3.218 \(\int \sec ^6(c+b x) \sin (a+b x) \, dx\)
Optimal. Leaf size=94 \[ \frac{3 \sin (a-c) \tanh ^{-1}(\sin (b x+c))}{8 b}+\frac{\cos (a-c) \sec ^5(b x+c)}{5 b}+\frac{\sin (a-c) \tan (b x+c) \sec ^3(b x+c)}{4 b}+\frac{3 \sin (a-c) \tan (b x+c) \sec (b x+c)}{8 b} \]
[Out]
(Cos[a - c]*Sec[c + b*x]^5)/(5*b) + (3*ArcTanh[Sin[c + b*x]]*Sin[a - c])/(8*b) + (3*Sec[c + b*x]*Sin[a - c]*Ta
n[c + b*x])/(8*b) + (Sec[c + b*x]^3*Sin[a - c]*Tan[c + b*x])/(4*b)
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Rubi [A] time = 0.0703124, antiderivative size = 94, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{4580, 2606, 30, 3768, 3770} \[ \frac{3 \sin (a-c) \tanh ^{-1}(\sin (b x+c))}{8 b}+\frac{\cos (a-c) \sec ^5(b x+c)}{5 b}+\frac{\sin (a-c) \tan (b x+c) \sec ^3(b x+c)}{4 b}+\frac{3 \sin (a-c) \tan (b x+c) \sec (b x+c)}{8 b} \]
Antiderivative was successfully verified.
[In]
Int[Sec[c + b*x]^6*Sin[a + b*x],x]
[Out]
(Cos[a - c]*Sec[c + b*x]^5)/(5*b) + (3*ArcTanh[Sin[c + b*x]]*Sin[a - c])/(8*b) + (3*Sec[c + b*x]*Sin[a - c]*Ta
n[c + b*x])/(8*b) + (Sec[c + b*x]^3*Sin[a - c]*Tan[c + b*x])/(4*b)
Rule 4580
Int[Sec[w_]^(n_.)*Sin[v_], x_Symbol] :> Dist[Cos[v - w], Int[Tan[w]*Sec[w]^(n - 1), x], x] + Dist[Sin[v - w],
Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Rule 2606
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 3768
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int \sec ^6(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \sec ^5(c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec ^5(c+b x) \, dx\\ &=\frac{\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}+\frac{\cos (a-c) \operatorname{Subst}\left (\int x^4 \, dx,x,\sec (c+b x)\right )}{b}+\frac{1}{4} (3 \sin (a-c)) \int \sec ^3(c+b x) \, dx\\ &=\frac{\cos (a-c) \sec ^5(c+b x)}{5 b}+\frac{3 \sec (c+b x) \sin (a-c) \tan (c+b x)}{8 b}+\frac{\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}+\frac{1}{8} (3 \sin (a-c)) \int \sec (c+b x) \, dx\\ &=\frac{\cos (a-c) \sec ^5(c+b x)}{5 b}+\frac{3 \tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{8 b}+\frac{3 \sec (c+b x) \sin (a-c) \tan (c+b x)}{8 b}+\frac{\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 1.00265, size = 78, normalized size = 0.83 \[ \frac{2 \sec ^5(b x+c) (5 \sin (a-c) (14 \sin (2 (b x+c))+3 \sin (4 (b x+c)))+64 \cos (a-c))+480 \sin (a-c) \tanh ^{-1}\left (\cos (c) \tan \left (\frac{b x}{2}\right )+\sin (c)\right )}{640 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Sec[c + b*x]^6*Sin[a + b*x],x]
[Out]
(480*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*Sin[a - c] + 2*Sec[c + b*x]^5*(64*Cos[a - c] + 5*Sin[a - c]*(14*Sin
[2*(c + b*x)] + 3*Sin[4*(c + b*x)])))/(640*b)
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Maple [B] time = 9.186, size = 97703, normalized size = 1039.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(b*x+c)^6*sin(b*x+a),x)
[Out]
result too large to display
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Maxima [B] time = 2.6606, size = 4180, normalized size = 44.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(b*x+c)^6*sin(b*x+a),x, algorithm="maxima")
[Out]
-1/80*(2*(15*cos(9*b*x + 2*a + 8*c) - 15*cos(9*b*x + 10*c) + 70*cos(7*b*x + 2*a + 6*c) - 70*cos(7*b*x + 8*c) -
128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*cos(3*b*x + 2*a + 2*c) + 70*cos(3*b*x + 4*c) - 15*cos(
b*x + 2*a) + 15*cos(b*x + 2*c))*cos(10*b*x + a + 10*c) + 30*(5*cos(8*b*x + a + 8*c) + 10*cos(6*b*x + a + 6*c)
+ 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(9*b*x + 2*a + 8*c) - 30*(5*cos(8*b*x + a + 8*
c) + 10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(9*b*x + 10*c) +
10*(70*cos(7*b*x + 2*a + 6*c) - 70*cos(7*b*x + 8*c) - 128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*c
os(3*b*x + 2*a + 2*c) + 70*cos(3*b*x + 4*c) - 15*cos(b*x + 2*a) + 15*cos(b*x + 2*c))*cos(8*b*x + a + 8*c) + 14
0*(10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(7*b*x + 2*a + 6*c)
- 140*(10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(7*b*x + 8*c)
- 20*(128*cos(5*b*x + 2*a + 4*c) + 128*cos(5*b*x + 6*c) + 70*cos(3*b*x + 2*a + 2*c) - 70*cos(3*b*x + 4*c) + 15
*cos(b*x + 2*a) - 15*cos(b*x + 2*c))*cos(6*b*x + a + 6*c) - 256*(10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2
*c) + cos(a))*cos(5*b*x + 2*a + 4*c) - 256*(10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b
*x + 6*c) - 100*(14*cos(3*b*x + 2*a + 2*c) - 14*cos(3*b*x + 4*c) + 3*cos(b*x + 2*a) - 3*cos(b*x + 2*c))*cos(4*
b*x + a + 4*c) - 140*(5*cos(2*b*x + a + 2*c) + cos(a))*cos(3*b*x + 2*a + 2*c) + 140*(5*cos(2*b*x + a + 2*c) +
cos(a))*cos(3*b*x + 4*c) - 150*(cos(b*x + 2*a) - cos(b*x + 2*c))*cos(2*b*x + a + 2*c) - 30*cos(b*x + 2*a)*cos(
a) + 30*cos(b*x + 2*c)*cos(a) - 15*(cos(10*b*x + a + 10*c)^2*sin(-a + c) + 25*cos(8*b*x + a + 8*c)^2*sin(-a +
c) + 100*cos(6*b*x + a + 6*c)^2*sin(-a + c) + 100*cos(4*b*x + a + 4*c)^2*sin(-a + c) + 25*cos(2*b*x + a + 2*c)
^2*sin(-a + c) + 10*cos(2*b*x + a + 2*c)*cos(a)*sin(-a + c) + sin(10*b*x + a + 10*c)^2*sin(-a + c) + 25*sin(8*
b*x + a + 8*c)^2*sin(-a + c) + 100*sin(6*b*x + a + 6*c)^2*sin(-a + c) + 100*sin(4*b*x + a + 4*c)^2*sin(-a + c)
+ 25*sin(2*b*x + a + 2*c)^2*sin(-a + c) + 10*sin(2*b*x + a + 2*c)*sin(a)*sin(-a + c) + 2*(5*cos(8*b*x + a + 8
*c)*sin(-a + c) + 10*cos(6*b*x + a + 6*c)*sin(-a + c) + 10*cos(4*b*x + a + 4*c)*sin(-a + c) + 5*cos(2*b*x + a
+ 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(10*b*x + a + 10*c) + 10*(10*cos(6*b*x + a + 6*c)*sin(-a + c) + 10
*cos(4*b*x + a + 4*c)*sin(-a + c) + 5*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(8*b*x + a + 8
*c) + 20*(10*cos(4*b*x + a + 4*c)*sin(-a + c) + 5*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(6
*b*x + a + 6*c) + 20*(5*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(4*b*x + a + 4*c) + 2*(5*sin
(8*b*x + a + 8*c)*sin(-a + c) + 10*sin(6*b*x + a + 6*c)*sin(-a + c) + 10*sin(4*b*x + a + 4*c)*sin(-a + c) + 5*
sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(-a + c))*sin(10*b*x + a + 10*c) + 10*(10*sin(6*b*x + a + 6*c)*si
n(-a + c) + 10*sin(4*b*x + a + 4*c)*sin(-a + c) + 5*sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(-a + c))*sin
(8*b*x + a + 8*c) + 20*(10*sin(4*b*x + a + 4*c)*sin(-a + c) + 5*sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(
-a + c))*sin(6*b*x + a + 6*c) + 20*(5*sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(-a + c))*sin(4*b*x + a + 4
*c) + (cos(a)^2 + sin(a)^2)*sin(-a + c))*log((cos(b*x + 2*c)^2 + cos(c)^2 - 2*cos(c)*sin(b*x + 2*c) + sin(b*x
+ 2*c)^2 + 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)/(cos(b*x + 2*c)^2 + cos(c)^2 + 2*cos(c)*sin(b*x + 2*c) + sin(b*
x + 2*c)^2 - 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)) + 2*(15*sin(9*b*x + 2*a + 8*c) - 15*sin(9*b*x + 10*c) + 70*s
in(7*b*x + 2*a + 6*c) - 70*sin(7*b*x + 8*c) - 128*sin(5*b*x + 2*a + 4*c) - 128*sin(5*b*x + 6*c) - 70*sin(3*b*x
+ 2*a + 2*c) + 70*sin(3*b*x + 4*c) - 15*sin(b*x + 2*a) + 15*sin(b*x + 2*c))*sin(10*b*x + a + 10*c) + 30*(5*si
n(8*b*x + a + 8*c) + 10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) + 5*sin(2*b*x + a + 2*c) + sin(a))*sin(
9*b*x + 2*a + 8*c) - 30*(5*sin(8*b*x + a + 8*c) + 10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) + 5*sin(2*
b*x + a + 2*c) + sin(a))*sin(9*b*x + 10*c) + 10*(70*sin(7*b*x + 2*a + 6*c) - 70*sin(7*b*x + 8*c) - 128*sin(5*b
*x + 2*a + 4*c) - 128*sin(5*b*x + 6*c) - 70*sin(3*b*x + 2*a + 2*c) + 70*sin(3*b*x + 4*c) - 15*sin(b*x + 2*a) +
15*sin(b*x + 2*c))*sin(8*b*x + a + 8*c) + 140*(10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) + 5*sin(2*b*
x + a + 2*c) + sin(a))*sin(7*b*x + 2*a + 6*c) - 140*(10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) + 5*sin
(2*b*x + a + 2*c) + sin(a))*sin(7*b*x + 8*c) - 20*(128*sin(5*b*x + 2*a + 4*c) + 128*sin(5*b*x + 6*c) + 70*sin(
3*b*x + 2*a + 2*c) - 70*sin(3*b*x + 4*c) + 15*sin(b*x + 2*a) - 15*sin(b*x + 2*c))*sin(6*b*x + a + 6*c) - 256*(
10*sin(4*b*x + a + 4*c) + 5*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*x + 2*a + 4*c) - 256*(10*sin(4*b*x + a + 4*
c) + 5*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*x + 6*c) - 100*(14*sin(3*b*x + 2*a + 2*c) - 14*sin(3*b*x + 4*c)
+ 3*sin(b*x + 2*a) - 3*sin(b*x + 2*c))*sin(4*b*x + a + 4*c) - 140*(5*sin(2*b*x + a + 2*c) + sin(a))*sin(3*b*x
+ 2*a + 2*c) + 140*(5*sin(2*b*x + a + 2*c) + sin(a))*sin(3*b*x + 4*c) - 150*(sin(b*x + 2*a) - sin(b*x + 2*c))*
sin(2*b*x + a + 2*c) - 30*sin(b*x + 2*a)*sin(a) + 30*sin(b*x + 2*c)*sin(a))/(b*cos(10*b*x + a + 10*c)^2 + 25*b
*cos(8*b*x + a + 8*c)^2 + 100*b*cos(6*b*x + a + 6*c)^2 + 100*b*cos(4*b*x + a + 4*c)^2 + 25*b*cos(2*b*x + a + 2
*c)^2 + 10*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(10*b*x + a + 10*c)^2 + 25*b*sin(8*b*x + a + 8*c)^2 + 100*b*si
n(6*b*x + a + 6*c)^2 + 100*b*sin(4*b*x + a + 4*c)^2 + 25*b*sin(2*b*x + a + 2*c)^2 + 10*b*sin(2*b*x + a + 2*c)*
sin(a) + (cos(a)^2 + sin(a)^2)*b + 2*(5*b*cos(8*b*x + a + 8*c) + 10*b*cos(6*b*x + a + 6*c) + 10*b*cos(4*b*x +
a + 4*c) + 5*b*cos(2*b*x + a + 2*c) + b*cos(a))*cos(10*b*x + a + 10*c) + 10*(10*b*cos(6*b*x + a + 6*c) + 10*b*
cos(4*b*x + a + 4*c) + 5*b*cos(2*b*x + a + 2*c) + b*cos(a))*cos(8*b*x + a + 8*c) + 20*(10*b*cos(4*b*x + a + 4*
c) + 5*b*cos(2*b*x + a + 2*c) + b*cos(a))*cos(6*b*x + a + 6*c) + 20*(5*b*cos(2*b*x + a + 2*c) + b*cos(a))*cos(
4*b*x + a + 4*c) + 2*(5*b*sin(8*b*x + a + 8*c) + 10*b*sin(6*b*x + a + 6*c) + 10*b*sin(4*b*x + a + 4*c) + 5*b*s
in(2*b*x + a + 2*c) + b*sin(a))*sin(10*b*x + a + 10*c) + 10*(10*b*sin(6*b*x + a + 6*c) + 10*b*sin(4*b*x + a +
4*c) + 5*b*sin(2*b*x + a + 2*c) + b*sin(a))*sin(8*b*x + a + 8*c) + 20*(10*b*sin(4*b*x + a + 4*c) + 5*b*sin(2*b
*x + a + 2*c) + b*sin(a))*sin(6*b*x + a + 6*c) + 20*(5*b*sin(2*b*x + a + 2*c) + b*sin(a))*sin(4*b*x + a + 4*c)
)
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Fricas [A] time = 0.547882, size = 294, normalized size = 3.13 \begin{align*} -\frac{15 \, \cos \left (b x + c\right )^{5} \log \left (\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 15 \, \cos \left (b x + c\right )^{5} \log \left (-\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) + 10 \,{\left (3 \, \cos \left (b x + c\right )^{3} + 2 \, \cos \left (b x + c\right )\right )} \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 16 \, \cos \left (-a + c\right )}{80 \, b \cos \left (b x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(b*x+c)^6*sin(b*x+a),x, algorithm="fricas")
[Out]
-1/80*(15*cos(b*x + c)^5*log(sin(b*x + c) + 1)*sin(-a + c) - 15*cos(b*x + c)^5*log(-sin(b*x + c) + 1)*sin(-a +
c) + 10*(3*cos(b*x + c)^3 + 2*cos(b*x + c))*sin(b*x + c)*sin(-a + c) - 16*cos(-a + c))/(b*cos(b*x + c)^5)
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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(sec(b*x+c)**6*sin(b*x+a),x)
[Out]
Timed out
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Giac [B] time = 1.21352, size = 1021, normalized size = 10.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(b*x+c)^6*sin(b*x+a),x, algorithm="giac")
[Out]
1/20*(15*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1
/2*c) + 1))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) - 15*(tan(1/2*a)^2*tan(1/2*c) - tan(
1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*c) - 1))/(tan(1/2*a)^2*tan(1/2*c)^2 +
tan(1/2*a)^2 + tan(1/2*c)^2 + 1) + 2*(25*tan(1/2*b*x + 1/2*c)^9*tan(1/2*a)^2*tan(1/2*c) - 25*tan(1/2*b*x + 1/
2*c)^9*tan(1/2*a)*tan(1/2*c)^2 - 20*tan(1/2*b*x + 1/2*c)^8*tan(1/2*a)^2*tan(1/2*c)^2 + 25*tan(1/2*b*x + 1/2*c)
^9*tan(1/2*a) + 20*tan(1/2*b*x + 1/2*c)^8*tan(1/2*a)^2 - 25*tan(1/2*b*x + 1/2*c)^9*tan(1/2*c) - 80*tan(1/2*b*x
+ 1/2*c)^8*tan(1/2*a)*tan(1/2*c) - 10*tan(1/2*b*x + 1/2*c)^7*tan(1/2*a)^2*tan(1/2*c) + 20*tan(1/2*b*x + 1/2*c
)^8*tan(1/2*c)^2 + 10*tan(1/2*b*x + 1/2*c)^7*tan(1/2*a)*tan(1/2*c)^2 - 20*tan(1/2*b*x + 1/2*c)^8 - 10*tan(1/2*
b*x + 1/2*c)^7*tan(1/2*a) + 10*tan(1/2*b*x + 1/2*c)^7*tan(1/2*c) - 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2*tan(
1/2*c)^2 + 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2 - 160*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)*tan(1/2*c) + 10*tan(
1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c) + 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*c)^2 - 10*tan(1/2*b*x + 1/2*c)^
3*tan(1/2*a)*tan(1/2*c)^2 - 40*tan(1/2*b*x + 1/2*c)^4 + 10*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a) - 10*tan(1/2*b*x
+ 1/2*c)^3*tan(1/2*c) - 25*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c) + 25*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*t
an(1/2*c)^2 - 4*tan(1/2*a)^2*tan(1/2*c)^2 - 25*tan(1/2*b*x + 1/2*c)*tan(1/2*a) + 4*tan(1/2*a)^2 + 25*tan(1/2*b
*x + 1/2*c)*tan(1/2*c) - 16*tan(1/2*a)*tan(1/2*c) + 4*tan(1/2*c)^2 - 4)/((tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*
a)^2 + tan(1/2*c)^2 + 1)*(tan(1/2*b*x + 1/2*c)^2 - 1)^5))/b