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

Optimal. Leaf size=33 \[ \frac {a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f} \]

[Out]

a^3*B*c^3*cos(f*x+e)^7*(a+a*sin(f*x+e))^(-3+m)/f

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Rubi [A]  time = 0.24, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2967, 2854} \[ \frac {a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]))

Rubi steps

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

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Mathematica [A]  time = 1.11, size = 66, normalized size = 2.00 \[ \frac {B c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (a (\sin (e+f x)+1))^m}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.49, size = 78, normalized size = 2.36 \[ -\frac {{\left (3 \, B c^{3} \cos \left (f x + e\right )^{3} - 4 \, B c^{3} \cos \left (f x + e\right ) - {\left (B c^{3} \cos \left (f x + e\right )^{3} - 4 \, B c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f} \]

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))^3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B {\left (m + 4\right )} \sin \left (f x + e\right ) - B {\left (m - 3\right )}\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

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))^3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x, algorithm="giac")

[Out]

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

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maple [F]  time = 7.86, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{3} \left (B \left (m -3\right )-B \left (4+m \right ) \sin \left (f x +e \right )\right )\, dx \]

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))^3*(B*(m-3)-B*(4+m)*sin(f*x+e)),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B {\left (m + 4\right )} \sin \left (f x + e\right ) - B {\left (m - 3\right )}\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

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))^3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 14.38, size = 61, normalized size = 1.85 \[ \frac {B\,c^3\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (14\,\cos \left (e+f\,x\right )-6\,\cos \left (3\,e+3\,f\,x\right )-14\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )\right )}{8\,f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*c^3*(a*(sin(e + f*x) + 1))^m*(14*cos(e + f*x) - 6*cos(3*e + 3*f*x) - 14*sin(2*e + 2*f*x) + sin(4*e + 4*f*x)
))/(8*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))**3*(B*(-3+m)-B*(4+m)*sin(f*x+e)),x)

[Out]

Timed out

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