3.177 \(\int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx\)
Optimal. Leaf size=58 \[ -\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{2-2 m}}{f g (-m+n+1)} \]
[Out]
-((a*(g*Cos[e + f*x])^(2 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(1 - m + n)))
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Rubi [A] time = 0.163097, antiderivative size = 58, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.025, Rules used =
{2844} \[ -\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{2-2 m}}{f g (-m+n+1)} \]
Antiderivative was successfully verified.
[In]
Int[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]
[Out]
-((a*(g*Cos[e + f*x])^(2 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(1 - m + n)))
Rule 2844
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e +
f*x])^n)/(f*g*(m - n - 1)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b
^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m - n - 1, 0]
Rubi steps
\begin{align*} \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx &=-\frac{a (g \cos (e+f x))^{2-2 m} (a+a \sin (e+f x))^{-1+m} (c-c \sin (e+f x))^n}{f g (1-m+n)}\\ \end{align*}
Mathematica [A] time = 0.672585, size = 96, normalized size = 1.66 \[ \frac{g (\sin (e+f x)-1) \cos ^{2 n}(e+f x) (g \cos (e+f x))^{-2 m} (a (\sin (e+f x)+1))^{m-n} \exp (n (\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x))-2 \log (\cos (e+f x))))}{f (-m+n+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]
[Out]
(E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*g*Cos[e + f*x]^(2*n)*(-1 +
Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(m - n))/(f*(1 - m + n)*(g*Cos[e + f*x])^(2*m))
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Maple [F] time = 20.304, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{1-2\,m} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)
[Out]
int((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)
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Maxima [B] time = 1.71773, size = 279, normalized size = 4.81 \begin{align*} \frac{{\left (a^{m} c^{n} g - \frac{2 \, a^{m} c^{n} g \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{a^{m} c^{n} g \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} e^{\left (2 \, n \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right ) - 2 \, m \log \left (-\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) + m \log \left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right ) - n \log \left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (g^{2 \, m}{\left (m - n - 1\right )} + \frac{g^{2 \, m}{\left (m - n - 1\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="maxima")
[Out]
(a^m*c^n*g - 2*a^m*c^n*g*sin(f*x + e)/(cos(f*x + e) + 1) + a^m*c^n*g*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*e^(2
*n*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1) - 2*m*log(-sin(f*x + e)/(cos(f*x + e) + 1) + 1) + m*log(sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 + 1) - n*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/((g^(2*m)*(m - n - 1) + g^(2*
m)*(m - n - 1)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*f)
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Fricas [B] time = 1.82133, size = 320, normalized size = 5.52 \begin{align*} \frac{\left (g \cos \left (f x + e\right )\right )^{-2 \, m + 1}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} e^{\left (2 \, n \log \left (g \cos \left (f x + e\right )\right ) - n \log \left (a \sin \left (f x + e\right ) + a\right ) + n \log \left (\frac{a c}{g^{2}}\right )\right )}}{f m - f n +{\left (f m - f n - f\right )} \cos \left (f x + e\right ) +{\left (f m - f n - f\right )} \sin \left (f x + e\right ) - f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="fricas")
[Out]
(g*cos(f*x + e))^(-2*m + 1)*(a*sin(f*x + e) + a)^m*(cos(f*x + e) - sin(f*x + e) + 1)*e^(2*n*log(g*cos(f*x + e)
) - n*log(a*sin(f*x + e) + a) + n*log(a*c/g^2))/(f*m - f*n + (f*m - f*n - f)*cos(f*x + e) + (f*m - f*n - f)*si
n(f*x + e) - f)
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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((g*cos(f*x+e))**(1-2*m)*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**n,x)
[Out]
Timed out
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Giac [B] time = 29.6986, size = 6257, normalized size = 107.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="giac")
[Out]
-(e^(m*log(2) - n*log(2) - 2*m*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1
)) + 2*n*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + m*log(abs(a)) + n
*log(abs(c)) - 2*m*log(abs(g)) - log(2) + log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1
/4*e)^2 + 1)) + log(abs(g)))*tan(1/8*pi - pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2
) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*m*floor(1/2*f*x/p
i + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*n*floor(-1/
4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m
*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*n + 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi - floor(
1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(tan(1/2*f*x + 1/2
*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)^2*tan(1/2*f*x + 1/2*e)^2 - 2*e^(m*log(2) - n*log(2) - 2*m*log(4*abs(tan(-1
/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + 2*n*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*
e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + m*log(abs(a)) + n*log(abs(c)) - 2*m*log(abs(g)) - log(2) + log(4
*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + log(abs(g)))*tan(1/8*pi - pi*m*
floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - f
loor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*m*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1
/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*
e)^2 - 1) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1
/4*pi*n + 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*fl
oor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)^2*tan(1/2*f
*x + 1/2*e) + 2*e^(m*log(2) - n*log(2) - 2*m*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x
+ 1/4*e)^2 + 1)) + 2*n*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + m*l
og(abs(a)) + n*log(abs(c)) - 2*m*log(abs(g)) - log(2) + log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi
+ 1/4*f*x + 1/4*e)^2 + 1)) + log(abs(g)))*tan(1/8*pi - pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi +
1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*m*f
loor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) +
pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^2 -
1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*n + 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi + 1/2
*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(tan
(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)*tan(1/2*f*x + 1/2*e)^2 + e^(m*log(2) - n*log(2) - 2*m*log(4*
abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + 2*n*log(4*abs(tan(-1/8*pi + 1/4*
f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + m*log(abs(a)) + n*log(abs(c)) - 2*m*log(abs(g)) - log(
2) + log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + log(abs(g)))*tan(1/8*
pi - pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/
2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*m*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*
f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*
f*x + 1/2*e)^2 - 1) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*
sgn(g) + 1/4*pi*n + 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) +
1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)^2
- e^(m*log(2) - n*log(2) - 2*m*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 +
1)) + 2*n*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + m*log(abs(a)) +
n*log(abs(c)) - 2*m*log(abs(g)) - log(2) + log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x +
1/4*e)^2 + 1)) + log(abs(g)))*tan(1/2*f*x + 1/2*e)^2 - 2*e^(m*log(2) - n*log(2) - 2*m*log(4*abs(tan(-1/8*pi +
1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + 2*n*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan
(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + m*log(abs(a)) + n*log(abs(c)) - 2*m*log(abs(g)) - log(2) + log(4*abs(tan
(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + log(abs(g)))*tan(1/8*pi - pi*m*floor(1/
2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2
*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*m*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi
+ 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1
) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*n
+ 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*floor(1/2*
f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e) + 2*e^(m*log(2) - n
*log(2) - 2*m*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + 2*n*log(4*ab
s(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + m*log(abs(a)) + n*log(abs(c)) - 2*
m*log(abs(g)) - log(2) + log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + l
og(abs(g)))*tan(1/2*f*x + 1/2*e) - e^(m*log(2) - n*log(2) - 2*m*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan
(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + 2*n*log(4*abs(tan(-1/8*pi + 1/4*f*x + 1/4*e))/(tan(-1/8*pi + 1/4*f*x + 1
/4*e)^2 + 1)) + m*log(abs(a)) + n*log(abs(c)) - 2*m*log(abs(g)) - log(2) + log(4*abs(tan(-1/8*pi + 1/4*f*x + 1
/4*e))/(tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1)) + log(abs(g))))/(f*m*tan(1/8*pi - pi*m*floor(1/2*f*x/pi + 1/2*e
/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/
pi + 1/2) + 1/4) - pi*m*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*fl
oor(-1/4*sgn(a) + 1/2) + pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/2*pi*n*sgn
(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*n + 1/4*f*x + 1/2*
pi*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/p
i + 1/2) + 1/4*pi*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)^2*tan(1/2*f*x + 1/2*e)^2 - f*n*tan(
1/8*pi - pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi
+ 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*m*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(
1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(
1/2*f*x + 1/2*e)^2 - 1) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*p
i*m*sgn(g) + 1/4*pi*n + 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/
4) + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*
e)^2*tan(1/2*f*x + 1/2*e)^2 - f*tan(1/8*pi - pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi +
1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*m*floor(1/2*f*
x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*n*floor(
-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*p
i*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*n + 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi - flo
or(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(tan(1/2*f*x +
1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)^2*tan(1/2*f*x + 1/2*e)^2 + f*m*tan(1/8*pi - pi*m*floor(1/2*f*x/pi + 1/2
*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*
e/pi + 1/2) + 1/4) - pi*m*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*
floor(-1/4*sgn(a) + 1/2) + pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/2*pi*n*s
gn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*n + 1/4*f*x + 1/
2*pi*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*floor(1/2*f*x/pi + 1/2*e
/pi + 1/2) + 1/4*pi*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)^2 - f*n*tan(1/8*pi - pi*m*floor(1
/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/
2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*m*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi
+ 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 -
1) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*n
+ 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*floor(1/2
*f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)^2 + f*m*tan(1/2*f*
x + 1/2*e)^2 - f*n*tan(1/2*f*x + 1/2*e)^2 - f*tan(1/8*pi - pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi
+ 1/2*e/pi + 1/2) + 1/4) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) - pi*
m*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*n*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2)
+ pi*n*floor(-1/4*sgn(c) + 1) - 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/2*pi*n*sgn(tan(1/2*f*x + 1/2*e)^
2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*n*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*n + 1/4*f*x + 1/2*pi*floor(1/2*f*x/pi +
1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4*pi*sgn(
tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*sgn(g) + 1/4*e)^2 - f*tan(1/2*f*x + 1/2*e)^2 + f*m - f*n - f)