3.1041 \(\int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx\)
Optimal. Leaf size=34 \[ -\frac{A (a-a \sin (e+f x))^m (g \cos (e+f x))^{p+1}}{f g} \]
[Out]
-((A*(g*Cos[e + f*x])^(1 + p)*(a - a*Sin[e + f*x])^m)/(f*g))
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Rubi [A] time = 0.115491, antiderivative size = 34, normalized size of antiderivative = 1.,
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 =
{2854} \[ -\frac{A (a-a \sin (e+f x))^m (g \cos (e+f x))^{p+1}}{f g} \]
Antiderivative was successfully verified.
[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 2854
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*} \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}\\ \end{align*}
Mathematica [A] time = 0.0613699, size = 35, normalized size = 1.03 \[ -\frac{A \cos (e+f x) (a-a \sin (e+f x))^m (g \cos (e+f x))^p}{f} \]
Antiderivative was successfully verified.
[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)
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Maple [F] time = 4.184, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{p} \left ( a-a\sin \left ( fx+e \right ) \right ) ^{m} \left ( Am+A \left ( 1+m+p \right ) \sin \left ( fx+e \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
[Out]
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)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Fricas [A] time = 1.42943, size = 84, normalized size = 2.47 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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]
Timed out
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Giac [B] time = 32.0174, size = 2515, normalized size = 73.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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*l
og(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*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/pi + 1/2) + pi*m*fl
oor(-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(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) + 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)*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*flo
or(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/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(t
an(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*fl
oor(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*t
an(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*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*flo
or(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)