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

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

[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 = 0.273913, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {2967, 2854} \[ \frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^{n-3}}{f} \]

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 = 0.52944, size = 63, normalized size = 1.85 \[ \frac{a^3 B (14 \sin (2 (e+f x))-\sin (4 (e+f x))+14 \cos (e+f x)-6 \cos (3 (e+f x))) (c-c \sin (e+f x))^n}{8 f} \]

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*(c - c*Sin[e + f*x])^n*(14*Cos[e + f*x] - 6*Cos[3*(e + f*x)] + 14*Sin[2*(e + f*x)] - Sin[4*(e + f*x)]))
/(8*f)

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Maple [F]  time = 2.334, 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 = 1.98062, size = 185, normalized size = 5.44 \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)