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(f*x+e))^(2-2*m)*(a+a*sin(f*x+e))^(-1+m)*(c-c*sin(f*x+e))^n/f/g/(1-m+n)

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 = 0.72, 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))

________________________________________________________________________________________

fricas [B]  time = 0.51, size = 129, normalized size = 2.22 \[ \frac {\left (g \cos \left (f x + e\right )\right )^{-2 \, m + 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} e^{\left (2 \, n \log \left (g \cos \left (f x + e\right )\right ) - n \log \left (a \sin \left (f x + e\right ) + a\right ) + n \log \left (\frac {a c}{g^{2}}\right )\right )}}{f m - f n + {\left (f m - f n - f\right )} \cos \left (f x + e\right ) + {\left (f m - f n - f\right )} \sin \left (f x + e\right ) - f} \]

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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (8*pi/x/2)>(-8*pi/
x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/
x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check si
gn: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check si
gn: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/
x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check si
gn: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/
x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)Unable to check si
gn: (8*pi/x/2)>(-8*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)(-exp(m*ln(2)-2*m*ln(4*abs(tan((2*f*x-pi
+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a))-2*m*ln(abs(g))-n*ln(2)+2*n*ln(4*abs(tan((2*f*x-p
i+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+n*ln(abs(c))-ln(2)+ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan
((2*f*x-pi+2*exp(1))/8)^2+1))+ln(abs(g)))*tan((2*f*x-8*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi
+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*m*pi*floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m*pi*sig
n(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*e
xp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*floor(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*sign(ta
n((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*pi*f
loor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x+exp(1))/2)^2-1)+pi+2*exp(1))/8)^2*tan((f*x+exp(1)
)/2)^2+2*exp(m*ln(2)-2*m*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a))-2
*m*ln(abs(g))-n*ln(2)+2*n*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+n*ln(abs(c))-
ln(2)+ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+ln(abs(g)))*tan((2*f*x-8*m*pi*flo
or((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*m*pi*
floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m*pi*sign(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floor((
2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*floo
r(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*sign(tan((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4*pi*floor((f*x
+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*pi*floor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x+ex
p(1))/2)^2-1)+pi+2*exp(1))/8)^2*tan((f*x+exp(1))/2)-exp(m*ln(2)-2*m*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan(
(2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a))-2*m*ln(abs(g))-n*ln(2)+2*n*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan
((2*f*x-pi+2*exp(1))/8)^2+1))+n*ln(abs(c))-ln(2)+ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))
/8)^2+1))+ln(abs(g)))*tan((2*f*x-8*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)-8
*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*m*pi*floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m*pi*sign(g)-4*m*pi*sign(tan
((f*x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+8*n*p
i*floor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*floor(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*sign(tan((f*x+exp(1))/2)^2-
1)+2*n*pi+4*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*pi*floor((f*x+pi+exp(1))
*1/2/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x+exp(1))/2)^2-1)+pi+2*exp(1))/8)^2-2*exp(m*ln(2)-2*m*ln(4*abs(tan((2*f
*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a))-2*m*ln(abs(g))-n*ln(2)+2*n*ln(4*abs(tan((2*
f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+n*ln(abs(c))-ln(2)+ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))
/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+ln(abs(g)))*tan((2*f*x-8*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/p
i)+pi+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*m*pi*floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m*p
i*sign(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+p
i+2*exp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*floor(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*si
gn(tan((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4
*pi*floor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x+exp(1))/2)^2-1)+pi+2*exp(1))/8)*tan((f*x+exp
(1))/2)^2+2*exp(m*ln(2)-2*m*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a)
)-2*m*ln(abs(g))-n*ln(2)+2*n*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+n*ln(abs(c
))-ln(2)+ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+ln(abs(g)))*tan((2*f*x-8*m*pi*
floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*m*
pi*floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m*pi*sign(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floo
r((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*f
loor(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*sign(tan((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4*pi*floor((
f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*pi*floor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x
+exp(1))/2)^2-1)+pi+2*exp(1))/8)+exp(m*ln(2)-2*m*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))
/8)^2+1))+m*ln(abs(a))-2*m*ln(abs(g))-n*ln(2)+2*n*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1)
)/8)^2+1))+n*ln(abs(c))-ln(2)+ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+ln(abs(g)
))*tan((f*x+exp(1))/2)^2-2*exp(m*ln(2)-2*m*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+
1))+m*ln(abs(a))-2*m*ln(abs(g))-n*ln(2)+2*n*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2
+1))+n*ln(abs(c))-ln(2)+ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+ln(abs(g)))*tan
((f*x+exp(1))/2)+exp(m*ln(2)-2*m*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(a
bs(a))-2*m*ln(abs(g))-n*ln(2)+2*n*ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+n*ln(
abs(c))-ln(2)+ln(4*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+ln(abs(g))))/(f*m*tan((2*
f*x-8*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1
/2/pi)+8*m*pi*floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m*pi*sign(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+
8*n*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/p
i)+8*n*pi*floor(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*sign(tan((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4
*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*pi*floor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*si
gn(tan((f*x+exp(1))/2)^2-1)+pi+2*exp(1))/8)^2*tan((f*x+exp(1))/2)^2+f*m*tan((2*f*x-8*m*pi*floor((2*f*x-4*pi*fl
oor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*m*pi*floor(-(sign(a)-2
)/4)+2*m*pi*sign(a)-4*m*pi*sign(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floor((2*f*x-4*pi*floor(
(f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*floor(-(sign(c)-4)/4)
+2*n*pi*sign(c)+4*n*pi*sign(tan((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/p
i)+pi+2*exp(1))*1/4/pi)+4*pi*floor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x+exp(1))/2)^2-1)+pi+
2*exp(1))/8)^2+f*m*tan((f*x+exp(1))/2)^2+f*m-f*n*tan((2*f*x-8*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2
/pi)+pi+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*m*pi*floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m
*pi*sign(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)
+pi+2*exp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*floor(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*
sign(tan((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)
+4*pi*floor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x+exp(1))/2)^2-1)+pi+2*exp(1))/8)^2*tan((f*x
+exp(1))/2)^2-f*n*tan((2*f*x-8*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)-8*m*p
i*floor((f*x+pi+exp(1))*1/2/pi)+8*m*pi*floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m*pi*sign(g)-4*m*pi*sign(tan((f*
x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+8*n*pi*fl
oor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*floor(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*sign(tan((f*x+exp(1))/2)^2-1)+2
*n*pi+4*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*pi*floor((f*x+pi+exp(1))*1/2
/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x+exp(1))/2)^2-1)+pi+2*exp(1))/8)^2-f*n*tan((f*x+exp(1))/2)^2-f*n-f*tan((2*
f*x-8*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1
/2/pi)+8*m*pi*floor(-(sign(a)-2)/4)+2*m*pi*sign(a)-4*m*pi*sign(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+
8*n*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/p
i)+8*n*pi*floor(-(sign(c)-4)/4)+2*n*pi*sign(c)+4*n*pi*sign(tan((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4
*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*pi*floor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*si
gn(tan((f*x+exp(1))/2)^2-1)+pi+2*exp(1))/8)^2*tan((f*x+exp(1))/2)^2-f*tan((2*f*x-8*m*pi*floor((2*f*x-4*pi*floo
r((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)-8*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*m*pi*floor(-(sign(a)-2)/
4)+2*m*pi*sign(a)-4*m*pi*sign(g)-4*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-2*m*pi+8*n*pi*floor((2*f*x-4*pi*floor((f
*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+8*n*pi*floor((f*x+pi+exp(1))*1/2/pi)+8*n*pi*floor(-(sign(c)-4)/4)+2
*n*pi*sign(c)+4*n*pi*sign(tan((f*x+exp(1))/2)^2-1)+2*n*pi+4*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)
+pi+2*exp(1))*1/4/pi)+4*pi*floor((f*x+pi+exp(1))*1/2/pi)+2*pi*sign(g)+2*pi*sign(tan((f*x+exp(1))/2)^2-1)+pi+2*
exp(1))/8)^2-f*tan((f*x+exp(1))/2)^2-f)

________________________________________________________________________________________

maple [F]  time = 37.18, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{1-2 m} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}\, dx \]

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)

________________________________________________________________________________________

maxima [B]  time = 1.26, size = 207, normalized size = 3.57 \[ \frac {{\left (a^{m} c^{n} g - \frac {2 \, a^{m} c^{n} g \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a^{m} c^{n} g \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} e^{\left (2 \, n \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right ) - 2 \, m \log \left (-\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) + m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right ) - n \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (g^{2 \, m} {\left (m - n - 1\right )} + \frac {g^{2 \, m} {\left (m - n - 1\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} f} \]

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)

________________________________________________________________________________________

mupad [B]  time = 9.42, size = 74, normalized size = 1.28 \[ -\frac {g\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^n}{2\,f\,{\left (g\,\cos \left (e+f\,x\right )\right )}^{2\,m}\,\left (\sin \left (e+f\,x\right )+1\right )\,\left (n-m+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

-(g*(cos(2*e + 2*f*x) + 1)*(a*(sin(e + f*x) + 1))^m*(-c*(sin(e + f*x) - 1))^n)/(2*f*(g*cos(e + f*x))^(2*m)*(si
n(e + f*x) + 1)*(n - m + 1))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________