\(\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx\) [220]
Optimal result
Integrand size = 44, antiderivative size = 33 \[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx=\frac {a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f}
\]
[Out]
a^3*B*c^3*cos(f*x+e)^7*(a+a*sin(f*x+e))^(-3+m)/f
Rubi [A] (verified)
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number
of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3046, 2933}
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx=\frac {a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f}
\]
[In]
Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3*(B*(-3 + m) - B*(4 + m)*Sin[e + f*x]),x]
[Out]
(a^3*B*c^3*Cos[e + f*x]^7*(a + a*Sin[e + f*x])^(-3 + m))/f
Rule 2933
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x
] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[a*d*m + b*c*(m + p + 1), 0]
Rule 3046
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Rubi steps \begin{align*}
\text {integral}& = \left (a^3 c^3\right ) \int \cos ^6(e+f x) (a+a \sin (e+f x))^{-3+m} (B (-3+m)-B (4+m) \sin (e+f x)) \, dx \\ & = \frac {a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f} \\
\end{align*}
Mathematica [A] (verified)
Time = 7.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx=\frac {B c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^m}{f}
\]
[In]
Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3*(B*(-3 + m) - B*(4 + m)*Sin[e + f*x]),x]
[Out]
(B*c^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m)
/f
Maple [A] (verified)
Time = 13.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88
| | |
| method | result | size |
| | |
| parallelrisch |
\(\frac {c^{3} B \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m} \left (14 \cos \left (f x +e \right )-6 \cos \left (3 f x +3 e \right )+\sin \left (4 f x +4 e \right )-14 \sin \left (2 f x +2 e \right )\right )}{8 f}\) |
\(62\) |
| | |
|
|
|
|
[In]
int((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x,method=_RETURNVERBOSE)
[Out]
1/8*c^3*B*(a*(1+sin(f*x+e)))^m*(14*cos(f*x+e)-6*cos(3*f*x+3*e)+sin(4*f*x+4*e)-14*sin(2*f*x+2*e))/f
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (33) = 66\).
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.36
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx=-\frac {{\left (3 \, B c^{3} \cos \left (f x + e\right )^{3} - 4 \, B c^{3} \cos \left (f x + e\right ) - {\left (B c^{3} \cos \left (f x + e\right )^{3} - 4 \, B c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f}
\]
[In]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x, algorithm="fricas")
[Out]
-(3*B*c^3*cos(f*x + e)^3 - 4*B*c^3*cos(f*x + e) - (B*c^3*cos(f*x + e)^3 - 4*B*c^3*cos(f*x + e))*sin(f*x + e))*
(a*sin(f*x + e) + a)^m/f
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 898 vs. \(2 (31) = 62\).
Time = 65.47 (sec) , antiderivative size = 898, normalized size of antiderivative = 27.21
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x)
[Out]
Piecewise((-B*c**3*(a + 2*a*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**m*tan(e/2 + f*x/2)**8/(f*tan(e/2 + f*
x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) + 6*B*c**3*(a + 2*a
*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**m*tan(e/2 + f*x/2)**7/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x
/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) - 14*B*c**3*(a + 2*a*tan(e/2 + f*x/2)/(tan(e/2
+ f*x/2)**2 + 1))**m*tan(e/2 + f*x/2)**6/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x
/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) + 14*B*c**3*(a + 2*a*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**m*tan(
e/2 + f*x/2)**5/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x
/2)**2 + f) - 14*B*c**3*(a + 2*a*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**m*tan(e/2 + f*x/2)**3/(f*tan(e/2
+ f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) + 14*B*c**3*(a
+ 2*a*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**m*tan(e/2 + f*x/2)**2/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2
+ f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) - 6*B*c**3*(a + 2*a*tan(e/2 + f*x/2)/(ta
n(e/2 + f*x/2)**2 + 1))**m*tan(e/2 + f*x/2)/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f
*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) + B*c**3*(a + 2*a*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**m/(f*ta
n(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f), Ne(f, 0)
), (x*(B*(m - 3) - B*(m + 4)*sin(e))*(a*sin(e) + a)**m*(-c*sin(e) + c)**3, True))
Maxima [F]
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx=\int { {\left (B {\left (m + 4\right )} \sin \left (f x + e\right ) - B {\left (m - 3\right )}\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x }
\]
[In]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x, algorithm="maxima")
[Out]
integrate((B*(m + 4)*sin(f*x + e) - B*(m - 3))*(c*sin(f*x + e) - c)^3*(a*sin(f*x + e) + a)^m, x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 9586 vs. \(2 (33) = 66\).
Time = 45.41 (sec) , antiderivative size = 9586, normalized size of antiderivative = 290.48
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x, algorithm="giac")
[Out]
(B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e
)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) +
8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/
2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan
(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)
^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3
- 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*
x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(
1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)
^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*
f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) -
1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 +
4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^8 - 6*B*c^3*(sqrt(2*tan
(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x +
1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*
tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 +
8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^
2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x +
1/2*e)^2 + 1))^m*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x
+ 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*
pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4
+ 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f
*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/
4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*
x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x +
1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^7 + 14*B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1
/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(
f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2
*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2
*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x
+ 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m
*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*s
gn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*
f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*
x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4
*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2
*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2
) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*
m*sgn(tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^6 - B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 +
4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f
*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x +
e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e
) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a
)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(1/2*f*x + 1/2*e
)^8 - 14*B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan
(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x +
1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2
+ tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2
+ 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(
f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1
/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*ta
n(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(
2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x
+ 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(
tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/
2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2
*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^5 + 6*B*c^3*(sq
rt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/
2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x
+ e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/
2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x +
1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1
/2*f*x + 1/2*e)^2 + 1))^m*tan(1/2*f*x + 1/2*e)^7 - 14*B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*
tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x
+ 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)
^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)
+ 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/
(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(1/2*f*x + 1/2*e)^
6 + 14*B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f
*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/
2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 +
tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 +
2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*
x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2
*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(
1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*
a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x +
1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(ta
n(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*
e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e
)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^3 + 14*B*c^3*(sqr
t(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2
*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x
+ e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2
*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x +
1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/
2*f*x + 1/2*e)^2 + 1))^m*tan(1/2*f*x + 1/2*e)^5 - 14*B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*t
an(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x
+ 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^
4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) +
4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(
tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(-1/4*pi*m*sgn(2*a*
tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/
2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn
(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(
1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)
) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*t
an(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*ta
n(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2
*e)))^2*tan(1/2*f*x + 1/2*e)^2 + 6*B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(
1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan
(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)
^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1
/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan
(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*
e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1
/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2
*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) -
2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/
2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e
)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)
^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*
x + 1/2*e) - 14*B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3
+ 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2
*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/
2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x
+ e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2
+ tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(1/2*f*x + 1/2*e)^3 - B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1
/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(
f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2
*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2
*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x
+ 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m
*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*s
gn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*
f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*
x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4
*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2
*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2
) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*
m*sgn(tan(1/2*f*x + 1/2*e)))^2 + 14*B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan
(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*ta
n(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e
)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x +
1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*ta
n(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m*tan(1/2*f*x + 1/2*e)^2 - 6*B*c^3*(sqrt(
2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f
*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x +
e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e
)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/
2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*
f*x + 1/2*e)^2 + 1))^m*tan(1/2*f*x + 1/2*e) + B*c^3*(sqrt(2*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^4 + 4*tan(f*x
+ e)^4*tan(1/2*f*x + 1/2*e)^3 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e)^2 + 3*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e
)^4 + 4*tan(f*x + e)^4*tan(1/2*f*x + 1/2*e) + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^3 + 2*tan(f*x + e)^4 + 6*t
an(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e)^4 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(
1/2*f*x + 1/2*e)^3 + 3*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 4*tan(1/2*f*x + 1/2*e) + 1)*abs(a)/(tan(f*x
+ e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + tan(1/2*f*x + 1/2*e)^2 + 1))^m)/(f*tan(-1/4*pi*m*sgn(2*a*tan
(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e
)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(ta
n(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2
*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) -
1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(
1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1
/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)
))^2*tan(1/2*f*x + 1/2*e)^8 + 4*f*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 -
4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x
+ 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/
2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3
+ 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*
x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/
4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*
a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^6 + f*tan(1/2*f*x + 1/2*e
)^8 + 6*f*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e)
- 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn
(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*t
an(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2
*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*
a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*
e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e))
+ 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^4 + 4*f*tan(1/2*f*x + 1/2*e)^6 + 4*f*tan(-1/4*pi*
m*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/
2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^
2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3
- 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*
x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)
^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*
sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2
*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^2 + 6*f*tan(1/2*f*x + 1/2*e)^4 + f*tan(-1/4*pi*m*sgn(2*a*tan(1/2*f*x +
1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*t
an(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x +
1/2*e)) + pi*m*floor(1/4*sgn(2*a*tan(1/2*f*x + 1/2*e)^4 + 4*a*tan(1/2*f*x + 1/2*e)^3 - 4*a*tan(1/2*f*x + 1/2*
e) - 2*a)*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(ta
n(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e)) + 1/4*sgn(4*a*tan(1/2*f*x + 1/2*e)^3 + 8*a*tan(1/2*f*x + 1
/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) - 1/4*sgn(tan(1/2*f*x + 1/2*e)) + 1/2) - 1/4*pi*m*sgn(4*a*tan(1/2*f*x + 1/
2*e)^3 + 8*a*tan(1/2*f*x + 1/2*e)^2 + 4*a*tan(1/2*f*x + 1/2*e)) + 1/4*pi*m*sgn(tan(1/2*f*x + 1/2*e)))^2 + 4*f*
tan(1/2*f*x + 1/2*e)^2 + f)
Mupad [B] (verification not implemented)
Time = 13.62 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.85
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx=\frac {B\,c^3\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (14\,\cos \left (e+f\,x\right )-6\,\cos \left (3\,e+3\,f\,x\right )-14\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )\right )}{8\,f}
\]
[In]
int((B*(m - 3) - B*sin(e + f*x)*(m + 4))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^3,x)
[Out]
(B*c^3*(a*(sin(e + f*x) + 1))^m*(14*cos(e + f*x) - 6*cos(3*e + 3*f*x) - 14*sin(2*e + 2*f*x) + sin(4*e + 4*f*x)
))/(8*f)