\(\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx\) [222]
Optimal result
Integrand size = 47, antiderivative size = 36 \[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx=\frac {B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{f}
\]
[Out]
B*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n/f
Rubi [A] (verified)
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {3049}
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx=\frac {B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f}
\]
[In]
Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(B*(m - n) - B*(1 + m + n)*Sin[e + f*x]),x]
[Out]
(B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/f
Rule 3049
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*
(m + n + 1))), x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ
[A*b*(m + n + 1) + a*B*(m - n), 0] && NeQ[m, -2^(-1)]
Rubi steps \begin{align*}
\text {integral}& = \frac {B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{f} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.87 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx=\frac {B \cos (e+f x) (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^n}{f}
\]
[In]
Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(B*(m - n) - B*(1 + m + n)*Sin[e + f*x]),x]
[Out]
(B*Cos[e + f*x]*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n)/f
Maple [A] (verified)
Time = 9.82 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03
| | |
| method | result | size |
| | |
| parallelrisch |
\(\frac {B \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m} \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{n} \cos \left (f x +e \right )}{f}\) |
\(37\) |
| | |
|
|
|
|
[In]
int((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n*(B*(m-n)-B*(1+m+n)*sin(f*x+e)),x,method=_RETURNVERBOSE)
[Out]
B/f*(a*(1+sin(f*x+e)))^m*(-c*(sin(f*x+e)-1))^n*cos(f*x+e)
Fricas [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx=\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n} B \cos \left (f x + e\right )}{f}
\]
[In]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n*(B*(m-n)-B*(1+m+n)*sin(f*x+e)),x, algorithm="fricas")
[Out]
(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n*B*cos(f*x + e)/f
Sympy [F]
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx=- B \left (\int \left (- m \left (a \sin {\left (e + f x \right )} + a\right )^{m} \left (- c \sin {\left (e + f x \right )} + c\right )^{n}\right )\, dx + \int n \left (a \sin {\left (e + f x \right )} + a\right )^{m} \left (- c \sin {\left (e + f x \right )} + c\right )^{n}\, dx + \int \left (a \sin {\left (e + f x \right )} + a\right )^{m} \left (- c \sin {\left (e + f x \right )} + c\right )^{n} \sin {\left (e + f x \right )}\, dx + \int m \left (a \sin {\left (e + f x \right )} + a\right )^{m} \left (- c \sin {\left (e + f x \right )} + c\right )^{n} \sin {\left (e + f x \right )}\, dx + \int n \left (a \sin {\left (e + f x \right )} + a\right )^{m} \left (- c \sin {\left (e + f x \right )} + c\right )^{n} \sin {\left (e + f x \right )}\, dx\right )
\]
[In]
integrate((a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**n*(B*(m-n)-B*(1+m+n)*sin(f*x+e)),x)
[Out]
-B*(Integral(-m*(a*sin(e + f*x) + a)**m*(-c*sin(e + f*x) + c)**n, x) + Integral(n*(a*sin(e + f*x) + a)**m*(-c*
sin(e + f*x) + c)**n, x) + Integral((a*sin(e + f*x) + a)**m*(-c*sin(e + f*x) + c)**n*sin(e + f*x), x) + Integr
al(m*(a*sin(e + f*x) + a)**m*(-c*sin(e + f*x) + c)**n*sin(e + f*x), x) + Integral(n*(a*sin(e + f*x) + a)**m*(-
c*sin(e + f*x) + c)**n*sin(e + f*x), x))
Maxima [F]
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx=\int { -{\left (B {\left (m + n + 1\right )} \sin \left (f x + e\right ) - B {\left (m - n\right )}\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n} \,d x }
\]
[In]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n*(B*(m-n)-B*(1+m+n)*sin(f*x+e)),x, algorithm="maxima")
[Out]
-integrate((B*(m + n + 1)*sin(f*x + e) - B*(m - n))*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n, x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 8947 vs. \(2 (36) = 72\).
Time = 53.15 (sec) , antiderivative size = 8947, normalized size of antiderivative = 248.53
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n*(B*(m-n)-B*(1+m+n)*sin(f*x+e)),x, algorithm="giac")
[Out]
(B*cos(2*pi*m*floor(-1/8*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) +
4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 2*pi*n*floor(-1/8*sgn(4*ta
n(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x
+ 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*ta
n(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)
+ 1/4*pi*n*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e
)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) - 1/4*pi*m - 1/4*pi*n)*e^(-m*log(2) - n*log(2) +
m*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan
(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + a
bs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(
1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2
+ 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e)
+ 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/
2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2))*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)) + n*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*ta
n(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8
*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2
- 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e
) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) +
4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f
*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x +
1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2))*abs(c)/(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)))*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*n*sgn(2*c*tan(1/2*f*x + 1/2*e)^4 - 4*c*tan(1/2*f*x + 1/2*e)^3 + 4*c*tan(1/2*f*x + 1/2*e) - 2*c)*sgn(
4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2 + 4*c*tan(1/2*f*x + 1/2*e)) + 1/2*pi*m*floor(f*x/pi +
e/pi + 1/2) + 1/2*pi*n*floor(f*x/pi + e/pi + 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*n*sgn(4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2
+ 4*c*tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^2 + 2*B*e^(-m*log(2) - n*log(2) + m*log(sqrt(2)*sqrt(abs(4
*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*
f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(
1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*t
an(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1
/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(1/2*f*x + 1/2*
e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2
+ 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2))*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)) + n*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8
*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) +
2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(
1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2 +
abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan
(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x +
1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x
+ 1/2*e) + 2))*abs(c)/(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)))
*sin(2*pi*m*floor(-1/8*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4
*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 2*pi*n*floor(-1/8*sgn(4*tan(
f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x +
1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(
f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2) +
1/4*pi*n*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^
2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) - 1/4*pi*m - 1/4*pi*n)*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*n*sgn(2*c*tan(1/2*f*x + 1/2*e)^4 - 4*c*tan(
1/2*f*x + 1/2*e)^3 + 4*c*tan(1/2*f*x + 1/2*e) - 2*c)*sgn(4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)
^2 + 4*c*tan(1/2*f*x + 1/2*e)) + 1/2*pi*m*floor(f*x/pi + e/pi + 1/2) + 1/2*pi*n*floor(f*x/pi + e/pi + 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*n*sgn
(4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2 + 4*c*tan(1/2*f*x + 1/2*e)))*tan(1/2*f*x + 1/2*e)^2 -
B*cos(2*pi*m*floor(-1/8*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) +
4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 2*pi*n*floor(-1/8*sgn(4*ta
n(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x
+ 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*ta
n(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)
+ 1/4*pi*n*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e
)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) - 1/4*pi*m - 1/4*pi*n)*e^(-m*log(2) - n*log(2) +
m*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan
(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + a
bs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(
1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2
+ 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e)
+ 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/
2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2))*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)) + n*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*ta
n(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8
*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2
- 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e
) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) +
4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f
*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x +
1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2))*abs(c)/(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)))*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*n*sgn(2*c*tan(1/2*f*x + 1/2*e)^4 - 4*c*tan(1/2*f*x + 1/2*e)^3 + 4*c*tan(1/2*f*x + 1/2*e) - 2*c)*sgn(
4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2 + 4*c*tan(1/2*f*x + 1/2*e)) + 1/2*pi*m*floor(f*x/pi +
e/pi + 1/2) + 1/2*pi*n*floor(f*x/pi + e/pi + 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*n*sgn(4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2
+ 4*c*tan(1/2*f*x + 1/2*e)))^2 - B*cos(2*pi*m*floor(-1/8*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f
*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2) + 5
/8) + 2*pi*n*floor(-1/8*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) +
4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(f*x + e)
^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^
2 + 8*tan(1/2*f*x + 1/2*e) + 2) + 1/4*pi*n*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(
1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) - 1/4*pi*m - 1/4*
pi*n)*e^(-m*log(2) - n*log(2) + m*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e
)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x +
e)^2*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/
2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*
x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1
/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8
*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) +
2))*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)) + n*log(sqrt
(2)*sqrt(abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^
2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f
*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x +
1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*
x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(
1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*t
an(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2))*abs(c)/(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)))*tan(1/2*f*x + 1/2*e)^2 - 2*B*e^(-m*log(2) - n*log(2)
+ m*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*t
an(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 +
abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*ta
n(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^
2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*
e) + 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x +
1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2))*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)) + n*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*
tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 -
8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)
^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2
*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)
+ 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan
(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x
+ 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2))*abs(c)/(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)))*sin(2*pi*m*floor(-1/8*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^
2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 2*
pi*n*floor(-1/8*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*
x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(f*x + e)^2*tan(1
/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*ta
n(1/2*f*x + 1/2*e) + 2) + 1/4*pi*n*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x
+ 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) - 1/4*pi*m - 1/4*pi*n)*ta
n(-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*n*sgn(2*c*tan(1/2
*f*x + 1/2*e)^4 - 4*c*tan(1/2*f*x + 1/2*e)^3 + 4*c*tan(1/2*f*x + 1/2*e) - 2*c)*sgn(4*c*tan(1/2*f*x + 1/2*e)^3
- 8*c*tan(1/2*f*x + 1/2*e)^2 + 4*c*tan(1/2*f*x + 1/2*e)) + 1/2*pi*m*floor(f*x/pi + e/pi + 1/2) + 1/2*pi*n*floo
r(f*x/pi + e/pi + 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*n*sgn(4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2 + 4*c*tan(1/2*f*x + 1/2*e))
) + B*cos(2*pi*m*floor(-1/8*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e
) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 2*pi*n*floor(-1/8*sgn(4
*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*
f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8
*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) +
2) + 1/4*pi*n*sgn(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x
+ e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2) - 1/4*pi*m - 1/4*pi*n)*e^(-m*log(2) - n*log(2)
+ m*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*
tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2
+ abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*t
an(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)
^2 + 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2
*e) + 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(f*x + e)^2*tan(1/2*f*x +
1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 + 8*tan(1/2*f*x + 1/2*e) + 2))*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)) + n*log(sqrt(2)*sqrt(abs(4*tan(f*x + e)^2
*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2
- 8*tan(1/2*f*x + 1/2*e) + 2)*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e
)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/
2*e) + 2)*tan(f*x + e)^2 + abs(4*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)
+ 4*tan(f*x + e)^2 + 2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2)*tan(1/2*f*x + 1/2*e)^2 + abs(4*ta
n(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 - 8*tan(f*x + e)^2*tan(1/2*f*x + 1/2*e) + 4*tan(f*x + e)^2 + 2*tan(1/2*f*x
+ 1/2*e)^2 - 8*tan(1/2*f*x + 1/2*e) + 2))*abs(c)/(tan(f*x + e)^2*tan(1/2*f*x + 1/2*e)^2 + tan(f*x + e)^2 + ta
n(1/2*f*x + 1/2*e)^2 + 1))))/(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*n*sgn(2*c*tan(1/2*f*x + 1/2*e)^4 - 4*c*tan(1/2*f*x + 1/2*e)^3 + 4*c*tan(1/2*f*x + 1/2*e) - 2*
c)*sgn(4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2 + 4*c*tan(1/2*f*x + 1/2*e)) + 1/2*pi*m*floor(f*
x/pi + e/pi + 1/2) + 1/2*pi*n*floor(f*x/pi + e/pi + 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*n*sgn(4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/
2*e)^2 + 4*c*tan(1/2*f*x + 1/2*e)))^2*tan(1/2*f*x + 1/2*e)^2 + 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*n*sgn(2*c*tan(1/2*f*x + 1/2*e)^4 - 4*c*tan(1/2*f*x + 1/2*e)^
3 + 4*c*tan(1/2*f*x + 1/2*e) - 2*c)*sgn(4*c*tan(1/2*f*x + 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2 + 4*c*tan(1/2*
f*x + 1/2*e)) + 1/2*pi*m*floor(f*x/pi + e/pi + 1/2) + 1/2*pi*n*floor(f*x/pi + e/pi + 1/2) - 1/4*pi*m*sgn(4*a*t
an(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*n*sgn(4*c*tan(1/2*f*x
+ 1/2*e)^3 - 8*c*tan(1/2*f*x + 1/2*e)^2 + 4*c*tan(1/2*f*x + 1/2*e)))^2 + f*tan(1/2*f*x + 1/2*e)^2 + f)
Mupad [B] (verification not implemented)
Time = 12.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
\[
\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx=\frac {B\,\cos \left (e+f\,x\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^n}{f}
\]
[In]
int((B*(m - n) - B*sin(e + f*x)*(m + n + 1))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n,x)
[Out]
(B*cos(e + f*x)*(a*(sin(e + f*x) + 1))^m*(-c*(sin(e + f*x) - 1))^n)/f