\(\int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx\) [1041]

Optimal result

Integrand size = 40, antiderivative size = 34 \[ \int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx=-\frac {A (g \cos (e+f x))^{1+p} (a-a \sin (e+f x))^m}{f g} \]

[Out]

-A*(g*cos(f*x+e))^(p+1)*(a-a*sin(f*x+e))^m/f/g

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 34, 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 = {2933} \[ \int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx=-\frac {A (a-a \sin (e+f x))^m (g \cos (e+f x))^{p+1}}{f g} \]

[In]

Int[(g*Cos[e + f*x])^p*(a - a*Sin[e + f*x])^m*(A*m + A*(1 + m + p)*Sin[e + f*x]),x]

[Out]

-((A*(g*Cos[e + f*x])^(1 + p)*(a - a*Sin[e + f*x])^m)/(f*g))

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {A (g \cos (e+f x))^{1+p} (a-a \sin (e+f x))^m}{f g} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx=-\frac {A \cos (e+f x) (g \cos (e+f x))^p (a-a \sin (e+f x))^m}{f} \]

[In]

Integrate[(g*Cos[e + f*x])^p*(a - a*Sin[e + f*x])^m*(A*m + A*(1 + m + p)*Sin[e + f*x]),x]

[Out]

-((A*Cos[e + f*x]*(g*Cos[e + f*x])^p*(a - a*Sin[e + f*x])^m)/f)

Maple [A] (verified)

Time = 4.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06

method	result	size
parallelrisch	\(-\frac {A \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{m} \left (g \cos \left (f x +e \right )\right )^{p} \cos \left (f x +e \right )}{f}\)	\(36\)

[In]

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

[Out]

-1/f*A*(-a*(sin(f*x+e)-1))^m*(g*cos(f*x+e))^p*cos(f*x+e)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx=-\frac {\left (g \cos \left (f x + e\right )\right )^{p} {\left (-a \sin \left (f x + e\right ) + a\right )}^{m} A \cos \left (f x + e\right )}{f} \]

[In]

integrate((g*cos(f*x+e))^p*(a-a*sin(f*x+e))^m*(A*m+A*(1+m+p)*sin(f*x+e)),x, algorithm="fricas")

[Out]

-(g*cos(f*x + e))^p*(-a*sin(f*x + e) + a)^m*A*cos(f*x + e)/f

Sympy [F]

\[ \int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx=A \left (\int m \left (g \cos {\left (e + f x \right )}\right )^{p} \left (- a \sin {\left (e + f x \right )} + a\right )^{m}\, dx + \int \left (g \cos {\left (e + f x \right )}\right )^{p} \left (- a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx + \int m \left (g \cos {\left (e + f x \right )}\right )^{p} \left (- a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx + \int p \left (g \cos {\left (e + f x \right )}\right )^{p} \left (- a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx\right ) \]

[In]

integrate((g*cos(f*x+e))**p*(a-a*sin(f*x+e))**m*(A*m+A*(1+m+p)*sin(f*x+e)),x)

[Out]

A*(Integral(m*(g*cos(e + f*x))**p*(-a*sin(e + f*x) + a)**m, x) + Integral((g*cos(e + f*x))**p*(-a*sin(e + f*x)
+ a)**m*sin(e + f*x), x) + Integral(m*(g*cos(e + f*x))**p*(-a*sin(e + f*x) + a)**m*sin(e + f*x), x) + Integra
l(p*(g*cos(e + f*x))**p*(-a*sin(e + f*x) + a)**m*sin(e + f*x), x))

Maxima [F]

\[ \int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx=\int { {\left (A {\left (m + p + 1\right )} \sin \left (f x + e\right ) + A m\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((g*cos(f*x+e))^p*(a-a*sin(f*x+e))^m*(A*m+A*(1+m+p)*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((A*(m + p + 1)*sin(f*x + e) + A*m)*(g*cos(f*x + e))^p*(-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. 1775 vs. \(2 (34) = 68\).

Time = 3.19 (sec) , antiderivative size = 1775, normalized size of antiderivative = 52.21 \[ \int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx=\text {Too large to display} \]

[In]

integrate((g*cos(f*x+e))^p*(a-a*sin(f*x+e))^m*(A*m+A*(1+m+p)*sin(f*x+e)),x, algorithm="giac")

[Out]

-(A*e^(-m*log(2) - p*log(2) + p*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)*abs(g)/(tan(1/8*pi - 1/4*f*x -
1/4*e)^2 + 1)) + 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)) + p*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)))*tan(-1/4*pi*p*sgn(
g*tan(1/2*f*x + 1/2*e)^2 + 2*g*tan(1/2*f*x + 1/2*e) + g)*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(g) + 1/2*pi*p*flo
or(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 3/4) + pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floo
r(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*p*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*p*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1
/4*sgn(a) + 1) - 1/4*pi*p*sgn(g*tan(1/2*f*x + 1/2*e)^2 + 2*g*tan(1/2*f*x + 1/2*e) + g) + 1/2*pi*m*sgn(tan(1/2*
f*x + 1/2*e)^2 - 1) + 1/4*pi*p*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*m + 1/4*pi*p)^2*tan(
1/2*f*x + 1/2*e)^2 - A*e^(-m*log(2) - p*log(2) + p*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)*abs(g)/(tan(
1/8*pi - 1/4*f*x - 1/4*e)^2 + 1)) + 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)) + p*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)))
*tan(-1/4*pi*p*sgn(g*tan(1/2*f*x + 1/2*e)^2 + 2*g*tan(1/2*f*x + 1/2*e) + g)*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sg
n(g) + 1/2*pi*p*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 3/4) + pi*m*floor(1/2*f*x/p
i + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 1/4) + 1/2*pi*p*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*p*floor(1/2*f*x/pi + 1/2*e/pi + 1
/2) + pi*m*floor(-1/4*sgn(a) + 1) - 1/4*pi*p*sgn(g*tan(1/2*f*x + 1/2*e)^2 + 2*g*tan(1/2*f*x + 1/2*e) + g) + 1/
2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*p*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*m
+ 1/4*pi*p)^2 - A*e^(-m*log(2) - p*log(2) + p*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)*abs(g)/(tan(1/8*
pi - 1/4*f*x - 1/4*e)^2 + 1)) + 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)) + p*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)))*tan
(1/2*f*x + 1/2*e)^2 + A*e^(-m*log(2) - p*log(2) + p*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)*abs(g)/(tan
(1/8*pi - 1/4*f*x - 1/4*e)^2 + 1)) + 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)) + p*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))
))/(f*tan(-1/4*pi*p*sgn(g*tan(1/2*f*x + 1/2*e)^2 + 2*g*tan(1/2*f*x + 1/2*e) + g)*sgn(tan(1/2*f*x + 1/2*e)^2 -
1)*sgn(g) + 1/2*pi*p*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 3/4) + 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) + 1/2*pi*p*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*p*floor(1/2*f*x/pi + 1/2*e/p
i + 1/2) + pi*m*floor(-1/4*sgn(a) + 1) - 1/4*pi*p*sgn(g*tan(1/2*f*x + 1/2*e)^2 + 2*g*tan(1/2*f*x + 1/2*e) + g)
+ 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*p*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4
*pi*m + 1/4*pi*p)^2*tan(1/2*f*x + 1/2*e)^2 + f*tan(-1/4*pi*p*sgn(g*tan(1/2*f*x + 1/2*e)^2 + 2*g*tan(1/2*f*x +
1/2*e) + g)*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(g) + 1/2*pi*p*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi +
1/2*e/pi + 1/2) + 3/4) + 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) + 1/2*p
i*p*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*p*floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1) - 1/4*pi*p*sgn(g*tan(1/2*f*x +
1/2*e)^2 + 2*g*tan(1/2*f*x + 1/2*e) + g) + 1/2*pi*m*sgn(tan(1/2*f*x + 1/2*e)^2 - 1) + 1/4*pi*p*sgn(tan(1/2*f*
x + 1/2*e)^2 - 1) + 1/4*pi*m*sgn(a) + 1/4*pi*m + 1/4*pi*p)^2 + f*tan(1/2*f*x + 1/2*e)^2 + f)

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx=-\frac {A\,\cos \left (e+f\,x\right )\,{\left (g\,\cos \left (e+f\,x\right )\right )}^p\,{\left (-a\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m}{f} \]

[In]

int((g*cos(e + f*x))^p*(A*m + A*sin(e + f*x)*(m + p + 1))*(a - a*sin(e + f*x))^m,x)

[Out]

-(A*cos(e + f*x)*(g*cos(e + f*x))^p*(-a*(sin(e + f*x) - 1))^m)/f