3.223 \(\int (a-a \sin (e+f x))^m (c+c \sin (e+f x))^n (B (m-n)+B (1+m+n) \sin (e+f x)) \, dx\)

Optimal. Leaf size=37 \[ -\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c \sin (e+f x)+c)^n}{f} \]

[Out]

-((B*Cos[e + f*x]*(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^n)/f)

________________________________________________________________________________________

Rubi [A]  time = 0.123455, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022, Rules used = {2970} \[ -\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c \sin (e+f x)+c)^n}{f} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^n*(B*(m - n) + B*(1 + m + n)*Sin[e + f*x]),x]

[Out]

-((B*Cos[e + f*x]*(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^n)/f)

Rule 2970

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> -Simp[(B*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(
m + n + 1)), x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[A
*b*(m + n + 1) + a*B*(m - n), 0] && NeQ[m, -2^(-1)]

Rubi steps

\begin{align*} \int (a-a \sin (e+f x))^m (c+c \sin (e+f x))^n (B (m-n)+B (1+m+n) \sin (e+f x)) \, dx &=-\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c+c \sin (e+f x))^n}{f}\\ \end{align*}

Mathematica [A]  time = 0.463715, size = 37, normalized size = 1. \[ -\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c (\sin (e+f x)+1))^n}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^n*(B*(m - n) + B*(1 + m + n)*Sin[e + f*x]),x]

[Out]

-((B*Cos[e + f*x]*(c*(1 + Sin[e + f*x]))^n*(a - a*Sin[e + f*x])^m)/f)

________________________________________________________________________________________

Maple [F]  time = 2.543, size = 0, normalized size = 0. \begin{align*} \int \left ( a-a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c+c\sin \left ( fx+e \right ) \right ) ^{n} \left ( B \left ( m-n \right ) +B \left ( m+n+1 \right ) \sin \left ( fx+e \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*sin(f*x+e))^m*(c+c*sin(f*x+e))^n*(B*(m-n)+B*(m+n+1)*sin(f*x+e)),x)

[Out]

int((a-a*sin(f*x+e))^m*(c+c*sin(f*x+e))^n*(B*(m-n)+B*(m+n+1)*sin(f*x+e)),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B{\left (m + n + 1\right )} \sin \left (f x + e\right ) + B{\left (m - n\right )}\right )}{\left (-a \sin \left (f x + e\right ) + a\right )}^{m}{\left (c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(f*x+e))^m*(c+c*sin(f*x+e))^n*(B*(m-n)+B*(1+m+n)*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*(m + n + 1)*sin(f*x + e) + B*(m - n))*(-a*sin(f*x + e) + a)^m*(c*sin(f*x + e) + c)^n, x)

________________________________________________________________________________________

Fricas [A]  time = 2.07714, size = 89, normalized size = 2.41 \begin{align*} -\frac{{\left (-a \sin \left (f x + e\right ) + a\right )}^{m}{\left (c \sin \left (f x + e\right ) + c\right )}^{n} B \cos \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(f*x+e))^m*(c+c*sin(f*x+e))^n*(B*(m-n)+B*(1+m+n)*sin(f*x+e)),x, algorithm="fricas")

[Out]

-(-a*sin(f*x + e) + a)^m*(c*sin(f*x + e) + c)^n*B*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((a-a*sin(f*x+e))**m*(c+c*sin(f*x+e))**n*(B*(m-n)+B*(1+m+n)*sin(f*x+e)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(f*x+e))^m*(c+c*sin(f*x+e))^n*(B*(m-n)+B*(1+m+n)*sin(f*x+e)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError