3.3.18 \(\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^n (B (3-n)-B (4+n) \sin (e+f x)) \, dx\) [218]

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

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Rubi [A]
time = 0.18, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {3046, 2933} \begin {gather*} \frac {a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^{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*}

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Mathematica [A]
time = 0.38, size = 63, normalized size = 1.85 \begin {gather*} \frac {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} \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*(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 = 1.09, 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)

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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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (36) = 72\).
time = 0.44, size = 84, normalized size = 2.47 \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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 898 vs. \(2 (31) = 62\).
time = 80.43, 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)/(ta
n(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*ta
n(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))

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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)

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Mupad [B]
time = 14.45, size = 64, normalized size = 1.88 \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)

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