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(f*x+e)*(a-a*sin(f*x+e))^m*(c+c*sin(f*x+e))^n/f

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Rubi [A]  time = 0.12, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.47, size = 37, normalized size = 1.00 \[ -\frac {B \cos (e+f x) (a-a \sin (e+f x))^m (c (\sin (e+f x)+1))^n}{f} \]

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)

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fricas [A]  time = 0.50, size = 37, normalized size = 1.00 \[ -\frac {{\left (-a \sin \left (f x + e\right ) + a\right )}^{m} {\left (c \sin \left (f x + e\right ) + c\right )}^{n} B \cos \left (f x + e\right )}{f} \]

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

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giac [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((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]

Timed out

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maple [F]  time = 8.40, size = 0, normalized size = 0.00 \[ \int \left (a -a \sin \left (f x +e \right )\right )^{m} \left (c +c \sin \left (f x +e \right )\right )^{n} \left (B \left (m -n \right )+B \left (n +m +1\right ) \sin \left (f x +e \right )\right )\, dx \]

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*(n+m+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*(n+m+1)*sin(f*x+e)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B {\left (m + n + 1\right )} \sin \left (f x + e\right ) + B {\left (m - n\right )}\right )} {\left (-a \sin \left (f x + e\right ) + a\right )}^{m} {\left (c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]

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)

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mupad [B]  time = 13.50, size = 37, normalized size = 1.00 \[ -\frac {B\,\cos \left (e+f\,x\right )\,{\left (-a\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m\,{\left (c\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^n}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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