3.449 \(\int \cos ^6(c+d x) (a+b \tan ^2(c+d x))^2 \, dx\)

Optimal. Leaf size=122 \[ \frac{\left (5 a^2+2 a b+b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left (5 a^2+2 a b+b^2\right )+\frac{(a-b) (5 a+3 b) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(a-b) \sin (c+d x) \cos ^5(c+d x) \left (a+b \tan ^2(c+d x)\right )}{6 d} \]

[Out]

((5*a^2 + 2*a*b + b^2)*x)/16 + ((5*a^2 + 2*a*b + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((a - b)*(5*a + 3*b)
*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((a - b)*Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Tan[c + d*x]^2))/(6*d)

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Rubi [A]  time = 0.131196, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3675, 413, 385, 199, 203} \[ \frac{\left (5 a^2+2 a b+b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left (5 a^2+2 a b+b^2\right )+\frac{(a-b) (5 a+3 b) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(a-b) \sin (c+d x) \cos ^5(c+d x) \left (a+b \tan ^2(c+d x)\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + b*Tan[c + d*x]^2)^2,x]

[Out]

((5*a^2 + 2*a*b + b^2)*x)/16 + ((5*a^2 + 2*a*b + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((a - b)*(5*a + 3*b)
*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((a - b)*Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Tan[c + d*x]^2))/(6*d)

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 385

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \cos ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^2}{\left (1+x^2\right )^4} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{(a-b) \cos ^5(c+d x) \sin (c+d x) \left (a+b \tan ^2(c+d x)\right )}{6 d}+\frac{\operatorname{Subst}\left (\int \frac{a (5 a+b)+3 b (a+b) x^2}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{6 d}\\ &=\frac{(a-b) (5 a+3 b) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{(a-b) \cos ^5(c+d x) \sin (c+d x) \left (a+b \tan ^2(c+d x)\right )}{6 d}+\frac{\left (5 a^2+2 a b+b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=\frac{\left (5 a^2+2 a b+b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{(a-b) (5 a+3 b) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{(a-b) \cos ^5(c+d x) \sin (c+d x) \left (a+b \tan ^2(c+d x)\right )}{6 d}+\frac{\left (5 a^2+2 a b+b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{16 d}\\ &=\frac{1}{16} \left (5 a^2+2 a b+b^2\right ) x+\frac{\left (5 a^2+2 a b+b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{(a-b) (5 a+3 b) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{(a-b) \cos ^5(c+d x) \sin (c+d x) \left (a+b \tan ^2(c+d x)\right )}{6 d}\\ \end{align*}

Mathematica [C]  time = 0.358395, size = 87, normalized size = 0.71 \[ \frac{12 (b+(1-2 i) a) (b+(1+2 i) a) (c+d x)+(a-b)^2 \sin (6 (c+d x))+3 (3 a+b) (a-b) \sin (4 (c+d x))+3 (5 a-b) (3 a+b) \sin (2 (c+d x))}{192 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + b*Tan[c + d*x]^2)^2,x]

[Out]

(12*((1 - 2*I)*a + b)*((1 + 2*I)*a + b)*(c + d*x) + 3*(5*a - b)*(3*a + b)*Sin[2*(c + d*x)] + 3*(a - b)*(3*a +
b)*Sin[4*(c + d*x)] + (a - b)^2*Sin[6*(c + d*x)])/(192*d)

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Maple [A]  time = 0.053, size = 166, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{6}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8}}+{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{16}}+{\frac{dx}{16}}+{\frac{c}{16}} \right ) +2\,ab \left ( -1/6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1/24\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) +{a}^{2} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^2*(-1/6*sin(d*x+c)^3*cos(d*x+c)^3-1/8*sin(d*x+c)*cos(d*x+c)^3+1/16*cos(d*x+c)*sin(d*x+c)+1/16*d*x+1/16*
c)+2*a*b*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)+a^2*(1/6
*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c))

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Maxima [A]  time = 1.55486, size = 177, normalized size = 1.45 \begin{align*} \frac{3 \,{\left (5 \, a^{2} + 2 \, a b + b^{2}\right )}{\left (d x + c\right )} + \frac{3 \,{\left (5 \, a^{2} + 2 \, a b + b^{2}\right )} \tan \left (d x + c\right )^{5} + 8 \,{\left (5 \, a^{2} + 2 \, a b - b^{2}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (11 \, a^{2} - 2 \, a b - b^{2}\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*tan(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/48*(3*(5*a^2 + 2*a*b + b^2)*(d*x + c) + (3*(5*a^2 + 2*a*b + b^2)*tan(d*x + c)^5 + 8*(5*a^2 + 2*a*b - b^2)*ta
n(d*x + c)^3 + 3*(11*a^2 - 2*a*b - b^2)*tan(d*x + c))/(tan(d*x + c)^6 + 3*tan(d*x + c)^4 + 3*tan(d*x + c)^2 +
1))/d

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Fricas [A]  time = 1.41763, size = 235, normalized size = 1.93 \begin{align*} \frac{3 \,{\left (5 \, a^{2} + 2 \, a b + b^{2}\right )} d x +{\left (8 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (5 \, a^{2} + 2 \, a b - 7 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (5 \, a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*tan(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/48*(3*(5*a^2 + 2*a*b + b^2)*d*x + (8*(a^2 - 2*a*b + b^2)*cos(d*x + c)^5 + 2*(5*a^2 + 2*a*b - 7*b^2)*cos(d*x
+ c)^3 + 3*(5*a^2 + 2*a*b + b^2)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*(a+b*tan(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B]  time = 61.6446, size = 6057, normalized size = 49.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*tan(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/48*(3*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*
tan(c))*tan(d*x)^6*tan(c)^6 + 15*a^2*d*x*tan(d*x)^6*tan(c)^6 + 6*a*b*d*x*tan(d*x)^6*tan(c)^6 + 3*b^2*d*x*tan(d
*x)^6*tan(c)^6 + 3*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^6*t
an(c)^6 + 9*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x)
- 2*tan(c))*tan(d*x)^6*tan(c)^4 + 9*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x
)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^6 + 6*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) -
1))*tan(d*x)^6*tan(c)^6 - 6*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^6*tan(c)^6 + 45*a^
2*d*x*tan(d*x)^6*tan(c)^4 + 18*a*b*d*x*tan(d*x)^6*tan(c)^4 + 9*b^2*d*x*tan(d*x)^6*tan(c)^4 + 9*pi*a*b*sgn(-2*t
an(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^6*tan(c)^4 + 45*a^2*d*x*tan(d*x)^4*ta
n(c)^6 + 18*a*b*d*x*tan(d*x)^4*tan(c)^6 + 9*b^2*d*x*tan(d*x)^4*tan(c)^6 + 9*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) +
2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^6 + 9*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn
(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^6*tan(c)^2 + 27*pi*a*b*sgn(2*tan
(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c
)^4 + 18*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^6*tan(c)^4 - 18*a*b*arctan(-(tan(d*x)
- tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^6*tan(c)^4 - 33*a^2*tan(d*x)^6*tan(c)^5 + 6*a*b*tan(d*x)^6*tan(c)^5
+ 3*b^2*tan(d*x)^6*tan(c)^5 + 9*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*ta
n(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^6 + 18*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))
*tan(d*x)^4*tan(c)^6 - 18*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^6 - 33*a^2*
tan(d*x)^5*tan(c)^6 + 6*a*b*tan(d*x)^5*tan(c)^6 + 3*b^2*tan(d*x)^5*tan(c)^6 + 45*a^2*d*x*tan(d*x)^6*tan(c)^2 +
18*a*b*d*x*tan(d*x)^6*tan(c)^2 + 9*b^2*d*x*tan(d*x)^6*tan(c)^2 + 9*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*
x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^6*tan(c)^2 + 135*a^2*d*x*tan(d*x)^4*tan(c)^4 + 54*a*b*d*x*tan(d*
x)^4*tan(c)^4 + 27*b^2*d*x*tan(d*x)^4*tan(c)^4 + 27*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*
tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^4 + 45*a^2*d*x*tan(d*x)^2*tan(c)^6 + 18*a*b*d*x*tan(d*x)^2*tan(c)^6 + 9
*b^2*d*x*tan(d*x)^2*tan(c)^6 + 9*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)
)*tan(d*x)^2*tan(c)^6 + 3*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2
+ 2*tan(d*x) - 2*tan(c))*tan(d*x)^6 + 27*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*t
an(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 + 18*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*ta
n(c) - 1))*tan(d*x)^6*tan(c)^2 - 18*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^6*tan(c)^2
- 40*a^2*tan(d*x)^6*tan(c)^3 - 16*a*b*tan(d*x)^6*tan(c)^3 + 8*b^2*tan(d*x)^6*tan(c)^3 + 27*pi*a*b*sgn(2*tan(d
*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^
4 + 54*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^4 - 54*a*b*arctan(-(tan(d*x) -
tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 + 45*a^2*tan(d*x)^5*tan(c)^4 - 78*a*b*tan(d*x)^5*tan(c)^4 +
9*b^2*tan(d*x)^5*tan(c)^4 + 45*a^2*tan(d*x)^4*tan(c)^5 - 78*a*b*tan(d*x)^4*tan(c)^5 + 9*b^2*tan(d*x)^4*tan(c)
^5 + 3*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*t
an(c))*tan(c)^6 + 18*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^6 - 18*a*b*arctan
(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^6 - 40*a^2*tan(d*x)^3*tan(c)^6 - 16*a*b*tan(d*x
)^3*tan(c)^6 + 8*b^2*tan(d*x)^3*tan(c)^6 + 15*a^2*d*x*tan(d*x)^6 + 6*a*b*d*x*tan(d*x)^6 + 3*b^2*d*x*tan(d*x)^6
+ 3*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^6 + 135*a^2*d*x*t
an(d*x)^4*tan(c)^2 + 54*a*b*d*x*tan(d*x)^4*tan(c)^2 + 27*b^2*d*x*tan(d*x)^4*tan(c)^2 + 27*pi*a*b*sgn(-2*tan(d*
x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 + 135*a^2*d*x*tan(d*x)^2*tan(c)
^4 + 54*a*b*d*x*tan(d*x)^2*tan(c)^4 + 27*b^2*d*x*tan(d*x)^2*tan(c)^4 + 27*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*
tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^4 + 15*a^2*d*x*tan(c)^6 + 6*a*b*d*x*tan(c)^6 + 3*
b^2*d*x*tan(c)^6 + 3*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^6 +
9*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c
))*tan(d*x)^4 + 6*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^6 - 6*a*b*arctan(-(tan(d*x) -
tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^6 - 15*a^2*tan(d*x)^6*tan(c) - 6*a*b*tan(d*x)^6*tan(c) - 3*b^2*tan(d*
x)^6*tan(c) + 27*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(
d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^2 + 54*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan
(c)^2 - 54*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^2 + 45*a^2*tan(d*x)^5*tan(
c)^2 + 18*a*b*tan(d*x)^5*tan(c)^2 - 39*b^2*tan(d*x)^5*tan(c)^2 - 120*a^2*tan(d*x)^4*tan(c)^3 + 144*a*b*tan(d*x
)^4*tan(c)^3 - 72*b^2*tan(d*x)^4*tan(c)^3 + 9*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) +
2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^4 + 54*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) -
1))*tan(d*x)^2*tan(c)^4 - 54*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^4 - 120*
a^2*tan(d*x)^3*tan(c)^4 + 144*a*b*tan(d*x)^3*tan(c)^4 - 72*b^2*tan(d*x)^3*tan(c)^4 + 45*a^2*tan(d*x)^2*tan(c)^
5 + 18*a*b*tan(d*x)^2*tan(c)^5 - 39*b^2*tan(d*x)^2*tan(c)^5 + 6*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c
) - 1))*tan(c)^6 - 6*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(c)^6 - 15*a^2*tan(d*x)*tan(c)^
6 - 6*a*b*tan(d*x)*tan(c)^6 - 3*b^2*tan(d*x)*tan(c)^6 + 45*a^2*d*x*tan(d*x)^4 + 18*a*b*d*x*tan(d*x)^4 + 9*b^2*
d*x*tan(d*x)^4 + 9*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4 +
135*a^2*d*x*tan(d*x)^2*tan(c)^2 + 54*a*b*d*x*tan(d*x)^2*tan(c)^2 + 27*b^2*d*x*tan(d*x)^2*tan(c)^2 + 27*pi*a*b
*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^2 + 45*a^2*d*x*tan(
c)^4 + 18*a*b*d*x*tan(c)^4 + 9*b^2*d*x*tan(c)^4 + 9*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*
tan(d*x) - 2*tan(c))*tan(c)^4 + 9*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*
tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2 + 18*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*
x)^4 - 18*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4 + 15*a^2*tan(d*x)^5 + 6*a*b*tan(d*
x)^5 + 3*b^2*tan(d*x)^5 - 45*a^2*tan(d*x)^4*tan(c) - 18*a*b*tan(d*x)^4*tan(c) + 39*b^2*tan(d*x)^4*tan(c) + 9*p
i*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*t
an(c)^2 + 54*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^2 - 54*a*b*arctan(-(tan(d
*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 + 120*a^2*tan(d*x)^3*tan(c)^2 - 144*a*b*tan(d*x)^3*ta
n(c)^2 + 72*b^2*tan(d*x)^3*tan(c)^2 + 120*a^2*tan(d*x)^2*tan(c)^3 - 144*a*b*tan(d*x)^2*tan(c)^3 + 72*b^2*tan(d
*x)^2*tan(c)^3 + 18*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(c)^4 - 18*a*b*arctan(-(tan(d*x)
- tan(c))/(tan(d*x)*tan(c) + 1))*tan(c)^4 - 45*a^2*tan(d*x)*tan(c)^4 - 18*a*b*tan(d*x)*tan(c)^4 + 39*b^2*tan(d
*x)*tan(c)^4 + 15*a^2*tan(c)^5 + 6*a*b*tan(c)^5 + 3*b^2*tan(c)^5 + 45*a^2*d*x*tan(d*x)^2 + 18*a*b*d*x*tan(d*x)
^2 + 9*b^2*d*x*tan(d*x)^2 + 9*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*t
an(d*x)^2 + 45*a^2*d*x*tan(c)^2 + 18*a*b*d*x*tan(c)^2 + 9*b^2*d*x*tan(c)^2 + 9*pi*a*b*sgn(-2*tan(d*x)^2*tan(c)
+ 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^2 + 3*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(
d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) + 18*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan
(c) - 1))*tan(d*x)^2 - 18*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2 + 40*a^2*tan(d*x)^
3 + 16*a*b*tan(d*x)^3 - 8*b^2*tan(d*x)^3 - 45*a^2*tan(d*x)^2*tan(c) + 78*a*b*tan(d*x)^2*tan(c) - 9*b^2*tan(d*x
)^2*tan(c) + 18*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(c)^2 - 18*a*b*arctan(-(tan(d*x) - ta
n(c))/(tan(d*x)*tan(c) + 1))*tan(c)^2 - 45*a^2*tan(d*x)*tan(c)^2 + 78*a*b*tan(d*x)*tan(c)^2 - 9*b^2*tan(d*x)*t
an(c)^2 + 40*a^2*tan(c)^3 + 16*a*b*tan(c)^3 - 8*b^2*tan(c)^3 + 15*a^2*d*x + 6*a*b*d*x + 3*b^2*d*x + 3*pi*a*b*s
gn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) + 6*a*b*arctan((tan(d*x) + tan(c))/(tan
(d*x)*tan(c) - 1)) - 6*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1)) + 33*a^2*tan(d*x) - 6*a*b*tan(d*
x) - 3*b^2*tan(d*x) + 33*a^2*tan(c) - 6*a*b*tan(c) - 3*b^2*tan(c))/(d*tan(d*x)^6*tan(c)^6 + 3*d*tan(d*x)^6*tan
(c)^4 + 3*d*tan(d*x)^4*tan(c)^6 + 3*d*tan(d*x)^6*tan(c)^2 + 9*d*tan(d*x)^4*tan(c)^4 + 3*d*tan(d*x)^2*tan(c)^6
+ d*tan(d*x)^6 + 9*d*tan(d*x)^4*tan(c)^2 + 9*d*tan(d*x)^2*tan(c)^4 + d*tan(c)^6 + 3*d*tan(d*x)^4 + 9*d*tan(d*x
)^2*tan(c)^2 + 3*d*tan(c)^4 + 3*d*tan(d*x)^2 + 3*d*tan(c)^2 + d)