\(\int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx\) [189]

Optimal result

Integrand size = 23, antiderivative size = 157 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{3/2} d}+\frac {2 \tan (c+d x)}{a d \sqrt {a+a \sec (c+d x)}}-\frac {2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {2 a^2 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}} \]

[Out]

-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(3/2)/d+2*tan(d*x+c)/a/d/(a+a*sec(d*x+c))^(1/2)-2/3*tan
(d*x+c)^3/d/(a+a*sec(d*x+c))^(3/2)+2/5*a*tan(d*x+c)^5/d/(a+a*sec(d*x+c))^(5/2)+2/7*a^2*tan(d*x+c)^7/d/(a+a*sec
(d*x+c))^(7/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3972, 470, 308, 209} \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{3/2} d}+\frac {2 a^2 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac {2 \tan (c+d x)}{a d \sqrt {a \sec (c+d x)+a}} \]

[In]

Int[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2),x]

[Out]

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

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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.93 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.58 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {32 \sqrt {2} \left (\frac {1}{1+\sec (c+d x)}\right )^{11/2} \left (\frac {\cos (c+d x) (11+7 \cos (c+d x)) \csc ^8\left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (105 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right ) \cos ^3(c+d x)+(76-198 \cos (c+d x)+61 \cos (2 (c+d x))-44 \cos (3 (c+d x))) \sqrt {1-\sec (c+d x)}\right )}{3360 \sqrt {1-\sec (c+d x)}}-\frac {4}{11} \operatorname {Hypergeometric2F1}\left (2,\frac {11}{2},\frac {13}{2},-2 \sec (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sec (c+d x) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan ^7(c+d x)}{7 d (a (1+\sec (c+d x)))^{3/2} \left (1-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^{9/2}} \]

[In]

Integrate[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2),x]

[Out]

(32*Sqrt[2]*((1 + Sec[c + d*x])^(-1))^(11/2)*((Cos[c + d*x]*(11 + 7*Cos[c + d*x])*Csc[(c + d*x)/2]^8*Sec[(c +
d*x)/2]^2*(105*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Cos[c + d*x]^3 + (76 - 198*Cos[c + d*x] + 61*Cos[2*(c + d*x)] -
44*Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]))/(3360*Sqrt[1 - Sec[c + d*x]]) - (4*Hypergeometric2F1[2, 11/2, 1
3/2, -2*Sec[c + d*x]*Sin[(c + d*x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]^2)/11)*Tan[c + d*x]^7)/(7*d*(a*(1 + Sec
[c + d*x]))^(3/2)*(1 - Tan[(c + d*x)/2]^2)^(9/2))

Maple [A] (warning: unable to verify)

Time = 3.74 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.48

method	result	size
default	\(-\frac {\left (105 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {7}{2}}-278 \left (1-\cos \left (d x +c \right )\right )^{7} \csc \left (d x +c \right )^{7}+1078 \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}-770 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+210 \csc \left (d x +c \right )-210 \cot \left (d x +c \right )\right ) \sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}{105 d \,a^{2} \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )-1\right )^{3} \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+1\right )^{3}}\)	\(233\)

[In]

int(tan(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/105/d/a^2*(105*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)^(7/2)-278*(1-cos(d*x+c))^7*csc(d*x+c)^7+1078*(1-cos(d*x+c))^5*csc(d*x+c)^5-770
*(1-cos(d*x+c))^3*csc(d*x+c)^3+210*csc(d*x+c)-210*cot(d*x+c))*(-2*a/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)/(
-cot(d*x+c)+csc(d*x+c)-1)^3/(csc(d*x+c)-cot(d*x+c)+1)^3

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.18 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [-\frac {105 \, {\left (\cos \left (d x + c\right )^{4} + \cos \left (d x + c\right )^{3}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) - 2 \, {\left (146 \, \cos \left (d x + c\right )^{3} - 32 \, \cos \left (d x + c\right )^{2} - 24 \, \cos \left (d x + c\right ) + 15\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{105 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}}, \frac {2 \, {\left (105 \, {\left (\cos \left (d x + c\right )^{4} + \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (146 \, \cos \left (d x + c\right )^{3} - 32 \, \cos \left (d x + c\right )^{2} - 24 \, \cos \left (d x + c\right ) + 15\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{105 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}}\right ] \]

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[-1/105*(105*(cos(d*x + c)^4 + cos(d*x + c)^3)*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*(146*cos(d*x +
c)^3 - 32*cos(d*x + c)^2 - 24*cos(d*x + c) + 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^2*d*
cos(d*x + c)^4 + a^2*d*cos(d*x + c)^3), 2/105*(105*(cos(d*x + c)^4 + cos(d*x + c)^3)*sqrt(a)*arctan(sqrt((a*co
s(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (146*cos(d*x + c)^3 - 32*cos(d*x + c)^2 -
24*cos(d*x + c) + 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^2*d*cos(d*x + c)^4 + a^2*d*cos
(d*x + c)^3)]

Sympy [F]

\[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\tan ^{6}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(tan(d*x+c)**6/(a+a*sec(d*x+c))**(3/2),x)

[Out]

Integral(tan(c + d*x)**6/(a*(sec(c + d*x) + 1))**(3/2), x)

Maxima [F]

\[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{6}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

1/210*(105*((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 + s
in(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)), (c
os(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^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)*integrate(-(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(3/4)*(((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*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*c
os(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(3/2*arctan
2(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*s
in(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(s
in(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(a^2*cos(14*d*x
+ 14*c)^2 + 36*a^2*cos(12*d*x + 12*c)^2 + 225*a^2*cos(10*d*x + 10*c)^2 + 400*a^2*cos(8*d*x + 8*c)^2 + 225*a^2
*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 12*a^2*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a^2*cos(2*d*x + 2
*c)^2 + a^2*sin(14*d*x + 14*c)^2 + 36*a^2*sin(12*d*x + 12*c)^2 + 225*a^2*sin(10*d*x + 10*c)^2 + 400*a^2*sin(8*
d*x + 8*c)^2 + 225*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 12*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*
c) + a^2*sin(2*d*x + 2*c)^2 + 2*(6*a^2*cos(12*d*x + 12*c) + 15*a^2*cos(10*d*x + 10*c) + 20*a^2*cos(8*d*x + 8*c
) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a^2*c
os(10*d*x + 10*c) + 20*a^2*cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x
+ 2*c))*cos(12*d*x + 12*c) + 30*(20*a^2*cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) +
a^2*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d
*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a^2*s
in(12*d*x + 12*c) + 15*a^2*sin(10*d*x + 10*c) + 20*a^2*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(
4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a^2*sin(10*d*x + 10*c) + 20*a^2*sin(8*d*x + 8
*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a^2
*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c
) + 40*(15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a^2*
sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - 12*(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)*integrate(-(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d
*x + 2*c) + 1)^(3/4)*(((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(1
0*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)*s
in(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 + 1
2*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(3/2*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)*s
in(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)*c
os(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(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c) + 1)))/(a^2*cos(14*d*x + 14*c)^2 + 36*a^2*cos(12*d*x + 12*c)^2 + 225*a^2*cos(10*d*x + 10*c)^2 + 400*a^2*
cos(8*d*x + 8*c)^2 + 225*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 12*a^2*cos(4*d*x + 4*c)*cos(2*d*
x + 2*c) + a^2*cos(2*d*x + 2*c)^2 + a^2*sin(14*d*x + 14*c)^2 + 36*a^2*sin(12*d*x + 12*c)^2 + 225*a^2*sin(10*d*
x + 10*c)^2 + 400*a^2*sin(8*d*x + 8*c)^2 + 225*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 12*a^2*sin
(4*d*x + 4*c)*sin(2*d*x + 2*c) + a^2*sin(2*d*x + 2*c)^2 + 2*(6*a^2*cos(12*d*x + 12*c) + 15*a^2*cos(10*d*x + 10
*c) + 20*a^2*cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(1
4*d*x + 14*c) + 12*(15*a^2*cos(10*d*x + 10*c) + 20*a^2*cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(
4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a^2*cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c
) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a^2*cos(6*d*x + 6*c) + 6*a^2*co
s(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*c
os(6*d*x + 6*c) + 2*(6*a^2*sin(12*d*x + 12*c) + 15*a^2*sin(10*d*x + 10*c) + 20*a^2*sin(8*d*x + 8*c) + 15*a^2*s
in(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a^2*sin(10*d*x +
10*c) + 20*a^2*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin
(12*d*x + 12*c) + 30*(20*a^2*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d
*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*s
in(8*d*x + 8*c) + 30*(6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) + 16*(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)*integrate(-(cos(2*d*x + 2*c)^2 + si
n(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(3/4)*(((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(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*c
os(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(3/2*arct
an2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(3/2*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)*si
n(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*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(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(3/2*arctan2(s
in(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(a^2*cos(14*d*x + 14*c)^2 + 36*a^2*cos(12*d*x + 12*c)^2 + 225*a^2*cos
(10*d*x + 10*c)^2 + 400*a^2*cos(8*d*x + 8*c)^2 + 225*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 12*a
^2*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a^2*cos(2*d*x + 2*c)^2 + a^2*sin(14*d*x + 14*c)^2 + 36*a^2*sin(12*d*x +
12*c)^2 + 225*a^2*sin(10*d*x + 10*c)^2 + 400*a^2*sin(8*d*x + 8*c)^2 + 225*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin
(4*d*x + 4*c)^2 + 12*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a^2*sin(2*d*x + 2*c)^2 + 2*(6*a^2*cos(12*d*x + 12
*c) + 15*a^2*cos(10*d*x + 10*c) + 20*a^2*cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) +
a^2*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a^2*cos(10*d*x + 10*c) + 20*a^2*cos(8*d*x + 8*c) + 15*a^2*c
os(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 30*(20*a^2*cos(8*d*x + 8
*c) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 40*(15*a^2
*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 30*(6*a^2*cos(4*d*x + 4*
c) + a^2*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a^2*sin(12*d*x + 12*c) + 15*a^2*sin(10*d*x + 10*c) + 20*a^2
*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(14*d*x + 14*c
) + 12*(15*a^2*sin(10*d*x + 10*c) + 20*a^2*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c)
+ a^2*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a^2*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*si
n(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*
c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6
*c)), x) - 6*(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)*integrat
e(-(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(3/4)*(((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)*co
s(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)*sin(10*d*x + 10*c) + 20*co
s(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(3/2*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*c
os(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(1
4*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) + 1
5*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(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(a^2*cos(14*d*x + 14*c)^2 + 36*a^2*cos(12*
d*x + 12*c)^2 + 225*a^2*cos(10*d*x + 10*c)^2 + 400*a^2*cos(8*d*x + 8*c)^2 + 225*a^2*cos(6*d*x + 6*c)^2 + 36*a^
2*cos(4*d*x + 4*c)^2 + 12*a^2*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + a^2*cos(2*d*x + 2*c)^2 + a^2*sin(14*d*x + 14
*c)^2 + 36*a^2*sin(12*d*x + 12*c)^2 + 225*a^2*sin(10*d*x + 10*c)^2 + 400*a^2*sin(8*d*x + 8*c)^2 + 225*a^2*sin(
6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 12*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + a^2*sin(2*d*x + 2*c)^2
+ 2*(6*a^2*cos(12*d*x + 12*c) + 15*a^2*cos(10*d*x + 10*c) + 20*a^2*cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c)
+ 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + 12*(15*a^2*cos(10*d*x + 10*c) + 20*a^2*
cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(12*d*x + 12*c)
+ 30*(20*a^2*cos(8*d*x + 8*c) + 15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(
10*d*x + 10*c) + 40*(15*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(8*d*x + 8*c)
+ 30*(6*a^2*cos(4*d*x + 4*c) + a^2*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(6*a^2*sin(12*d*x + 12*c) + 15*a^2*
sin(10*d*x + 10*c) + 20*a^2*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*
x + 2*c))*sin(14*d*x + 14*c) + 12*(15*a^2*sin(10*d*x + 10*c) + 20*a^2*sin(8*d*x + 8*c) + 15*a^2*sin(6*d*x + 6*
c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 30*(20*a^2*sin(8*d*x + 8*c) + 15*a^2*
sin(6*d*x + 6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 40*(15*a^2*sin(6*d*x +
6*c) + 6*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 30*(6*a^2*sin(4*d*x + 4*c) + a^2*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)^(3/4)*
sqrt(a) - 8*(7*(15*sin(6*d*x + 6*c) + 25*sin(4*d*x + 4*c) + 29*sin(2*d*x + 2*c))*cos(7/2*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c) + 1)) - (105*cos(6*d*x + 6*c) + 175*cos(4*d*x + 4*c) + 203*cos(2*d*x + 2*c) + 73)*sin(7/
2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))*sqrt(a))/((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)^(3
/4))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (137) = 274\).

Time = 3.75 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.89 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {105 \, \sqrt {-a} {\left (\frac {\log \left ({\left | {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a {\left (2 \, \sqrt {2} + 3\right )} \right |}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {\log \left ({\left | {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + a {\left (2 \, \sqrt {2} - 3\right )} \right |}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} + \frac {2 \, {\left ({\left ({\left (\frac {139 \, \sqrt {2} a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{\mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {539 \, \sqrt {2} a^{2}}{\mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {385 \, \sqrt {2} a^{2}}{\mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {105 \, \sqrt {2} a^{2}}{\mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}}{105 \, d} \]

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")

[Out]

1/105*(105*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*sq
rt(2) + 3)))/(a^2*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^2*sgn(cos(d*x + c)))) + 2*(((139*sqrt(2)*a^2*tan(1/2*d*x + 1/2*c)^2/sgn(cos(
d*x + c)) - 539*sqrt(2)*a^2/sgn(cos(d*x + c)))*tan(1/2*d*x + 1/2*c)^2 + 385*sqrt(2)*a^2/sgn(cos(d*x + c)))*tan
(1/2*d*x + 1/2*c)^2 - 105*sqrt(2)*a^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)^
3*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^6}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(tan(c + d*x)^6/(a + a/cos(c + d*x))^(3/2),x)

[Out]

int(tan(c + d*x)^6/(a + a/cos(c + d*x))^(3/2), x)