3.375 \(\int (a+b \tan ^3(c+d x))^3 \, dx\)
Optimal. Leaf size=168 \[ \frac{b \left (3 a^2-b^2\right ) \tan ^2(c+d x)}{2 d}+\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+a x \left (a^2-3 b^2\right )+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}-\frac{a b^2 \tan ^3(c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^8(c+d x)}{8 d}-\frac{b^3 \tan ^6(c+d x)}{6 d}+\frac{b^3 \tan ^4(c+d x)}{4 d} \]
[Out]
a*(a^2 - 3*b^2)*x + (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b*(3*a^2 - b^2)*Tan[c
+ d*x]^2)/(2*d) - (a*b^2*Tan[c + d*x]^3)/d + (b^3*Tan[c + d*x]^4)/(4*d) + (3*a*b^2*Tan[c + d*x]^5)/(5*d) - (b^
3*Tan[c + d*x]^6)/(6*d) + (b^3*Tan[c + d*x]^8)/(8*d)
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Rubi [A] time = 0.0977967, antiderivative size = 168, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used =
{3661, 1810, 635, 203, 260} \[ \frac{b \left (3 a^2-b^2\right ) \tan ^2(c+d x)}{2 d}+\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+a x \left (a^2-3 b^2\right )+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}-\frac{a b^2 \tan ^3(c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^8(c+d x)}{8 d}-\frac{b^3 \tan ^6(c+d x)}{6 d}+\frac{b^3 \tan ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Tan[c + d*x]^3)^3,x]
[Out]
a*(a^2 - 3*b^2)*x + (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b*(3*a^2 - b^2)*Tan[c
+ d*x]^2)/(2*d) - (a*b^2*Tan[c + d*x]^3)/d + (b^3*Tan[c + d*x]^4)/(4*d) + (3*a*b^2*Tan[c + d*x]^5)/(5*d) - (b^
3*Tan[c + d*x]^6)/(6*d) + (b^3*Tan[c + d*x]^8)/(8*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 1810
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rule 635
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[-(a*c)]
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])
Rule 260
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rubi steps
\begin{align*} \int \left (a+b \tan ^3(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^3\right )^3}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 a b^2+b \left (3 a^2-b^2\right ) x-3 a b^2 x^2+b^3 x^3+3 a b^2 x^4-b^3 x^5+b^3 x^7+\frac{a^3-3 a b^2-b \left (3 a^2-b^2\right ) x}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b \left (3 a^2-b^2\right ) \tan ^2(c+d x)}{2 d}-\frac{a b^2 \tan ^3(c+d x)}{d}+\frac{b^3 \tan ^4(c+d x)}{4 d}+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}-\frac{b^3 \tan ^6(c+d x)}{6 d}+\frac{b^3 \tan ^8(c+d x)}{8 d}+\frac{\operatorname{Subst}\left (\int \frac{a^3-3 a b^2-b \left (3 a^2-b^2\right ) x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b \left (3 a^2-b^2\right ) \tan ^2(c+d x)}{2 d}-\frac{a b^2 \tan ^3(c+d x)}{d}+\frac{b^3 \tan ^4(c+d x)}{4 d}+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}-\frac{b^3 \tan ^6(c+d x)}{6 d}+\frac{b^3 \tan ^8(c+d x)}{8 d}+\frac{\left (a \left (a^2-3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}-\frac{\left (b \left (3 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=a \left (a^2-3 b^2\right ) x+\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b \left (3 a^2-b^2\right ) \tan ^2(c+d x)}{2 d}-\frac{a b^2 \tan ^3(c+d x)}{d}+\frac{b^3 \tan ^4(c+d x)}{4 d}+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}-\frac{b^3 \tan ^6(c+d x)}{6 d}+\frac{b^3 \tan ^8(c+d x)}{8 d}\\ \end{align*}
Mathematica [C] time = 0.428701, size = 160, normalized size = 0.95 \[ \frac{-60 b \left (b^2-3 a^2\right ) \tan ^2(c+d x)+72 a b^2 \tan ^5(c+d x)-120 a b^2 \tan ^3(c+d x)+360 a b^2 \tan (c+d x)+60 \left (i (a+i b)^3 \log (\tan (c+d x)+i)-i (a-i b)^3 \log (-\tan (c+d x)+i)\right )+15 b^3 \tan ^8(c+d x)-20 b^3 \tan ^6(c+d x)+30 b^3 \tan ^4(c+d x)}{120 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Tan[c + d*x]^3)^3,x]
[Out]
(60*((-I)*(a - I*b)^3*Log[I - Tan[c + d*x]] + I*(a + I*b)^3*Log[I + Tan[c + d*x]]) + 360*a*b^2*Tan[c + d*x] -
60*b*(-3*a^2 + b^2)*Tan[c + d*x]^2 - 120*a*b^2*Tan[c + d*x]^3 + 30*b^3*Tan[c + d*x]^4 + 72*a*b^2*Tan[c + d*x]^
5 - 20*b^3*Tan[c + d*x]^6 + 15*b^3*Tan[c + d*x]^8)/(120*d)
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Maple [A] time = 0.007, size = 201, normalized size = 1.2 \begin{align*}{\frac{{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{8}}{8\,d}}-{\frac{{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,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 ) ^{4}}{4\,d}}-{\frac{a{b}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{3\, \left ( \tan \left ( dx+c \right ) \right ) ^{2}{a}^{2}b}{2\,d}}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{2\,d}}+3\,{\frac{a{b}^{2}\tan \left ( dx+c \right ) }{d}}-{\frac{3\,\ln \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ){a}^{2}b}{2\,d}}+{\frac{\ln \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ){b}^{3}}{2\,d}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}}{d}}-3\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*tan(d*x+c)^3)^3,x)
[Out]
1/8*b^3*tan(d*x+c)^8/d-1/6*b^3*tan(d*x+c)^6/d+3/5*a*b^2*tan(d*x+c)^5/d+1/4*b^3*tan(d*x+c)^4/d-a*b^2*tan(d*x+c)
^3/d+3/2/d*tan(d*x+c)^2*a^2*b-1/2/d*tan(d*x+c)^2*b^3+3*a*b^2*tan(d*x+c)/d-3/2/d*ln(tan(d*x+c)^2+1)*a^2*b+1/2/d
*ln(tan(d*x+c)^2+1)*b^3+1/d*arctan(tan(d*x+c))*a^3-3/d*arctan(tan(d*x+c))*a*b^2
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Maxima [A] time = 1.52337, size = 247, normalized size = 1.47 \begin{align*} a^{3} x + \frac{{\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a b^{2}}{5 \, d} + \frac{b^{3}{\left (\frac{48 \, \sin \left (d x + c\right )^{6} - 108 \, \sin \left (d x + c\right )^{4} + 88 \, \sin \left (d x + c\right )^{2} - 25}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 12 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{24 \, d} - \frac{3 \, a^{2} b{\left (\frac{1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(d*x+c)^3)^3,x, algorithm="maxima")
[Out]
a^3*x + 1/5*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c))*a*b^2/d + 1/24*b^3*((48*si
n(d*x + c)^6 - 108*sin(d*x + c)^4 + 88*sin(d*x + c)^2 - 25)/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c
)^4 - 4*sin(d*x + c)^2 + 1) - 12*log(sin(d*x + c)^2 - 1))/d - 3/2*a^2*b*(1/(sin(d*x + c)^2 - 1) - log(sin(d*x
+ c)^2 - 1))/d
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Fricas [A] time = 1.672, size = 365, normalized size = 2.17 \begin{align*} \frac{15 \, b^{3} \tan \left (d x + c\right )^{8} - 20 \, b^{3} \tan \left (d x + c\right )^{6} + 72 \, a b^{2} \tan \left (d x + c\right )^{5} + 30 \, b^{3} \tan \left (d x + c\right )^{4} - 120 \, a b^{2} \tan \left (d x + c\right )^{3} + 360 \, a b^{2} \tan \left (d x + c\right ) + 120 \,{\left (a^{3} - 3 \, a b^{2}\right )} d x + 60 \,{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{2} + 60 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(d*x+c)^3)^3,x, algorithm="fricas")
[Out]
1/120*(15*b^3*tan(d*x + c)^8 - 20*b^3*tan(d*x + c)^6 + 72*a*b^2*tan(d*x + c)^5 + 30*b^3*tan(d*x + c)^4 - 120*a
*b^2*tan(d*x + c)^3 + 360*a*b^2*tan(d*x + c) + 120*(a^3 - 3*a*b^2)*d*x + 60*(3*a^2*b - b^3)*tan(d*x + c)^2 + 6
0*(3*a^2*b - b^3)*log(1/(tan(d*x + c)^2 + 1)))/d
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Sympy [A] time = 1.90791, size = 194, normalized size = 1.15 \begin{align*} \begin{cases} a^{3} x - \frac{3 a^{2} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{3 a^{2} b \tan ^{2}{\left (c + d x \right )}}{2 d} - 3 a b^{2} x + \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} + \frac{b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b^{3} \tan ^{8}{\left (c + d x \right )}}{8 d} - \frac{b^{3} \tan ^{6}{\left (c + d x \right )}}{6 d} + \frac{b^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{b^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan ^{3}{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(d*x+c)**3)**3,x)
[Out]
Piecewise((a**3*x - 3*a**2*b*log(tan(c + d*x)**2 + 1)/(2*d) + 3*a**2*b*tan(c + d*x)**2/(2*d) - 3*a*b**2*x + 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*log(tan(c + d*x)**2 +
1)/(2*d) + b**3*tan(c + d*x)**8/(8*d) - b**3*tan(c + d*x)**6/(6*d) + b**3*tan(c + d*x)**4/(4*d) - b**3*tan(c
+ d*x)**2/(2*d), Ne(d, 0)), (x*(a + b*tan(c)**3)**3, True))
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Giac [B] time = 17.7387, size = 4219, normalized size = 25.11 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(d*x+c)^3)^3,x, algorithm="giac")
[Out]
1/120*(120*a^3*d*x*tan(d*x)^8*tan(c)^8 - 360*a*b^2*d*x*tan(d*x)^8*tan(c)^8 + 180*a^2*b*log(4*(tan(c)^2 + 1)/(t
an(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)
^8*tan(c)^8 - 60*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + t
an(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^8*tan(c)^8 - 960*a^3*d*x*tan(d*x)^7*tan(c)^7 + 2880*a*b^2*d*x*tan
(d*x)^7*tan(c)^7 + 180*a^2*b*tan(d*x)^8*tan(c)^8 - 125*b^3*tan(d*x)^8*tan(c)^8 - 1440*a^2*b*log(4*(tan(c)^2 +
1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan
(d*x)^7*tan(c)^7 + 480*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)
^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^7*tan(c)^7 - 360*a*b^2*tan(d*x)^8*tan(c)^7 - 360*a*b^2*tan(
d*x)^7*tan(c)^8 + 3360*a^3*d*x*tan(d*x)^6*tan(c)^6 - 10080*a*b^2*d*x*tan(d*x)^6*tan(c)^6 + 180*a^2*b*tan(d*x)^
8*tan(c)^6 - 60*b^3*tan(d*x)^8*tan(c)^6 - 1080*a^2*b*tan(d*x)^7*tan(c)^7 + 880*b^3*tan(d*x)^7*tan(c)^7 + 180*a
^2*b*tan(d*x)^6*tan(c)^8 - 60*b^3*tan(d*x)^6*tan(c)^8 + 120*a*b^2*tan(d*x)^8*tan(c)^5 + 5040*a^2*b*log(4*(tan(
c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) +
1))*tan(d*x)^6*tan(c)^6 - 1680*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^
2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^6*tan(c)^6 + 2880*a*b^2*tan(d*x)^7*tan(c)^6 + 2880*
a*b^2*tan(d*x)^6*tan(c)^7 + 120*a*b^2*tan(d*x)^5*tan(c)^8 + 30*b^3*tan(d*x)^8*tan(c)^4 - 6720*a^3*d*x*tan(d*x)
^5*tan(c)^5 + 20160*a*b^2*d*x*tan(d*x)^5*tan(c)^5 - 1080*a^2*b*tan(d*x)^7*tan(c)^5 + 480*b^3*tan(d*x)^7*tan(c)
^5 + 2880*a^2*b*tan(d*x)^6*tan(c)^6 - 2600*b^3*tan(d*x)^6*tan(c)^6 - 1080*a^2*b*tan(d*x)^5*tan(c)^7 + 480*b^3*
tan(d*x)^5*tan(c)^7 + 30*b^3*tan(d*x)^4*tan(c)^8 - 72*a*b^2*tan(d*x)^8*tan(c)^3 - 960*a*b^2*tan(d*x)^7*tan(c)^
4 - 10080*a^2*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*
x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 + 3360*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*ta
n(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 - 10080*a*b^2
*tan(d*x)^6*tan(c)^5 - 10080*a*b^2*tan(d*x)^5*tan(c)^6 - 960*a*b^2*tan(d*x)^4*tan(c)^7 - 72*a*b^2*tan(d*x)^3*t
an(c)^8 - 20*b^3*tan(d*x)^8*tan(c)^2 - 240*b^3*tan(d*x)^7*tan(c)^3 + 8400*a^3*d*x*tan(d*x)^4*tan(c)^4 - 25200*
a*b^2*d*x*tan(d*x)^4*tan(c)^4 + 2700*a^2*b*tan(d*x)^6*tan(c)^4 - 1680*b^3*tan(d*x)^6*tan(c)^4 - 4680*a^2*b*tan
(d*x)^5*tan(c)^5 + 4080*b^3*tan(d*x)^5*tan(c)^5 + 2700*a^2*b*tan(d*x)^4*tan(c)^6 - 1680*b^3*tan(d*x)^4*tan(c)^
6 - 240*b^3*tan(d*x)^3*tan(c)^7 - 20*b^3*tan(d*x)^2*tan(c)^8 + 216*a*b^2*tan(d*x)^7*tan(c)^2 + 2280*a*b^2*tan(
d*x)^6*tan(c)^3 + 12600*a^2*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan
(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 - 4200*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*t
an(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4
+ 18360*a*b^2*tan(d*x)^5*tan(c)^4 + 18360*a*b^2*tan(d*x)^4*tan(c)^5 + 2280*a*b^2*tan(d*x)^3*tan(c)^6 + 216*a*
b^2*tan(d*x)^2*tan(c)^7 + 15*b^3*tan(d*x)^8 + 160*b^3*tan(d*x)^7*tan(c) + 840*b^3*tan(d*x)^6*tan(c)^2 - 6720*a
^3*d*x*tan(d*x)^3*tan(c)^3 + 20160*a*b^2*d*x*tan(d*x)^3*tan(c)^3 - 3600*a^2*b*tan(d*x)^5*tan(c)^3 + 3360*b^3*t
an(d*x)^5*tan(c)^3 + 5400*a^2*b*tan(d*x)^4*tan(c)^4 - 3420*b^3*tan(d*x)^4*tan(c)^4 - 3600*a^2*b*tan(d*x)^3*tan
(c)^5 + 3360*b^3*tan(d*x)^3*tan(c)^5 + 840*b^3*tan(d*x)^2*tan(c)^6 + 160*b^3*tan(d*x)*tan(c)^7 + 15*b^3*tan(c)
^8 - 216*a*b^2*tan(d*x)^6*tan(c) - 2280*a*b^2*tan(d*x)^5*tan(c)^2 - 10080*a^2*b*log(4*(tan(c)^2 + 1)/(tan(d*x)
^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(
c)^3 + 3360*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*
x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 - 18360*a*b^2*tan(d*x)^4*tan(c)^3 - 18360*a*b^2*tan(d*x)^3*
tan(c)^4 - 2280*a*b^2*tan(d*x)^2*tan(c)^5 - 216*a*b^2*tan(d*x)*tan(c)^6 - 20*b^3*tan(d*x)^6 - 240*b^3*tan(d*x)
^5*tan(c) + 3360*a^3*d*x*tan(d*x)^2*tan(c)^2 - 10080*a*b^2*d*x*tan(d*x)^2*tan(c)^2 + 2700*a^2*b*tan(d*x)^4*tan
(c)^2 - 1680*b^3*tan(d*x)^4*tan(c)^2 - 4680*a^2*b*tan(d*x)^3*tan(c)^3 + 4080*b^3*tan(d*x)^3*tan(c)^3 + 2700*a^
2*b*tan(d*x)^2*tan(c)^4 - 1680*b^3*tan(d*x)^2*tan(c)^4 - 240*b^3*tan(d*x)*tan(c)^5 - 20*b^3*tan(c)^6 + 72*a*b^
2*tan(d*x)^5 + 960*a*b^2*tan(d*x)^4*tan(c) + 5040*a^2*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)
^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 - 1680*b^3*log(4*(t
an(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c)
+ 1))*tan(d*x)^2*tan(c)^2 + 10080*a*b^2*tan(d*x)^3*tan(c)^2 + 10080*a*b^2*tan(d*x)^2*tan(c)^3 + 960*a*b^2*tan
(d*x)*tan(c)^4 + 72*a*b^2*tan(c)^5 + 30*b^3*tan(d*x)^4 - 960*a^3*d*x*tan(d*x)*tan(c) + 2880*a*b^2*d*x*tan(d*x)
*tan(c) - 1080*a^2*b*tan(d*x)^3*tan(c) + 480*b^3*tan(d*x)^3*tan(c) + 2880*a^2*b*tan(d*x)^2*tan(c)^2 - 2600*b^3
*tan(d*x)^2*tan(c)^2 - 1080*a^2*b*tan(d*x)*tan(c)^3 + 480*b^3*tan(d*x)*tan(c)^3 + 30*b^3*tan(c)^4 - 120*a*b^2*
tan(d*x)^3 - 1440*a^2*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2
+ tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) + 480*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2
*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) - 2880*a*b^2*t
an(d*x)^2*tan(c) - 2880*a*b^2*tan(d*x)*tan(c)^2 - 120*a*b^2*tan(c)^3 + 120*a^3*d*x - 360*a*b^2*d*x + 180*a^2*b
*tan(d*x)^2 - 60*b^3*tan(d*x)^2 - 1080*a^2*b*tan(d*x)*tan(c) + 880*b^3*tan(d*x)*tan(c) + 180*a^2*b*tan(c)^2 -
60*b^3*tan(c)^2 + 180*a^2*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c
)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) - 60*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*t
an(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) + 360*a*b^2*tan(d*x) + 360*a*b^2*tan(c) + 1
80*a^2*b - 125*b^3)/(d*tan(d*x)^8*tan(c)^8 - 8*d*tan(d*x)^7*tan(c)^7 + 28*d*tan(d*x)^6*tan(c)^6 - 56*d*tan(d*x
)^5*tan(c)^5 + 70*d*tan(d*x)^4*tan(c)^4 - 56*d*tan(d*x)^3*tan(c)^3 + 28*d*tan(d*x)^2*tan(c)^2 - 8*d*tan(d*x)*t
an(c) + d)