3.1040 \(\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A m-A (1+m+p) \sin (e+f x)) \, dx\)
Optimal. Leaf size=32 \[ \frac {A (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g} \]
[Out]
A*(g*cos(f*x+e))^(1+p)*(a+a*sin(f*x+e))^m/f/g
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Rubi [A] time = 0.12, antiderivative size = 32, 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 =
{2854} \[ \frac {A (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g} \]
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*}
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Mathematica [A] time = 0.17, size = 33, normalized size = 1.03 \[ \frac {A \cos (e+f x) (a (\sin (e+f x)+1))^m (g \cos (e+f x))^p}{f} \]
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*(1 + Sin[e + f*x]))^m)/f
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fricas [A] time = 0.63, size = 33, normalized size = 1.03 \[ \frac {\left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} A \cos \left (f x + e\right )}{f} \]
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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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))^p*(a+a*sin(f*x+e))^m*(A*m-A*(1+m+p)*sin(f*x+e)),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: (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: (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: (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: (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: (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: (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)-(-A*exp(-m*ln(2)+
2*m*ln(2*abs(tan((2*f*x-pi+2*exp(1))/8)^2-1)/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a))-p*ln(2)+p*ln(2*abs
(tan((2*f*x-pi+2*exp(1))/8)^2-1)/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+p*ln(4*abs(g)*abs(tan((2*f*x-pi+2*exp(1))/8
))/(tan((2*f*x-pi+2*exp(1))/8)^2+1)))*tan((4*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+3*pi+2*exp(1
))*1/4/pi)+4*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+4*m*pi*floor(-(sign(a)-2)/4)+m*pi*sign(a)-2*m*pi*sign(tan((f*x
+exp(1))/2)^2-1)-3*m*pi+2*p*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+3*pi+2*exp(1))*1/4/pi)+2*p*pi*f
loor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*p*pi*floor((f*x+pi+exp(1))*1/2/pi)+p*pi*
sign(g)*sign(g*tan((f*x+exp(1))/2)^2-2*g*tan((f*x+exp(1))/2)+g)*sign(tan((f*x+exp(1))/2)^2-1)+p*pi*sign(g*tan(
(f*x+exp(1))/2)^2-2*g*tan((f*x+exp(1))/2)+g)-p*pi*sign(tan((f*x+exp(1))/2)^2-1)-p*pi)/4)^2*tan((f*x+exp(1))/2)
^2+A*exp(-m*ln(2)+2*m*ln(2*abs(tan((2*f*x-pi+2*exp(1))/8)^2-1)/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a))-
p*ln(2)+p*ln(2*abs(tan((2*f*x-pi+2*exp(1))/8)^2-1)/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+p*ln(4*abs(g)*abs(tan((2*
f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1)))*tan((4*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/
2/pi)+3*pi+2*exp(1))*1/4/pi)+4*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+4*m*pi*floor(-(sign(a)-2)/4)+m*pi*sign(a)-2*
m*pi*sign(tan((f*x+exp(1))/2)^2-1)-3*m*pi+2*p*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+3*pi+2*exp(1)
)*1/4/pi)+2*p*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*p*pi*floor((f*x+pi+exp
(1))*1/2/pi)+p*pi*sign(g)*sign(g*tan((f*x+exp(1))/2)^2-2*g*tan((f*x+exp(1))/2)+g)*sign(tan((f*x+exp(1))/2)^2-1
)+p*pi*sign(g*tan((f*x+exp(1))/2)^2-2*g*tan((f*x+exp(1))/2)+g)-p*pi*sign(tan((f*x+exp(1))/2)^2-1)-p*pi)/4)^2+A
*exp(-m*ln(2)+2*m*ln(2*abs(tan((2*f*x-pi+2*exp(1))/8)^2-1)/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a))-p*ln
(2)+p*ln(2*abs(tan((2*f*x-pi+2*exp(1))/8)^2-1)/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+p*ln(4*abs(g)*abs(tan((2*f*x-
pi+2*exp(1))/8))/(tan((2*f*x-pi+2*exp(1))/8)^2+1)))*tan((f*x+exp(1))/2)^2-A*exp(-m*ln(2)+2*m*ln(2*abs(tan((2*f
*x-pi+2*exp(1))/8)^2-1)/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+m*ln(abs(a))-p*ln(2)+p*ln(2*abs(tan((2*f*x-pi+2*exp(
1))/8)^2-1)/(tan((2*f*x-pi+2*exp(1))/8)^2+1))+p*ln(4*abs(g)*abs(tan((2*f*x-pi+2*exp(1))/8))/(tan((2*f*x-pi+2*e
xp(1))/8)^2+1))))/(f*tan((4*m*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+3*pi+2*exp(1))*1/4/pi)+4*m*pi
*floor((f*x+pi+exp(1))*1/2/pi)+4*m*pi*floor(-(sign(a)-2)/4)+m*pi*sign(a)-2*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-
3*m*pi+2*p*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+3*pi+2*exp(1))*1/4/pi)+2*p*pi*floor((2*f*x-4*pi*
floor((f*x+pi+exp(1))*1/2/pi)+pi+2*exp(1))*1/4/pi)+4*p*pi*floor((f*x+pi+exp(1))*1/2/pi)+p*pi*sign(g)*sign(g*ta
n((f*x+exp(1))/2)^2-2*g*tan((f*x+exp(1))/2)+g)*sign(tan((f*x+exp(1))/2)^2-1)+p*pi*sign(g*tan((f*x+exp(1))/2)^2
-2*g*tan((f*x+exp(1))/2)+g)-p*pi*sign(tan((f*x+exp(1))/2)^2-1)-p*pi)/4)^2*tan((f*x+exp(1))/2)^2+f*tan((4*m*pi*
floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+3*pi+2*exp(1))*1/4/pi)+4*m*pi*floor((f*x+pi+exp(1))*1/2/pi)+4*
m*pi*floor(-(sign(a)-2)/4)+m*pi*sign(a)-2*m*pi*sign(tan((f*x+exp(1))/2)^2-1)-3*m*pi+2*p*pi*floor((2*f*x-4*pi*f
loor((f*x+pi+exp(1))*1/2/pi)+3*pi+2*exp(1))*1/4/pi)+2*p*pi*floor((2*f*x-4*pi*floor((f*x+pi+exp(1))*1/2/pi)+pi+
2*exp(1))*1/4/pi)+4*p*pi*floor((f*x+pi+exp(1))*1/2/pi)+p*pi*sign(g)*sign(g*tan((f*x+exp(1))/2)^2-2*g*tan((f*x+
exp(1))/2)+g)*sign(tan((f*x+exp(1))/2)^2-1)+p*pi*sign(g*tan((f*x+exp(1))/2)^2-2*g*tan((f*x+exp(1))/2)+g)-p*pi*
sign(tan((f*x+exp(1))/2)^2-1)-p*pi)/4)^2+f*tan((f*x+exp(1))/2)^2+f)
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maple [F] time = 10.90, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{p} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A m -A \left (1+m +p \right ) \sin \left (f x +e \right )\right )\, dx \]
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 = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (A {\left (m + p + 1\right )} \sin \left (f x + e\right ) - A m\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
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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mupad [B] time = 9.70, size = 33, normalized size = 1.03 \[ \frac {A\,\cos \left (e+f\,x\right )\,{\left (g\,\cos \left (e+f\,x\right )\right )}^p\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[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
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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))**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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