\(\int \frac {\tan ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx\) [177]

Optimal result

Integrand size = 23, antiderivative size = 189 \[ \int \frac {\tan ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {2 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {6 a^3 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {2 a^4 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}} \]

[Out]

-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d/a^(1/2)+2*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-2/3*a*tan
(d*x+c)^3/d/(a+a*sec(d*x+c))^(3/2)+2/5*a^2*tan(d*x+c)^5/d/(a+a*sec(d*x+c))^(5/2)+6/7*a^3*tan(d*x+c)^7/d/(a+a*s
ec(d*x+c))^(7/2)+2/9*a^4*tan(d*x+c)^9/d/(a+a*sec(d*x+c))^(9/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 189, 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} \[ \int \frac {\tan ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 a^4 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac {6 a^3 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac {2 \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]

[In]

Int[Tan[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]

[Out]

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

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^3\right ) \text {Subst}\left (\int \frac {x^6 \left (2+a x^2\right )^2}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {\left (2 a^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^3}-\frac {x^2}{a^2}+\frac {x^4}{a}+3 x^6+a x^8-\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)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {6 a^3 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {2 a^4 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\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 )}{d} \\ & = -\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {2 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {6 a^3 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {2 a^4 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 14.08 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {-5040 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {\sec (c+d x)}{(1+\sec (c+d x))^2}} \sqrt {1+\sec (c+d x)}+(901+164 \cos (c+d x)+1004 \cos (2 (c+d x))+68 \cos (3 (c+d x))+383 \cos (4 (c+d x))) \sec ^4(c+d x) \tan (c+d x)}{1260 d \sqrt {a (1+\sec (c+d x))}} \]

[In]

Integrate[Tan[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]

[Out]

(-5040*ArcTan[Tan[(c + d*x)/2]/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]]*Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*Sq
rt[Sec[c + d*x]/(1 + Sec[c + d*x])^2]*Sqrt[1 + Sec[c + d*x]] + (901 + 164*Cos[c + d*x] + 1004*Cos[2*(c + d*x)]
+ 68*Cos[3*(c + d*x)] + 383*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Tan[c + d*x])/(1260*d*Sqrt[a*(1 + Sec[c + d*x])]
)

Maple [A] (verified)

Time = 3.72 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {2 \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (315 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+315 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-383 \sin \left (d x +c \right )-34 \tan \left (d x +c \right )+132 \sec \left (d x +c \right ) \tan \left (d x +c \right )+5 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}-35 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}\right )}{315 d a \left (\cos \left (d x +c \right )+1\right )}\)	\(220\)

[In]

int(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/315/d/a*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)*(315*(-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))*cos(d*x+c)+315*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(s
in(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-383*sin(d*x+c)-34*tan(d*x+c)+132*sec(d*x+c)*tan(d
*x+c)+5*tan(d*x+c)*sec(d*x+c)^2-35*tan(d*x+c)*sec(d*x+c)^3)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.88 \[ \int \frac {\tan ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\left [-\frac {315 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) - 2 \, {\left (383 \, \cos \left (d x + c\right )^{4} + 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}}, \frac {2 \, {\left (315 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (383 \, \cos \left (d x + c\right )^{4} + 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{315 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}}\right ] \]

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[-1/315*(315*(cos(d*x + c)^5 + cos(d*x + c)^4)*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*(383*cos(d*x +
c)^4 + 34*cos(d*x + c)^3 - 132*cos(d*x + c)^2 - 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*s
in(d*x + c))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4), 2/315*(315*(cos(d*x + c)^5 + cos(d*x + c)^4)*sqrt(a)*a
rctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (383*cos(d*x + c)^4 + 34*
cos(d*x + c)^3 - 132*cos(d*x + c)^2 - 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c
))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-1/630*(12*(105*sin(8*d*x + 8*c) + 280*sin(6*d*x + 6*c) + 546*sin(4*d*x + 4*c) + 312*sin(2*d*x + 2*c))*cos(9/2
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - 4*(315*cos(8*d*x + 8*c) + 840*cos(6*d*x + 6*c) + 1638*cos(
4*d*x + 4*c) + 936*cos(2*d*x + 2*c) + 383)*sin(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - 315*((co
s(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*(a*d*cos(2*d*x + 2*c)^4 + a*d*sin(2*d*x + 2*c)^4 + 4*a*d*cos(2*d*x + 2*c)^3
+ 6*a*d*cos(2*d*x + 2*c)^2 + 4*a*d*cos(2*d*x + 2*c) + 2*(a*d*cos(2*d*x + 2*c)^2 + 2*a*d*cos(2*d*x + 2*c) + a*
d)*sin(2*d*x + 2*c)^2 + a*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(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(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))))*co
s(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(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)))/(
a*cos(14*d*x + 14*c)^2 + 36*a*cos(12*d*x + 12*c)^2 + 225*a*cos(10*d*x + 10*c)^2 + 400*a*cos(8*d*x + 8*c)^2 + 2
25*a*cos(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 12*a*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a*cos(2*d*x + 2*c
)^2 + a*sin(14*d*x + 14*c)^2 + 36*a*sin(12*d*x + 12*c)^2 + 225*a*sin(10*d*x + 10*c)^2 + 400*a*sin(8*d*x + 8*c)
^2 + 225*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 12*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a*sin(2*d*x
+ 2*c)^2 + 2*(6*a*cos(12*d*x + 12*c) + 15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c
) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d
*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a*c
os(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(
15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a*cos(4*d*x + 4*c)
+ a*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a*sin(12*d*x + 12*c) + 15*a*sin(10*d*x + 10*c) + 20*a*sin(8*d*x
+ 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a*sin
(10*d*x + 10*c) + 20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*s
in(12*d*x + 12*c) + 30*(20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2
*c))*sin(10*d*x + 10*c) + 40*(15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8
*c) + 30*(6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - 10*(a*d*cos(2*d*x + 2*c)^4 + a*d*
sin(2*d*x + 2*c)^4 + 4*a*d*cos(2*d*x + 2*c)^3 + 6*a*d*cos(2*d*x + 2*c)^2 + 4*a*d*cos(2*d*x + 2*c) + 2*(a*d*cos
(2*d*x + 2*c)^2 + 2*a*d*cos(2*d*x + 2*c) + a*d)*sin(2*d*x + 2*c)^2 + a*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(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*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))*sin(9/2*arcta
n2(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) + 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)*s
in(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(si
n(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(a*cos(14*d*x + 14*c)^2 + 36*a*cos(12*d*x + 12*c)^2 + 225*a*cos(10*d*x
+ 10*c)^2 + 400*a*cos(8*d*x + 8*c)^2 + 225*a*cos(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 12*a*cos(4*d*x +
4*c)*cos(2*d*x + 2*c) + a*cos(2*d*x + 2*c)^2 + a*sin(14*d*x + 14*c)^2 + 36*a*sin(12*d*x + 12*c)^2 + 225*a*sin(
10*d*x + 10*c)^2 + 400*a*sin(8*d*x + 8*c)^2 + 225*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 12*a*sin(4*
d*x + 4*c)*sin(2*d*x + 2*c) + a*sin(2*d*x + 2*c)^2 + 2*(6*a*cos(12*d*x + 12*c) + 15*a*cos(10*d*x + 10*c) + 20*
a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 1
2*(15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*
x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*co
s(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*co
s(8*d*x + 8*c) + 30*(6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a*sin(12*d*x + 12*c) +
15*a*sin(10*d*x + 10*c) + 20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x
+ 2*c))*sin(14*d*x + 14*c) + 12*(15*a*sin(10*d*x + 10*c) + 20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a
*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c)
+ 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x +
4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)
), x) + 20*(a*d*cos(2*d*x + 2*c)^4 + a*d*sin(2*d*x + 2*c)^4 + 4*a*d*cos(2*d*x + 2*c)^3 + 6*a*d*cos(2*d*x + 2*c
)^2 + 4*a*d*cos(2*d*x + 2*c) + 2*(a*d*cos(2*d*x + 2*c)^2 + 2*a*d*cos(2*d*x + 2*c) + a*d)*sin(2*d*x + 2*c)^2 +
a*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(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(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (c
os(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(7/2*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(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)*co
s(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)))/(a*cos(14*d*x + 14*c)^2 + 36*
a*cos(12*d*x + 12*c)^2 + 225*a*cos(10*d*x + 10*c)^2 + 400*a*cos(8*d*x + 8*c)^2 + 225*a*cos(6*d*x + 6*c)^2 + 36
*a*cos(4*d*x + 4*c)^2 + 12*a*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a*cos(2*d*x + 2*c)^2 + a*sin(14*d*x + 14*c)^2
+ 36*a*sin(12*d*x + 12*c)^2 + 225*a*sin(10*d*x + 10*c)^2 + 400*a*sin(8*d*x + 8*c)^2 + 225*a*sin(6*d*x + 6*c)^
2 + 36*a*sin(4*d*x + 4*c)^2 + 12*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a*sin(2*d*x + 2*c)^2 + 2*(6*a*cos(12*d*
x + 12*c) + 15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a
*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x +
6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a*cos(8*d*x + 8*c) + 15*a*cos(6
*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a*cos(6*d*x + 6*c) + 6*a*
cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(6
*d*x + 6*c) + 2*(6*a*sin(12*d*x + 12*c) + 15*a*sin(10*d*x + 10*c) + 20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6
*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a*sin(10*d*x + 10*c) + 20*a*sin(8
*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a
*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40
*(15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a*sin(4*d*x + 4*
c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - 20*(a*d*cos(2*d*x + 2*c)^4 + a*d*sin(2*d*x + 2*c)^4 + 4*a*d*c
os(2*d*x + 2*c)^3 + 6*a*d*cos(2*d*x + 2*c)^2 + 4*a*d*cos(2*d*x + 2*c) + 2*(a*d*cos(2*d*x + 2*c)^2 + 2*a*d*cos(
2*d*x + 2*c) + a*d)*sin(2*d*x + 2*c)^2 + a*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(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)))/(a*cos(14*d*x + 14*c)^2 + 36*a*cos(12*d*x + 12*c)^2 + 225*a*cos(10*d*x + 10*c)^2 + 400*a*cos(8*d*x
+ 8*c)^2 + 225*a*cos(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 12*a*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a*co
s(2*d*x + 2*c)^2 + a*sin(14*d*x + 14*c)^2 + 36*a*sin(12*d*x + 12*c)^2 + 225*a*sin(10*d*x + 10*c)^2 + 400*a*sin
(8*d*x + 8*c)^2 + 225*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 12*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c)
+ a*sin(2*d*x + 2*c)^2 + 2*(6*a*cos(12*d*x + 12*c) + 15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a*co
s(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a*cos(10*d*x + 10*c) +
20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(12*d*x + 12*c)
+ 30*(20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(10*d*x +
10*c) + 40*(15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a*cos
(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a*sin(12*d*x + 12*c) + 15*a*sin(10*d*x + 10*c) + 2
0*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) +
12*(15*a*sin(10*d*x + 10*c) + 20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*
d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*
sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*
sin(8*d*x + 8*c) + 30*(6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) + 10*(a*d*cos(2*d*x +
2*c)^4 + a*d*sin(2*d*x + 2*c)^4 + 4*a*d*cos(2*d*x + 2*c)^3 + 6*a*d*cos(2*d*x + 2*c)^2 + 4*a*d*cos(2*d*x + 2*c)
+ 2*(a*d*cos(2*d*x + 2*c)^2 + 2*a*d*cos(2*d*x + 2*c) + a*d)*sin(2*d*x + 2*c)^2 + a*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(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) + 2
0*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 + 1
4*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*co
s(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*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*c
os(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*c
os(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(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)))/(a*cos(14*d*x + 14*c)^2 + 36*a*cos(12*d*x + 12*c)^2 + 225
*a*cos(10*d*x + 10*c)^2 + 400*a*cos(8*d*x + 8*c)^2 + 225*a*cos(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 12*a
*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a*cos(2*d*x + 2*c)^2 + a*sin(14*d*x + 14*c)^2 + 36*a*sin(12*d*x + 12*c)^2
+ 225*a*sin(10*d*x + 10*c)^2 + 400*a*sin(8*d*x + 8*c)^2 + 225*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2
+ 12*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a*sin(2*d*x + 2*c)^2 + 2*(6*a*cos(12*d*x + 12*c) + 15*a*cos(10*d*x
+ 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(14*d*
x + 14*c) + 12*(15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c)
+ a*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x
+ 4*c) + a*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d
*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a*sin(12*
d*x + 12*c) + 15*a*sin(10*d*x + 10*c) + 20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) +
a*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a*sin(10*d*x + 10*c) + 20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x
+ 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a*sin(8*d*x + 8*c) + 15*a*sin
(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a*sin(6*d*x + 6*c) + 6*
a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin
(6*d*x + 6*c)), x) - 2*(a*d*cos(2*d*x + 2*c)^4 + a*d*sin(2*d*x + 2*c)^4 + 4*a*d*cos(2*d*x + 2*c)^3 + 6*a*d*cos
(2*d*x + 2*c)^2 + 4*a*d*cos(2*d*x + 2*c) + 2*(a*d*cos(2*d*x + 2*c)^2 + 2*a*d*cos(2*d*x + 2*c) + a*d)*sin(2*d*x
+ 2*c)^2 + a*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(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*cos(2*d*x + 2*c)*s
in(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) + 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(1/2*arctan2(sin(2*d*x + 2*c), co
s(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*c
os(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 + 1
2*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)))/(a*cos(14*d*x + 1
4*c)^2 + 36*a*cos(12*d*x + 12*c)^2 + 225*a*cos(10*d*x + 10*c)^2 + 400*a*cos(8*d*x + 8*c)^2 + 225*a*cos(6*d*x +
6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 12*a*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a*cos(2*d*x + 2*c)^2 + a*sin(14*d
*x + 14*c)^2 + 36*a*sin(12*d*x + 12*c)^2 + 225*a*sin(10*d*x + 10*c)^2 + 400*a*sin(8*d*x + 8*c)^2 + 225*a*sin(6
*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 12*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a*sin(2*d*x + 2*c)^2 + 2*(6
*a*cos(12*d*x + 12*c) + 15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*
x + 4*c) + a*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a*cos(10*d*x + 10*c) + 20*a*cos(8*d*x + 8*c) + 15*a
*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a*cos(8*d*x + 8*c)
+ 15*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a*cos(6*d*x +
6*c) + 6*a*cos(4*d*x + 4*c) + a*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a*cos(4*d*x + 4*c) + a*cos(2*d*x +
2*c))*cos(6*d*x + 6*c) + 2*(6*a*sin(12*d*x + 12*c) + 15*a*sin(10*d*x + 10*c) + 20*a*sin(8*d*x + 8*c) + 15*a*s
in(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a*sin(10*d*x + 10*c)
+ 20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(12*d*x + 12*c
) + 30*(20*a*sin(8*d*x + 8*c) + 15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(10*d*x
+ 10*c) + 40*(15*a*sin(6*d*x + 6*c) + 6*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a*si
n(4*d*x + 4*c) + a*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))/((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*cos(2*d*x + 2*c) + 1)^(1/4)*sqrt(a))

Giac [A] (verification not implemented)

none

Time = 2.84 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.41 \[ \int \frac {\tan ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\sqrt {2} {\left (\frac {315 \, \sqrt {2} \sqrt {-a} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{{\left | a \right |}} + \frac {4 \, {\left (315 \, a^{4} - {\left (1470 \, a^{4} - {\left (2772 \, a^{4} + {\left (257 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1314 \, a^{4}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}\right )}}{630 \, d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \]

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/630*sqrt(2)*(315*sqrt(2)*sqrt(-a)*log(abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2
+ a))^2 - 4*sqrt(2)*abs(a) - 6*a)/abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^
2 + 4*sqrt(2)*abs(a) - 6*a))/abs(a) + 4*(315*a^4 - (1470*a^4 - (2772*a^4 + (257*a^4*tan(1/2*d*x + 1/2*c)^2 - 1
314*a^4)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)/((a*tan(
1/2*d*x + 1/2*c)^2 - a)^4*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)))/(d*sgn(cos(d*x + c)))

Mupad [F(-1)]

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

[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)