3.3.19 \(\int (a-a \sin (e+f x))^3 (c+c \sin (e+f x))^n (B (3-n)+B (4+n) \sin (e+f x)) \, dx\) [219]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {3046, 2933} \begin {gather*} -\frac {a^3 B c^3 \cos ^7(e+f x) (c \sin (e+f x)+c)^{n-3}}{f} \end {gather*}

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 2933

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 3046

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))^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 = 0.78, size = 67, normalized size = 1.97 \begin {gather*} -\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 (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c (1+\sin (e+f x)))^n}{f} \end {gather*}

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)

________________________________________________________________________________________

Maple [F]
time = 1.07, size = 0, normalized size = 0.00 \[\int \left (a -a \sin \left (f x +e \right )\right )^{3} \left (c +c \sin \left (f x +e \right )\right )^{n} \left (B \left (3-n \right )+B \left (4+n \right ) \sin \left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*sin(f*x+e))^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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (36) = 72\).
time = 0.39, size = 83, normalized size = 2.44 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 898 vs. \(2 (32) = 64\).
time = 80.81, size = 898, normalized size = 26.41 \begin {gather*} \begin {cases} \frac {B a^{3} \left (c + \frac {2 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{n} \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} - \frac {6 B a^{3} \left (c + \frac {2 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{n} \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} + \frac {14 B a^{3} \left (c + \frac {2 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{n} \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} - \frac {14 B a^{3} \left (c + \frac {2 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{n} \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} + \frac {14 B a^{3} \left (c + \frac {2 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{n} \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} - \frac {14 B a^{3} \left (c + \frac {2 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{n} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} + \frac {6 B a^{3} \left (c + \frac {2 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{n} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} - \frac {B a^{3} \left (c + \frac {2 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{n}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} & \text {for}\: f \neq 0 \\x \left (B \left (3 - n\right ) + B \left (n + 4\right ) \sin {\left (e \right )}\right ) \left (- a \sin {\left (e \right )} + a\right )^{3} \left (c \sin {\left (e \right )} + c\right )^{n} & \text {otherwise} \end {cases} \end {gather*}

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]

Piecewise((B*a**3*(c + 2*c*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**n*tan(e/2 + f*x/2)**8/(f*tan(e/2 + f*x
/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) - 6*B*a**3*(c + 2*c*
tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**n*tan(e/2 + f*x/2)**7/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/
2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) + 14*B*a**3*(c + 2*c*tan(e/2 + f*x/2)/(tan(e/2
+ f*x/2)**2 + 1))**n*tan(e/2 + f*x/2)**6/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/
2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) - 14*B*a**3*(c + 2*c*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**n*tan(e
/2 + f*x/2)**5/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/
2)**2 + f) + 14*B*a**3*(c + 2*c*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**n*tan(e/2 + f*x/2)**3/(f*tan(e/2
+ f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) - 14*B*a**3*(c
+ 2*c*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**n*tan(e/2 + f*x/2)**2/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2
+ f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) + 6*B*a**3*(c + 2*c*tan(e/2 + f*x/2)/(tan
(e/2 + f*x/2)**2 + 1))**n*tan(e/2 + f*x/2)/(f*tan(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*
x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f) - B*a**3*(c + 2*c*tan(e/2 + f*x/2)/(tan(e/2 + f*x/2)**2 + 1))**n/(f*tan
(e/2 + f*x/2)**8 + 4*f*tan(e/2 + f*x/2)**6 + 6*f*tan(e/2 + f*x/2)**4 + 4*f*tan(e/2 + f*x/2)**2 + f), Ne(f, 0))
, (x*(B*(3 - n) + B*(n + 4)*sin(e))*(-a*sin(e) + a)**3*(c*sin(e) + c)**n, True))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 14.46, size = 61, normalized size = 1.79 \begin {gather*} -\frac {B\,a^3\,{\left (c\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^n\,\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*a^3*(c*(sin(e + f*x) + 1))^n*(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)

________________________________________________________________________________________