3.382 \(\int (a+b \tan ^4(c+d x))^3 \, dx\)

Optimal. Leaf size=144 \[ \frac{b \left (3 a^2+3 a b+b^2\right ) \tan ^3(c+d x)}{3 d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \tan (c+d x)}{d}+\frac{b^2 (3 a+b) \tan ^7(c+d x)}{7 d}-\frac{b^2 (3 a+b) \tan ^5(c+d x)}{5 d}+x (a+b)^3+\frac{b^3 \tan ^{11}(c+d x)}{11 d}-\frac{b^3 \tan ^9(c+d x)}{9 d} \]

[Out]

(a + b)^3*x - (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x])/d + (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x]^3)/(3*d) - (b^2
*(3*a + b)*Tan[c + d*x]^5)/(5*d) + (b^2*(3*a + b)*Tan[c + d*x]^7)/(7*d) - (b^3*Tan[c + d*x]^9)/(9*d) + (b^3*Ta
n[c + d*x]^11)/(11*d)

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Rubi [A]  time = 0.0824736, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3661, 1154, 203} \[ \frac{b \left (3 a^2+3 a b+b^2\right ) \tan ^3(c+d x)}{3 d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \tan (c+d x)}{d}+\frac{b^2 (3 a+b) \tan ^7(c+d x)}{7 d}-\frac{b^2 (3 a+b) \tan ^5(c+d x)}{5 d}+x (a+b)^3+\frac{b^3 \tan ^{11}(c+d x)}{11 d}-\frac{b^3 \tan ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x]^4)^3,x]

[Out]

(a + b)^3*x - (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x])/d + (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x]^3)/(3*d) - (b^2
*(3*a + b)*Tan[c + d*x]^5)/(5*d) + (b^2*(3*a + b)*Tan[c + d*x]^7)/(7*d) - (b^3*Tan[c + d*x]^9)/(9*d) + (b^3*Ta
n[c + d*x]^11)/(11*d)

Rule 3661

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

Rule 1154

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a
+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

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 \left (a+b \tan ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^4\right )^3}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b \left (3 a^2+3 a b+b^2\right )+b \left (3 a^2+3 a b+b^2\right ) x^2-b^2 (3 a+b) x^4+b^2 (3 a+b) x^6-b^3 x^8+b^3 x^{10}+\frac{(a+b)^3}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{b \left (3 a^2+3 a b+b^2\right ) \tan (c+d x)}{d}+\frac{b \left (3 a^2+3 a b+b^2\right ) \tan ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tan ^5(c+d x)}{5 d}+\frac{b^2 (3 a+b) \tan ^7(c+d x)}{7 d}-\frac{b^3 \tan ^9(c+d x)}{9 d}+\frac{b^3 \tan ^{11}(c+d x)}{11 d}+\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=(a+b)^3 x-\frac{b \left (3 a^2+3 a b+b^2\right ) \tan (c+d x)}{d}+\frac{b \left (3 a^2+3 a b+b^2\right ) \tan ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tan ^5(c+d x)}{5 d}+\frac{b^2 (3 a+b) \tan ^7(c+d x)}{7 d}-\frac{b^3 \tan ^9(c+d x)}{9 d}+\frac{b^3 \tan ^{11}(c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 0.973732, size = 128, normalized size = 0.89 \[ \frac{b \tan (c+d x) \left (1155 \left (3 a^2+3 a b+b^2\right ) \tan ^2(c+d x)-3465 \left (3 a^2+3 a b+b^2\right )+495 b (3 a+b) \tan ^6(c+d x)-693 b (3 a+b) \tan ^4(c+d x)+315 b^2 \tan ^{10}(c+d x)-385 b^2 \tan ^8(c+d x)\right )}{3465 d}+\frac{(a+b)^3 \tan ^{-1}(\tan (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x]^4)^3,x]

[Out]

((a + b)^3*ArcTan[Tan[c + d*x]])/d + (b*Tan[c + d*x]*(-3465*(3*a^2 + 3*a*b + b^2) + 1155*(3*a^2 + 3*a*b + b^2)
*Tan[c + d*x]^2 - 693*b*(3*a + b)*Tan[c + d*x]^4 + 495*b*(3*a + b)*Tan[c + d*x]^6 - 385*b^2*Tan[c + d*x]^8 + 3
15*b^2*Tan[c + d*x]^10))/(3465*d)

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Maple [A]  time = 0.006, size = 252, normalized size = 1.8 \begin{align*}{\frac{{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{11}}{11\,d}}-{\frac{{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{9}}{9\,d}}+{\frac{3\, \left ( \tan \left ( dx+c \right ) \right ) ^{7}a{b}^{2}}{7\,d}}+{\frac{{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{3\,a{b}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}{a}^{2}b}{d}}+{\frac{a{b}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}{b}^{3}}{3\,d}}-3\,{\frac{\tan \left ( dx+c \right ){a}^{2}b}{d}}-3\,{\frac{a{b}^{2}\tan \left ( dx+c \right ) }{d}}-{\frac{{b}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}}{d}}+3\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}b}{d}}+3\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{d}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{3}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(d*x+c)^4)^3,x)

[Out]

1/11*b^3*tan(d*x+c)^11/d-1/9*b^3*tan(d*x+c)^9/d+3/7/d*tan(d*x+c)^7*a*b^2+1/7/d*b^3*tan(d*x+c)^7-3/5*a*b^2*tan(
d*x+c)^5/d-1/5*b^3*tan(d*x+c)^5/d+1/d*tan(d*x+c)^3*a^2*b+a*b^2*tan(d*x+c)^3/d+1/3/d*tan(d*x+c)^3*b^3-3/d*tan(d
*x+c)*a^2*b-3*a*b^2*tan(d*x+c)/d-1/d*b^3*tan(d*x+c)+1/d*arctan(tan(d*x+c))*a^3+3/d*arctan(tan(d*x+c))*a^2*b+3/
d*arctan(tan(d*x+c))*a*b^2+1/d*arctan(tan(d*x+c))*b^3

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Maxima [A]  time = 1.53418, size = 225, normalized size = 1.56 \begin{align*} a^{3} x + \frac{{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{2} b}{d} + \frac{{\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} a b^{2}}{35 \, d} + \frac{{\left (315 \, \tan \left (d x + c\right )^{11} - 385 \, \tan \left (d x + c\right )^{9} + 495 \, \tan \left (d x + c\right )^{7} - 693 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3} + 3465 \, d x + 3465 \, c - 3465 \, \tan \left (d x + c\right )\right )} b^{3}}{3465 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+tan(d*x+c)^4*b)^3,x, algorithm="maxima")

[Out]

a^3*x + (tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^2*b/d + 1/35*(15*tan(d*x + c)^7 - 21*tan(d*x + c)^5
+ 35*tan(d*x + c)^3 + 105*d*x + 105*c - 105*tan(d*x + c))*a*b^2/d + 1/3465*(315*tan(d*x + c)^11 - 385*tan(d*x
+ c)^9 + 495*tan(d*x + c)^7 - 693*tan(d*x + c)^5 + 1155*tan(d*x + c)^3 + 3465*d*x + 3465*c - 3465*tan(d*x + c)
)*b^3/d

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Fricas [A]  time = 1.37365, size = 367, normalized size = 2.55 \begin{align*} \frac{315 \, b^{3} \tan \left (d x + c\right )^{11} - 385 \, b^{3} \tan \left (d x + c\right )^{9} + 495 \,{\left (3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )^{7} - 693 \,{\left (3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )^{5} + 1155 \,{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )^{3} + 3465 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x - 3465 \,{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )}{3465 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+tan(d*x+c)^4*b)^3,x, algorithm="fricas")

[Out]

1/3465*(315*b^3*tan(d*x + c)^11 - 385*b^3*tan(d*x + c)^9 + 495*(3*a*b^2 + b^3)*tan(d*x + c)^7 - 693*(3*a*b^2 +
b^3)*tan(d*x + c)^5 + 1155*(3*a^2*b + 3*a*b^2 + b^3)*tan(d*x + c)^3 + 3465*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*
x - 3465*(3*a^2*b + 3*a*b^2 + b^3)*tan(d*x + c))/d

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Sympy [A]  time = 4.19558, size = 224, normalized size = 1.56 \begin{align*} \begin{cases} a^{3} x + 3 a^{2} b x + \frac{a^{2} b \tan ^{3}{\left (c + d x \right )}}{d} - \frac{3 a^{2} b \tan{\left (c + d x \right )}}{d} + 3 a b^{2} x + \frac{3 a b^{2} \tan ^{7}{\left (c + d x \right )}}{7 d} - \frac{3 a b^{2} \tan ^{5}{\left (c + d x \right )}}{5 d} + \frac{a b^{2} \tan ^{3}{\left (c + d x \right )}}{d} - \frac{3 a b^{2} \tan{\left (c + d x \right )}}{d} + b^{3} x + \frac{b^{3} \tan ^{11}{\left (c + d x \right )}}{11 d} - \frac{b^{3} \tan ^{9}{\left (c + d x \right )}}{9 d} + \frac{b^{3} \tan ^{7}{\left (c + d x \right )}}{7 d} - \frac{b^{3} \tan ^{5}{\left (c + d x \right )}}{5 d} + \frac{b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{3} \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tan ^{4}{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+tan(d*x+c)**4*b)**3,x)

[Out]

Piecewise((a**3*x + 3*a**2*b*x + a**2*b*tan(c + d*x)**3/d - 3*a**2*b*tan(c + d*x)/d + 3*a*b**2*x + 3*a*b**2*ta
n(c + d*x)**7/(7*d) - 3*a*b**2*tan(c + d*x)**5/(5*d) + a*b**2*tan(c + d*x)**3/d - 3*a*b**2*tan(c + d*x)/d + b*
*3*x + b**3*tan(c + d*x)**11/(11*d) - b**3*tan(c + d*x)**9/(9*d) + b**3*tan(c + d*x)**7/(7*d) - b**3*tan(c + d
*x)**5/(5*d) + b**3*tan(c + d*x)**3/(3*d) - b**3*tan(c + d*x)/d, Ne(d, 0)), (x*(a + b*tan(c)**4)**3, True))

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Giac [B]  time = 34.4993, size = 4724, normalized size = 32.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+tan(d*x+c)^4*b)^3,x, algorithm="giac")

[Out]

1/3465*(3465*a^3*d*x*tan(d*x)^11*tan(c)^11 + 10395*a^2*b*d*x*tan(d*x)^11*tan(c)^11 + 10395*a*b^2*d*x*tan(d*x)^
11*tan(c)^11 + 3465*b^3*d*x*tan(d*x)^11*tan(c)^11 - 38115*a^3*d*x*tan(d*x)^10*tan(c)^10 - 114345*a^2*b*d*x*tan
(d*x)^10*tan(c)^10 - 114345*a*b^2*d*x*tan(d*x)^10*tan(c)^10 - 38115*b^3*d*x*tan(d*x)^10*tan(c)^10 + 10395*a^2*
b*tan(d*x)^11*tan(c)^10 + 10395*a*b^2*tan(d*x)^11*tan(c)^10 + 3465*b^3*tan(d*x)^11*tan(c)^10 + 10395*a^2*b*tan
(d*x)^10*tan(c)^11 + 10395*a*b^2*tan(d*x)^10*tan(c)^11 + 3465*b^3*tan(d*x)^10*tan(c)^11 + 190575*a^3*d*x*tan(d
*x)^9*tan(c)^9 + 571725*a^2*b*d*x*tan(d*x)^9*tan(c)^9 + 571725*a*b^2*d*x*tan(d*x)^9*tan(c)^9 + 190575*b^3*d*x*
tan(d*x)^9*tan(c)^9 - 3465*a^2*b*tan(d*x)^11*tan(c)^8 - 3465*a*b^2*tan(d*x)^11*tan(c)^8 - 1155*b^3*tan(d*x)^11
*tan(c)^8 - 114345*a^2*b*tan(d*x)^10*tan(c)^9 - 114345*a*b^2*tan(d*x)^10*tan(c)^9 - 38115*b^3*tan(d*x)^10*tan(
c)^9 - 114345*a^2*b*tan(d*x)^9*tan(c)^10 - 114345*a*b^2*tan(d*x)^9*tan(c)^10 - 38115*b^3*tan(d*x)^9*tan(c)^10
- 3465*a^2*b*tan(d*x)^8*tan(c)^11 - 3465*a*b^2*tan(d*x)^8*tan(c)^11 - 1155*b^3*tan(d*x)^8*tan(c)^11 - 571725*a
^3*d*x*tan(d*x)^8*tan(c)^8 - 1715175*a^2*b*d*x*tan(d*x)^8*tan(c)^8 - 1715175*a*b^2*d*x*tan(d*x)^8*tan(c)^8 - 5
71725*b^3*d*x*tan(d*x)^8*tan(c)^8 + 2079*a*b^2*tan(d*x)^11*tan(c)^6 + 693*b^3*tan(d*x)^11*tan(c)^6 + 27720*a^2
*b*tan(d*x)^10*tan(c)^7 + 38115*a*b^2*tan(d*x)^10*tan(c)^7 + 12705*b^3*tan(d*x)^10*tan(c)^7 + 550935*a^2*b*tan
(d*x)^9*tan(c)^8 + 571725*a*b^2*tan(d*x)^9*tan(c)^8 + 190575*b^3*tan(d*x)^9*tan(c)^8 + 550935*a^2*b*tan(d*x)^8
*tan(c)^9 + 571725*a*b^2*tan(d*x)^8*tan(c)^9 + 190575*b^3*tan(d*x)^8*tan(c)^9 + 27720*a^2*b*tan(d*x)^7*tan(c)^
10 + 38115*a*b^2*tan(d*x)^7*tan(c)^10 + 12705*b^3*tan(d*x)^7*tan(c)^10 + 2079*a*b^2*tan(d*x)^6*tan(c)^11 + 693
*b^3*tan(d*x)^6*tan(c)^11 + 1143450*a^3*d*x*tan(d*x)^7*tan(c)^7 + 3430350*a^2*b*d*x*tan(d*x)^7*tan(c)^7 + 3430
350*a*b^2*d*x*tan(d*x)^7*tan(c)^7 + 1143450*b^3*d*x*tan(d*x)^7*tan(c)^7 - 1485*a*b^2*tan(d*x)^11*tan(c)^4 - 49
5*b^3*tan(d*x)^11*tan(c)^4 - 22869*a*b^2*tan(d*x)^10*tan(c)^5 - 7623*b^3*tan(d*x)^10*tan(c)^5 - 97020*a^2*b*ta
n(d*x)^9*tan(c)^6 - 190575*a*b^2*tan(d*x)^9*tan(c)^6 - 63525*b^3*tan(d*x)^9*tan(c)^6 - 1538460*a^2*b*tan(d*x)^
8*tan(c)^7 - 1715175*a*b^2*tan(d*x)^8*tan(c)^7 - 571725*b^3*tan(d*x)^8*tan(c)^7 - 1538460*a^2*b*tan(d*x)^7*tan
(c)^8 - 1715175*a*b^2*tan(d*x)^7*tan(c)^8 - 571725*b^3*tan(d*x)^7*tan(c)^8 - 97020*a^2*b*tan(d*x)^6*tan(c)^9 -
190575*a*b^2*tan(d*x)^6*tan(c)^9 - 63525*b^3*tan(d*x)^6*tan(c)^9 - 22869*a*b^2*tan(d*x)^5*tan(c)^10 - 7623*b^
3*tan(d*x)^5*tan(c)^10 - 1485*a*b^2*tan(d*x)^4*tan(c)^11 - 495*b^3*tan(d*x)^4*tan(c)^11 - 1600830*a^3*d*x*tan(
d*x)^6*tan(c)^6 - 4802490*a^2*b*d*x*tan(d*x)^6*tan(c)^6 - 4802490*a*b^2*d*x*tan(d*x)^6*tan(c)^6 - 1600830*b^3*
d*x*tan(d*x)^6*tan(c)^6 + 385*b^3*tan(d*x)^11*tan(c)^2 + 5940*a*b^2*tan(d*x)^10*tan(c)^3 + 5445*b^3*tan(d*x)^1
0*tan(c)^3 + 72765*a*b^2*tan(d*x)^9*tan(c)^4 + 38115*b^3*tan(d*x)^9*tan(c)^4 + 194040*a^2*b*tan(d*x)^8*tan(c)^
5 + 474705*a*b^2*tan(d*x)^8*tan(c)^5 + 190575*b^3*tan(d*x)^8*tan(c)^5 + 2765070*a^2*b*tan(d*x)^7*tan(c)^6 + 32
84820*a*b^2*tan(d*x)^7*tan(c)^6 + 1143450*b^3*tan(d*x)^7*tan(c)^6 + 2765070*a^2*b*tan(d*x)^6*tan(c)^7 + 328482
0*a*b^2*tan(d*x)^6*tan(c)^7 + 1143450*b^3*tan(d*x)^6*tan(c)^7 + 194040*a^2*b*tan(d*x)^5*tan(c)^8 + 474705*a*b^
2*tan(d*x)^5*tan(c)^8 + 190575*b^3*tan(d*x)^5*tan(c)^8 + 72765*a*b^2*tan(d*x)^4*tan(c)^9 + 38115*b^3*tan(d*x)^
4*tan(c)^9 + 5940*a*b^2*tan(d*x)^3*tan(c)^10 + 5445*b^3*tan(d*x)^3*tan(c)^10 + 385*b^3*tan(d*x)^2*tan(c)^11 +
1600830*a^3*d*x*tan(d*x)^5*tan(c)^5 + 4802490*a^2*b*d*x*tan(d*x)^5*tan(c)^5 + 4802490*a*b^2*d*x*tan(d*x)^5*tan
(c)^5 + 1600830*b^3*d*x*tan(d*x)^5*tan(c)^5 - 315*b^3*tan(d*x)^11 - 4235*b^3*tan(d*x)^10*tan(c) - 8910*a*b^2*t
an(d*x)^9*tan(c)^2 - 27225*b^3*tan(d*x)^9*tan(c)^2 - 103950*a*b^2*tan(d*x)^8*tan(c)^3 - 114345*b^3*tan(d*x)^8*
tan(c)^3 - 242550*a^2*b*tan(d*x)^7*tan(c)^4 - 637560*a*b^2*tan(d*x)^7*tan(c)^4 - 381150*b^3*tan(d*x)^7*tan(c)^
4 - 3347190*a^2*b*tan(d*x)^6*tan(c)^5 - 4074840*a*b^2*tan(d*x)^6*tan(c)^5 - 1600830*b^3*tan(d*x)^6*tan(c)^5 -
3347190*a^2*b*tan(d*x)^5*tan(c)^6 - 4074840*a*b^2*tan(d*x)^5*tan(c)^6 - 1600830*b^3*tan(d*x)^5*tan(c)^6 - 2425
50*a^2*b*tan(d*x)^4*tan(c)^7 - 637560*a*b^2*tan(d*x)^4*tan(c)^7 - 381150*b^3*tan(d*x)^4*tan(c)^7 - 103950*a*b^
2*tan(d*x)^3*tan(c)^8 - 114345*b^3*tan(d*x)^3*tan(c)^8 - 8910*a*b^2*tan(d*x)^2*tan(c)^9 - 27225*b^3*tan(d*x)^2
*tan(c)^9 - 4235*b^3*tan(d*x)*tan(c)^10 - 315*b^3*tan(c)^11 - 1143450*a^3*d*x*tan(d*x)^4*tan(c)^4 - 3430350*a^
2*b*d*x*tan(d*x)^4*tan(c)^4 - 3430350*a*b^2*d*x*tan(d*x)^4*tan(c)^4 - 1143450*b^3*d*x*tan(d*x)^4*tan(c)^4 + 38
5*b^3*tan(d*x)^9 + 5940*a*b^2*tan(d*x)^8*tan(c) + 5445*b^3*tan(d*x)^8*tan(c) + 72765*a*b^2*tan(d*x)^7*tan(c)^2
+ 38115*b^3*tan(d*x)^7*tan(c)^2 + 194040*a^2*b*tan(d*x)^6*tan(c)^3 + 474705*a*b^2*tan(d*x)^6*tan(c)^3 + 19057
5*b^3*tan(d*x)^6*tan(c)^3 + 2765070*a^2*b*tan(d*x)^5*tan(c)^4 + 3284820*a*b^2*tan(d*x)^5*tan(c)^4 + 1143450*b^
3*tan(d*x)^5*tan(c)^4 + 2765070*a^2*b*tan(d*x)^4*tan(c)^5 + 3284820*a*b^2*tan(d*x)^4*tan(c)^5 + 1143450*b^3*ta
n(d*x)^4*tan(c)^5 + 194040*a^2*b*tan(d*x)^3*tan(c)^6 + 474705*a*b^2*tan(d*x)^3*tan(c)^6 + 190575*b^3*tan(d*x)^
3*tan(c)^6 + 72765*a*b^2*tan(d*x)^2*tan(c)^7 + 38115*b^3*tan(d*x)^2*tan(c)^7 + 5940*a*b^2*tan(d*x)*tan(c)^8 +
5445*b^3*tan(d*x)*tan(c)^8 + 385*b^3*tan(c)^9 + 571725*a^3*d*x*tan(d*x)^3*tan(c)^3 + 1715175*a^2*b*d*x*tan(d*x
)^3*tan(c)^3 + 1715175*a*b^2*d*x*tan(d*x)^3*tan(c)^3 + 571725*b^3*d*x*tan(d*x)^3*tan(c)^3 - 1485*a*b^2*tan(d*x
)^7 - 495*b^3*tan(d*x)^7 - 22869*a*b^2*tan(d*x)^6*tan(c) - 7623*b^3*tan(d*x)^6*tan(c) - 97020*a^2*b*tan(d*x)^5
*tan(c)^2 - 190575*a*b^2*tan(d*x)^5*tan(c)^2 - 63525*b^3*tan(d*x)^5*tan(c)^2 - 1538460*a^2*b*tan(d*x)^4*tan(c)
^3 - 1715175*a*b^2*tan(d*x)^4*tan(c)^3 - 571725*b^3*tan(d*x)^4*tan(c)^3 - 1538460*a^2*b*tan(d*x)^3*tan(c)^4 -
1715175*a*b^2*tan(d*x)^3*tan(c)^4 - 571725*b^3*tan(d*x)^3*tan(c)^4 - 97020*a^2*b*tan(d*x)^2*tan(c)^5 - 190575*
a*b^2*tan(d*x)^2*tan(c)^5 - 63525*b^3*tan(d*x)^2*tan(c)^5 - 22869*a*b^2*tan(d*x)*tan(c)^6 - 7623*b^3*tan(d*x)*
tan(c)^6 - 1485*a*b^2*tan(c)^7 - 495*b^3*tan(c)^7 - 190575*a^3*d*x*tan(d*x)^2*tan(c)^2 - 571725*a^2*b*d*x*tan(
d*x)^2*tan(c)^2 - 571725*a*b^2*d*x*tan(d*x)^2*tan(c)^2 - 190575*b^3*d*x*tan(d*x)^2*tan(c)^2 + 2079*a*b^2*tan(d
*x)^5 + 693*b^3*tan(d*x)^5 + 27720*a^2*b*tan(d*x)^4*tan(c) + 38115*a*b^2*tan(d*x)^4*tan(c) + 12705*b^3*tan(d*x
)^4*tan(c) + 550935*a^2*b*tan(d*x)^3*tan(c)^2 + 571725*a*b^2*tan(d*x)^3*tan(c)^2 + 190575*b^3*tan(d*x)^3*tan(c
)^2 + 550935*a^2*b*tan(d*x)^2*tan(c)^3 + 571725*a*b^2*tan(d*x)^2*tan(c)^3 + 190575*b^3*tan(d*x)^2*tan(c)^3 + 2
7720*a^2*b*tan(d*x)*tan(c)^4 + 38115*a*b^2*tan(d*x)*tan(c)^4 + 12705*b^3*tan(d*x)*tan(c)^4 + 2079*a*b^2*tan(c)
^5 + 693*b^3*tan(c)^5 + 38115*a^3*d*x*tan(d*x)*tan(c) + 114345*a^2*b*d*x*tan(d*x)*tan(c) + 114345*a*b^2*d*x*ta
n(d*x)*tan(c) + 38115*b^3*d*x*tan(d*x)*tan(c) - 3465*a^2*b*tan(d*x)^3 - 3465*a*b^2*tan(d*x)^3 - 1155*b^3*tan(d
*x)^3 - 114345*a^2*b*tan(d*x)^2*tan(c) - 114345*a*b^2*tan(d*x)^2*tan(c) - 38115*b^3*tan(d*x)^2*tan(c) - 114345
*a^2*b*tan(d*x)*tan(c)^2 - 114345*a*b^2*tan(d*x)*tan(c)^2 - 38115*b^3*tan(d*x)*tan(c)^2 - 3465*a^2*b*tan(c)^3
- 3465*a*b^2*tan(c)^3 - 1155*b^3*tan(c)^3 - 3465*a^3*d*x - 10395*a^2*b*d*x - 10395*a*b^2*d*x - 3465*b^3*d*x +
10395*a^2*b*tan(d*x) + 10395*a*b^2*tan(d*x) + 3465*b^3*tan(d*x) + 10395*a^2*b*tan(c) + 10395*a*b^2*tan(c) + 34
65*b^3*tan(c))/(d*tan(d*x)^11*tan(c)^11 - 11*d*tan(d*x)^10*tan(c)^10 + 55*d*tan(d*x)^9*tan(c)^9 - 165*d*tan(d*
x)^8*tan(c)^8 + 330*d*tan(d*x)^7*tan(c)^7 - 462*d*tan(d*x)^6*tan(c)^6 + 462*d*tan(d*x)^5*tan(c)^5 - 330*d*tan(
d*x)^4*tan(c)^4 + 165*d*tan(d*x)^3*tan(c)^3 - 55*d*tan(d*x)^2*tan(c)^2 + 11*d*tan(d*x)*tan(c) - d)