d*x + c)^3 + cos(d*x + c)^2)*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*(23*cos(d*x + c)^
2 - 11*cos(d*x + c) + 3)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + a^3*d*c
os(d*x + c)^2), 2/15*(15*(cos(d*x + c)^3 + cos(d*x + c)^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x +
c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (23*cos(d*x + c)^2 - 11*cos(d*x + c) + 3)*sqrt((a*cos(d*x + c) + a)
/cos(d*x + c))*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2)]

Sympy [F]

\[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\tan ^{6}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(tan(d*x+c)**6/(a+a*sec(d*x+c))**(5/2),x)

[Out]

Integral(tan(c + d*x)**6/(a*(sec(c + d*x) + 1))**(5/2), x)

Maxima [F]

\[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{6}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-1/30*(20*(9*sin(4*d*x + 4*c) + 16*sin(2*d*x + 2*c))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))
- 4*(45*cos(4*d*x + 4*c) + 80*cos(2*d*x + 2*c) + 23)*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))
- 15*((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*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)^2 + sin(2*d*x + 2*c)^2 + 2*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*(a^3*d*cos(2*d*x + 2*c)^2 + a^3*d*sin(2*d*x + 2*c)^2 + 2*a^3*d*cos(2*d*x + 2
*c) + a^3*d)*integrate(-(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(2*d*x
+ 2*c)^4 + sin(2*d*x + 2*c)^4 + 2*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 + (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2
*c))*cos(14*d*x + 14*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 +
cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2
*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2
*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*
sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2
*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(4*d*x + 4*c) + cos(2*d*x + 2*c)^
2 + (sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) +
6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) +
15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) +
20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 15
*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(s
in(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c))*cos(7/2*
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - ((sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)
+ 1)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)
+ 1)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)
+ 1)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)
+ 1)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) +
1)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*
sin(2*d*x + 2*c))*cos(4*d*x + 4*c) - (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c)^2 + cos(2*d*x + 2*c))*sin(14*d*x + 14*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 +
2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(12*d*x + 12*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2
*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - 20*(cos(2*d*x + 2*c)^3 + cos(2*d
*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - 15*(cos(2*d*x + 2*c
)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(6*d*x + 6*c) - 6*(cos
(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(4*d*x + 4
*c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)
+ 1)) - (((sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(14*d*x + 1
4*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(12*d*x + 12
*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(10*d*x + 10
*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c
) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c)
+ 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(4*d*x + 4*c) - (
cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(14*d*x
+ 14*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*
c))*sin(12*d*x + 12*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 +
cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2
*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c
)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(6*d*x + 6*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin
(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(4*d*x + 4*c))*cos(7/2*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 2*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 + (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*c
os(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x
+ 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x +
2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(cos(2*d*x + 2*c)^3
+ cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(cos(2
*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c
) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos
(4*d*x + 4*c) + cos(2*d*x + 2*c)^2 + (sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2
*d*x + 2*c))*sin(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*
d*x + 2*c))*sin(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*
d*x + 2*c))*sin(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*
d*x + 2*c))*sin(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*
x + 2*c))*sin(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x +
2*c))*sin(4*d*x + 4*c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(5/2*arctan2(sin(2*d*x + 2*c
), cos(2*d*x + 2*c) + 1)))/(a^3*cos(14*d*x + 14*c)^2 + 36*a^3*cos(12*d*x + 12*c)^2 + 225*a^3*cos(10*d*x + 10*c
)^2 + 400*a^3*cos(8*d*x + 8*c)^2 + 225*a^3*cos(6*d*x + 6*c)^2 + 36*a^3*cos(4*d*x + 4*c)^2 + 12*a^3*cos(4*d*x +
4*c)*cos(2*d*x + 2*c) + a^3*cos(2*d*x + 2*c)^2 + a^3*sin(14*d*x + 14*c)^2 + 36*a^3*sin(12*d*x + 12*c)^2 + 225
*a^3*sin(10*d*x + 10*c)^2 + 400*a^3*sin(8*d*x + 8*c)^2 + 225*a^3*sin(6*d*x + 6*c)^2 + 36*a^3*sin(4*d*x + 4*c)^
2 + 12*a^3*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a^3*sin(2*d*x + 2*c)^2 + 2*(6*a^3*cos(12*d*x + 12*c) + 15*a^3*c
os(10*d*x + 10*c) + 20*a^3*cos(8*d*x + 8*c) + 15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x
+ 2*c))*cos(14*d*x + 14*c) + 12*(15*a^3*cos(10*d*x + 10*c) + 20*a^3*cos(8*d*x + 8*c) + 15*a^3*cos(6*d*x + 6*c
) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a^3*cos(8*d*x + 8*c) + 15*a^3*c
os(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a^3*cos(6*d*x + 6
*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a^3*cos(4*d*x + 4*c) + a^3*cos(2
*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a^3*sin(12*d*x + 12*c) + 15*a^3*sin(10*d*x + 10*c) + 20*a^3*sin(8*d*x + 8
*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a^3
*sin(10*d*x + 10*c) + 20*a^3*sin(8*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d
*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a^3*sin(8*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c)
+ a^3*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2
*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) + 18*
(a^3*d*cos(2*d*x + 2*c)^2 + a^3*d*sin(2*d*x + 2*c)^2 + 2*a^3*d*cos(2*d*x + 2*c) + a^3*d)*integrate(-(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(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 2*
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 + (cos(2*d*x + 2*c)^3
+ cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(cos(2
*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(12*d*x + 12
*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*
cos(10*d*x + 10*c) + 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos
(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2
*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(4*d*x + 4*c) + cos(2*d*x + 2*c)^2 + (sin(2*d*x + 2*c)^3 + (cos(2*d
*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*
x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*
x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*
x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x
+ 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2
*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*
x + 2*c))) - ((sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(14*d*x
+ 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(12*d*x
+ 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(10*d*x
+ 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(8*d*x +
8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(6*d*x + 6
*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(4*d*x + 4*c)
- (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(14
*d*x + 14*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x
+ 2*c))*sin(12*d*x + 12*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)
^2 + cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*c
os(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x +
2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(6*d*x + 6*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)
*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(4*d*x + 4*c))*sin(5/2*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c))))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - (((sin(2*d*x + 2*c)^3 + (c
os(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (co
s(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (co
s(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (co
s(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(
2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d
*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(4*d*x + 4*c) - (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2
*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(14*d*x + 14*c) - 6*(cos(2*d*x + 2*c)^3 +
cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(12*d*x + 12*c) - 15*(cos(2
*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(10*d*x + 10
*c) - 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*
sin(8*d*x + 8*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2
*d*x + 2*c))*sin(6*d*x + 6*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*
c)^2 + cos(2*d*x + 2*c))*sin(4*d*x + 4*c))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x +
2*c)^4 + sin(2*d*x + 2*c)^4 + 2*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 + (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*
c))*cos(14*d*x + 14*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 +
cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*
d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*
c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*s
in(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*
d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(4*d*x + 4*c) + cos(2*d*x + 2*c)^2
+ (sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) +
6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 1
5*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 2
0*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 15*
(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(si
n(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c))*sin(5/2*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(a^3*co
s(14*d*x + 14*c)^2 + 36*a^3*cos(12*d*x + 12*c)^2 + 225*a^3*cos(10*d*x + 10*c)^2 + 400*a^3*cos(8*d*x + 8*c)^2 +
225*a^3*cos(6*d*x + 6*c)^2 + 36*a^3*cos(4*d*x + 4*c)^2 + 12*a^3*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a^3*cos(2
*d*x + 2*c)^2 + a^3*sin(14*d*x + 14*c)^2 + 36*a^3*sin(12*d*x + 12*c)^2 + 225*a^3*sin(10*d*x + 10*c)^2 + 400*a^
3*sin(8*d*x + 8*c)^2 + 225*a^3*sin(6*d*x + 6*c)^2 + 36*a^3*sin(4*d*x + 4*c)^2 + 12*a^3*sin(4*d*x + 4*c)*sin(2*
d*x + 2*c) + a^3*sin(2*d*x + 2*c)^2 + 2*(6*a^3*cos(12*d*x + 12*c) + 15*a^3*cos(10*d*x + 10*c) + 20*a^3*cos(8*d
*x + 8*c) + 15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(
15*a^3*cos(10*d*x + 10*c) + 20*a^3*cos(8*d*x + 8*c) + 15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*c
os(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a^3*cos(8*d*x + 8*c) + 15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x
+ 4*c) + a^3*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3
*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*
(6*a^3*sin(12*d*x + 12*c) + 15*a^3*sin(10*d*x + 10*c) + 20*a^3*sin(8*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*
a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a^3*sin(10*d*x + 10*c) + 20*a^3*sin(8
*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30
*(20*a^3*sin(8*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(10*d*
x + 10*c) + 40*(15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30
*(6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - 10*(a^3*d*cos(2*d*x + 2*c)^2 + a^3*d*
sin(2*d*x + 2*c)^2 + 2*a^3*d*cos(2*d*x + 2*c) + a^3*d)*integrate(-(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(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 2*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 + (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c
)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*s
in(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(cos(2*d*x + 2*c)^3 + cos
(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(cos(2*d*x
+ 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) +
15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*
d*x + 6*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x +
2*c))*cos(4*d*x + 4*c) + cos(2*d*x + 2*c)^2 + (sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) +
1)*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) +
1)*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) +
1)*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) +
1)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)
*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*si
n(2*d*x + 2*c))*sin(4*d*x + 4*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - ((sin(2*d*x + 2*c)^3
+ (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 +
(cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 +
(cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 +
(cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (
cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos
(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(4*d*x + 4*c) - (cos(2*d*x + 2*c)^3 + cos(2*d*x
+ 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(14*d*x + 14*c) - 6*(cos(2*d*x + 2*c)
^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(12*d*x + 12*c) - 15*(c
os(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(10*d*x
+ 10*c) - 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*
c))*sin(8*d*x + 8*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + c
os(2*d*x + 2*c))*sin(6*d*x + 6*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x
+ 2*c)^2 + cos(2*d*x + 2*c))*sin(4*d*x + 4*c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(5/2*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - (((sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c) + 1)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2
*c) + 1)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2
*c) + 1)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2
*c) + 1)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c
) + 1)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) +
1)*sin(2*d*x + 2*c))*cos(4*d*x + 4*c) - (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d
*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(14*d*x + 14*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)
^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(12*d*x + 12*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*s
in(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - 20*(cos(2*d*x + 2*c)^3 + cos
(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - 15*(cos(2*d*x +
2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(6*d*x + 6*c) - 6*
(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(4*d*x
+ 4*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 2*c
os(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 + (cos(2*d*x + 2*c)^3 +
cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(cos(2*
d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(12*d*x + 12*
c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*c
os(10*d*x + 10*c) + 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(
2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*
cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(4*d*x + 4*c) + cos(2*d*x + 2*c)^2 + (sin(2*d*x + 2*c)^3 + (cos(2*d*
x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x
+ 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x
+ 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x
+ 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x +
2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*
c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c))))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(a^3*cos(14*d*x + 14*c)^2 + 36*a^3*cos(12
*d*x + 12*c)^2 + 225*a^3*cos(10*d*x + 10*c)^2 + 400*a^3*cos(8*d*x + 8*c)^2 + 225*a^3*cos(6*d*x + 6*c)^2 + 36*a
^3*cos(4*d*x + 4*c)^2 + 12*a^3*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a^3*cos(2*d*x + 2*c)^2 + a^3*sin(14*d*x + 1
4*c)^2 + 36*a^3*sin(12*d*x + 12*c)^2 + 225*a^3*sin(10*d*x + 10*c)^2 + 400*a^3*sin(8*d*x + 8*c)^2 + 225*a^3*sin
(6*d*x + 6*c)^2 + 36*a^3*sin(4*d*x + 4*c)^2 + 12*a^3*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a^3*sin(2*d*x + 2*c)^
2 + 2*(6*a^3*cos(12*d*x + 12*c) + 15*a^3*cos(10*d*x + 10*c) + 20*a^3*cos(8*d*x + 8*c) + 15*a^3*cos(6*d*x + 6*c
) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a^3*cos(10*d*x + 10*c) + 20*a^3
*cos(8*d*x + 8*c) + 15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(12*d*x + 12*c
) + 30*(20*a^3*cos(8*d*x + 8*c) + 15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos
(10*d*x + 10*c) + 40*(15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(8*d*x + 8*c
) + 30*(6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a^3*sin(12*d*x + 12*c) + 15*a^3
*sin(10*d*x + 10*c) + 20*a^3*sin(8*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d
*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a^3*sin(10*d*x + 10*c) + 20*a^3*sin(8*d*x + 8*c) + 15*a^3*sin(6*d*x + 6
*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a^3*sin(8*d*x + 8*c) + 15*a^3
*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a^3*sin(6*d*x +
6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a^3*sin(4*d*x + 4*c) + a^3*sin
(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - 10*(a^3*d*cos(2*d*x + 2*c)^2 + a^3*d*sin(2*d*x + 2*c)^2 + 2*a^3*d*cos(2
*d*x + 2*c) + a^3*d)*integrate(-(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(((co
s(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 2*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 + (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2
*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c)^2 + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 +
2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2
*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x
+ 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(cos(2*d*x + 2*c)^3
+ cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(4*d*x + 4*c) + cos(2*d*x
+ 2*c)^2 + (sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(14*d*x +
14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(12*d*x +
12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(10*d*x +
10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(8*d*x + 8
*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c
) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c))*
cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - ((sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*
x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x
+ 2*c) + 1)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x
+ 2*c) + 1)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x
+ 2*c) + 1)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c) + 1)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*
c) + 1)*sin(2*d*x + 2*c))*cos(4*d*x + 4*c) - (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos
(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(14*d*x + 14*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x +
2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(12*d*x + 12*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*
c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - 20*(cos(2*d*x + 2*c)^3 +
cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - 15*(cos(2*d
*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(6*d*x + 6*c)
- 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(4
*d*x + 4*c))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c) + 1)) - (((sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(14
*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(12*
d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(10*
d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(8*d
*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(6*d*x
+ 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*cos(4*d*x +
4*c) - (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*si
n(14*d*x + 14*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*
d*x + 2*c))*sin(12*d*x + 12*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c)^2 + cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - 20*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 +
2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - 15*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d
*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(6*d*x + 6*c) - 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x +
2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*sin(4*d*x + 4*c))*cos(1/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 2*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 + (cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)
^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*si
n(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 15*(cos(2*d*x + 2*c)^3 + cos(
2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 20*(cos(2*d*x
+ 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 1
5*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*d
*x + 6*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2
*c))*cos(4*d*x + 4*c) + cos(2*d*x + 2*c)^2 + (sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) +
1)*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1
)*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1
)*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 20*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1
)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 15*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*
sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin
(2*d*x + 2*c))*sin(4*d*x + 4*c))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(5/2*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c) + 1)))/(a^3*cos(14*d*x + 14*c)^2 + 36*a^3*cos(12*d*x + 12*c)^2 + 225*a^3*cos(10*d*
x + 10*c)^2 + 400*a^3*cos(8*d*x + 8*c)^2 + 225*a^3*cos(6*d*x + 6*c)^2 + 36*a^3*cos(4*d*x + 4*c)^2 + 12*a^3*cos
(4*d*x + 4*c)*cos(2*d*x + 2*c) + a^3*cos(2*d*x + 2*c)^2 + a^3*sin(14*d*x + 14*c)^2 + 36*a^3*sin(12*d*x + 12*c)
^2 + 225*a^3*sin(10*d*x + 10*c)^2 + 400*a^3*sin(8*d*x + 8*c)^2 + 225*a^3*sin(6*d*x + 6*c)^2 + 36*a^3*sin(4*d*x
+ 4*c)^2 + 12*a^3*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a^3*sin(2*d*x + 2*c)^2 + 2*(6*a^3*cos(12*d*x + 12*c) +
15*a^3*cos(10*d*x + 10*c) + 20*a^3*cos(8*d*x + 8*c) + 15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*c
os(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a^3*cos(10*d*x + 10*c) + 20*a^3*cos(8*d*x + 8*c) + 15*a^3*cos(6*d
*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a^3*cos(8*d*x + 8*c) +
15*a^3*cos(6*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a^3*cos(6
*d*x + 6*c) + 6*a^3*cos(4*d*x + 4*c) + a^3*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a^3*cos(4*d*x + 4*c) + a
^3*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a^3*sin(12*d*x + 12*c) + 15*a^3*sin(10*d*x + 10*c) + 20*a^3*sin(8
*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12
*(15*a^3*sin(10*d*x + 10*c) + 20*a^3*sin(8*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a^3
*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a^3*sin(8*d*x + 8*c) + 15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*
x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a^3*sin(6*d*x + 6*c) + 6*a^3*sin(4*d*x + 4*c) + a
^3*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a^3*sin(4*d*x + 4*c) + a^3*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)),
x))*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4))/((a^2*d*cos(2*d*x + 2*c)^2 + a^2
*d*sin(2*d*x + 2*c)^2 + 2*a^2*d*cos(2*d*x + 2*c) + a^2*d)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d
*x + 2*c) + 1)^(1/4)*sqrt(a))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (111) = 222\).

Time = 4.29 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.02 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {15 \, \sqrt {-a} {\left (\frac {\log \left ({\left | {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a {\left (2 \, \sqrt {2} + 3\right )} \right |}\right )}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {\log \left ({\left | {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + a {\left (2 \, \sqrt {2} - 3\right )} \right |}\right )}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} + \frac {2 \, {\left ({\left (\frac {37 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{\mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {40 \, \sqrt {2}}{\mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {15 \, \sqrt {2}}{\mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}}{15 \, d} \]

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/15*(15*sqrt(-a)*(log(abs((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a*(2*sqrt
(2) + 3)))/(a^3*sgn(cos(d*x + c))) - log(abs((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 +
a))^2 + a*(2*sqrt(2) - 3)))/(a^3*sgn(cos(d*x + c)))) + 2*((37*sqrt(2)*tan(1/2*d*x + 1/2*c)^2/sgn(cos(d*x + c)
) - 40*sqrt(2)/sgn(cos(d*x + c)))*tan(1/2*d*x + 1/2*c)^2 + 15*sqrt(2)/sgn(cos(d*x + c)))*tan(1/2*d*x + 1/2*c)/
((a*tan(1/2*d*x + 1/2*c)^2 - a)^2*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^6}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int(tan(c + d*x)^6/(a + a/cos(c + d*x))^(5/2),x)

[Out]

int(tan(c + d*x)^6/(a + a/cos(c + d*x))^(5/2), x)