\(\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A m-A (1+m+p) \sin (e+f x)) \, dx\) [1040]

Optimal result

Integrand size = 40, antiderivative size = 32 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A m-A (1+m+p) \sin (e+f x)) \, dx=\frac {A (g \cos (e+f x))^{1+p} (a+a \sin (e+f x))^m}{f g} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 32, 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} \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A m-A (1+m+p) \sin (e+f x)) \, dx=\frac {A (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g} \]

[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 = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A m-A (1+m+p) \sin (e+f x)) \, dx=\frac {A \cos (e+f x) (g \cos (e+f x))^p (a (1+\sin (e+f x)))^m}{f} \]

[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*(1 + Sin[e + f*x]))^m)/f

Maple [A] (verified)

Time = 4.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06

method	result	size
parallelrisch	\(\frac {A \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m} \left (g \cos \left (f x +e \right )\right )^{p} \cos \left (f x +e \right )}{f}\)	\(34\)

[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*(1+sin(f*x+e)))^m*(g*cos(f*x+e))^p*cos(f*x+e)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 33, 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]

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

[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) + Integral
(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. 1790 vs. \(2 (32) = 64\).

Time = 3.81 (sec) , antiderivative size = 1790, normalized size of antiderivative = 55.94 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A m-A (1+m+p) \sin (e+f x)) \, dx=\text {Too large to display} \]

[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(4*abs(g)*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*m*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)/(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 + 1)) + p*log(2*
abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)/(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) + pi*m*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) + 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) + 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*floo
r(-1/4*sgn(a) + 1/2) + 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(ta
n(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) - 3/4*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(4*abs(g)*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*m*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)/(tan(1/8*pi - 1/4*f*x -
1/4*e)^2 + 1)) + p*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)/(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) + pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 3/4) + 1/2*pi*p*floo
r(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 3/4) + 1/2*pi*p*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*p*floor(1/2*f*x/pi + 1
/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + 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) - 3/4*pi*m - 1/4*pi*p)^2 - A*e^(-m*log(2) - p*log(2) + p*log(4*abs(g)*abs(tan(1/8*pi - 1/4*f*x - 1/4*e))/(t
an(1/8*pi - 1/4*f*x - 1/4*e)^2 + 1)) + 2*m*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)/(tan(1/8*pi - 1/4*f*
x - 1/4*e)^2 + 1)) + p*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)/(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(4*abs(g)*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*m*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)/(tan(1/8*
pi - 1/4*f*x - 1/4*e)^2 + 1)) + p*log(2*abs(tan(1/8*pi - 1/4*f*x - 1/4*e)^2 - 1)/(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(t
an(1/2*f*x + 1/2*e)^2 - 1)*sgn(g) + pi*m*floor(1/2*f*x/pi + 1/2*e/pi - floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 3/
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) + 3/4) + 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*fl
oor(1/2*f*x/pi + 1/2*e/pi + 1/2) + pi*m*floor(-1/4*sgn(a) + 1/2) + 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) - 3/4*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) + pi*m*floor(1/2*f*x/pi + 1/2*e/pi
- floor(1/2*f*x/pi + 1/2*e/pi + 1/2) + 3/4) + 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) + 3/4) + 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*fl
oor(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) + 1/2) + 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) - 3/4*pi*m - 1/4*pi*p)^2 + f*tan(1/2*f*x + 1/2*
e)^2 + f)

Mupad [B] (verification not implemented)

Time = 10.50 (sec) , antiderivative size = 33, 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