3.2.0 ∫ (g cos(e + fx))1−2m(a + a sin(e + fx))m(c − c sin(e + fx))n dx [177]

Integrand size = 40, antiderivative size = 58 \[ \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=-\frac {a (g \cos (e+f x))^{2-2 m} (a+a \sin (e+f x))^{-1+m} (c-c \sin (e+f x))^n}{f g (1-m+n)} \]

-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] (veriﬁed)

Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2923} \[ \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=-\frac {a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{2-2 m}}{f g (-m+n+1)} \]

Int[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]

-((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)))

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

Rubi steps \begin{align*} \text {integral}& = -\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] (veriﬁed)

Time = 1.80 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.66 \[ \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\frac {e^{n (-2 \log (\cos (e+f x))+\log (a (1+\sin (e+f x)))+\log (c-c \sin (e+f x)))} g \cos ^{2 n}(e+f x) (g \cos (e+f x))^{-2 m} (-1+\sin (e+f x)) (a (1+\sin (e+f x)))^{m-n}}{f (1-m+n)} \]

Integrate[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]

(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))

Maple [A] (veriﬁed)

Time = 39.98 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03


method	result	size

parallelrisch	\(-\frac {g \left (g \cos \left (f x +e \right )\right )^{-2 m} \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m} \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{n} \left (\sin \left (f x +e \right )-1\right )}{f \left (m -n -1\right )}\)	\(60\)

int((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x,method=_RETURNVERBOSE)

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

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (57) = 114\).

Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.22 \[ \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\frac {\left (g \cos \left (f x + e\right )\right )^{-2 \, m + 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} e^{\left (2 \, n \log \left (g \cos \left (f x + e\right )\right ) - n \log \left (a \sin \left (f x + e\right ) + a\right ) + n \log \left (\frac {a c}{g^{2}}\right )\right )}}{f m - f n + {\left (f m - f n - f\right )} \cos \left (f x + e\right ) + {\left (f m - f n - f\right )} \sin \left (f x + e\right ) - f} \]

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")

(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

Sympy [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\text {Timed out} \]

integrate((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] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (57) = 114\).

Time = 0.33 (sec) , antiderivative size = 207, normalized size of antiderivative = 3.57 \[ \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\frac {{\left (a^{m} c^{n} g - \frac {2 \, a^{m} c^{n} g \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a^{m} c^{n} g \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} e^{\left (2 \, n \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right ) - 2 \, m \log \left (-\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) + m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right ) - n \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (g^{2 \, m} {\left (m - n - 1\right )} + \frac {g^{2 \, m} {\left (m - n - 1\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} f} \]

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")

(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*

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4499 vs. \(2 (57) = 114\).

Time = 2.05 (sec) , antiderivative size = 4499, normalized size of antiderivative = 77.57 \[ \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\text {Too large to display} \]

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")

-(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

a) - 1/4*pi*n*sgn(c) + 1/2*pi*m*sgn(g) + 1/4*pi*m - 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 -

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*m - 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*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*m - 1/4*pi*n - 1/4*f*x - 1/2*pi*floo

r(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*l

og(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(ta

n(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*m - 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/p

i - 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/p

i + 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*m - 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) - 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*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*m - 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 - 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*m - 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*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*f

loor(-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*sg

n(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*m - 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*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*floo

r(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*s

gn(c) + 1/2*pi*m*sgn(g) + 1/4*pi*m - 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/p

i + 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/p

i + 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*fl

oor(-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*m - 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*floo

r(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*s

gn(c) + 1/2*pi*m*sgn(g) + 1/4*pi*m - 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/p

i + 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)

Mupad [B] (veriﬁcation not implemented)

Time = 9.97 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28 \[ \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=-\frac {g\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^n}{2\,f\,{\left (g\,\cos \left (e+f\,x\right )\right )}^{2\,m}\,\left (\sin \left (e+f\,x\right )+1\right )\,\left (n-m+1\right )} \]

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

-(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

\(\int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx\) [177]

Optimal result