3.219 \(\int (a-a \sin (e+f x))^3 (c+c \sin (e+f x))^n (B (3-n)+B (4+n) \sin (e+f x)) \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^3 B c^3 \cos ^7(e+f x) (c \sin (e+f x)+c)^{n-3}}{f} \]

[Out]

-((a^3*B*c^3*Cos[e + f*x]^7*(c + c*Sin[e + f*x])^(-3 + n))/f)

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Rubi [A]  time = 0.238354, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.044, Rules used = {2967, 2854} \[ -\frac{a^3 B c^3 \cos ^7(e+f x) (c \sin (e+f x)+c)^{n-3}}{f} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Sin[e + f*x])^3*(c + c*Sin[e + f*x])^n*(B*(3 - n) + B*(4 + n)*Sin[e + f*x]),x]

[Out]

-((a^3*B*c^3*Cos[e + f*x]^7*(c + c*Sin[e + f*x])^(-3 + n))/f)

Rule 2967

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

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 (a-a \sin (e+f x))^3 (c+c \sin (e+f x))^n (B (3-n)+B (4+n) \sin (e+f x)) \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (c+c \sin (e+f x))^{-3+n} (B (3-n)+B (4+n) \sin (e+f x)) \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x) (c+c \sin (e+f x))^{-3+n}}{f}\\ \end{align*}

Mathematica [A]  time = 1.1436, size = 67, normalized size = 1.97 \[ -\frac{a^3 B \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (c (\sin (e+f x)+1))^n}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Sin[e + f*x])^3*(c + c*Sin[e + f*x])^n*(B*(3 - n) + B*(4 + n)*Sin[e + f*x]),x]

[Out]

-((a^3*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^
n)/f)

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Maple [F]  time = 2.428, size = 0, normalized size = 0. \begin{align*} \int \left ( a-a\sin \left ( fx+e \right ) \right ) ^{3} \left ( c+c\sin \left ( fx+e \right ) \right ) ^{n} \left ( B \left ( 3-n \right ) +B \left ( 4+n \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))^3*(c+c*sin(f*x+e))^n*(B*(3-n)+B*(4+n)*sin(f*x+e)),x)

[Out]

int((a-a*sin(f*x+e))^3*(c+c*sin(f*x+e))^n*(B*(3-n)+B*(4+n)*sin(f*x+e)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (B{\left (n + 4\right )} \sin \left (f x + e\right ) - B{\left (n - 3\right )}\right )}{\left (a \sin \left (f x + e\right ) - a\right )}^{3}{\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))^3*(c+c*sin(f*x+e))^n*(B*(3-n)+B*(4+n)*sin(f*x+e)),x, algorithm="maxima")

[Out]

-integrate((B*(n + 4)*sin(f*x + e) - B*(n - 3))*(a*sin(f*x + e) - a)^3*(c*sin(f*x + e) + c)^n, x)

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Fricas [B]  time = 2.02568, size = 182, normalized size = 5.35 \begin{align*} \frac{{\left (3 \, B a^{3} \cos \left (f x + e\right )^{3} - 4 \, B a^{3} \cos \left (f x + e\right ) -{\left (B a^{3} \cos \left (f x + e\right )^{3} - 4 \, B a^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}{\left (c \sin \left (f x + e\right ) + c\right )}^{n}}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(f*x+e))^3*(c+c*sin(f*x+e))^n*(B*(3-n)+B*(4+n)*sin(f*x+e)),x, algorithm="fricas")

[Out]

(3*B*a^3*cos(f*x + e)^3 - 4*B*a^3*cos(f*x + e) - (B*a^3*cos(f*x + e)^3 - 4*B*a^3*cos(f*x + e))*sin(f*x + e))*(
c*sin(f*x + e) + c)^n/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((a-a*sin(f*x+e))**3*(c+c*sin(f*x+e))**n*(B*(3-n)+B*(4+n)*sin(f*x+e)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (B{\left (n + 4\right )} \sin \left (f x + e\right ) - B{\left (n - 3\right )}\right )}{\left (a \sin \left (f x + e\right ) - a\right )}^{3}{\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))^3*(c+c*sin(f*x+e))^n*(B*(3-n)+B*(4+n)*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate(-(B*(n + 4)*sin(f*x + e) - B*(n - 3))*(a*sin(f*x + e) - a)^3*(c*sin(f*x + e) + c)^n, x)