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

________________________________________________________________________________________

Rubi [A]  time = 0.117022, antiderivative size = 32, 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} \[ \frac{A (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g} \]

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*}

Mathematica [A]  time = 0.172178, 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

________________________________________________________________________________________

Maple [F]  time = 4.252, 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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [A]  time = 1.55306, size = 81, normalized size = 2.53 \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

________________________________________________________________________________________

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \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]

sage2