3.19 \(\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx\)

Optimal. Leaf size=97 \[ \frac {a c^2 (A-B) \cos ^3(e+f x)}{3 f}+\frac {a c^2 (4 A-B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a c^2 x (4 A-B)+\frac {a B c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 105, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2967, 2860, 2669, 2635, 8} \[ \frac {a c^2 (4 A-B) \cos ^3(e+f x)}{12 f}+\frac {a c^2 (4 A-B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a c^2 x (4 A-B)-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2,x]

[Out]

(a*(4*A - B)*c^2*x)/8 + (a*(4*A - B)*c^2*Cos[e + f*x]^3)/(12*f) + (a*(4*A - B)*c^2*Cos[e + f*x]*Sin[e + f*x])/
(8*f) - (a*B*Cos[e + f*x]^3*(c^2 - c^2*Sin[e + f*x]))/(4*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2860

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]
+ Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; Fre
eQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[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)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx &=(a c) \int \cos ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx\\ &=-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac {1}{4} (a (4 A-B) c) \int \cos ^2(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac {1}{4} \left (a (4 A-B) c^2\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}+\frac {a (4 A-B) c^2 \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac {1}{8} \left (a (4 A-B) c^2\right ) \int 1 \, dx\\ &=\frac {1}{8} a (4 A-B) c^2 x+\frac {a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}+\frac {a (4 A-B) c^2 \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}\\ \end {align*}

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Mathematica [A]  time = 0.67, size = 74, normalized size = 0.76 \[ \frac {a c^2 (3 (8 A \sin (2 (e+f x))+16 A f x+B \sin (4 (e+f x))-4 B f x)+24 (A-B) \cos (e+f x)+8 (A-B) \cos (3 (e+f x)))}{96 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2,x]

[Out]

(a*c^2*(24*(A - B)*Cos[e + f*x] + 8*(A - B)*Cos[3*(e + f*x)] + 3*(16*A*f*x - 4*B*f*x + 8*A*Sin[2*(e + f*x)] +
B*Sin[4*(e + f*x)])))/(96*f)

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fricas [A]  time = 0.45, size = 82, normalized size = 0.85 \[ \frac {8 \, {\left (A - B\right )} a c^{2} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, A - B\right )} a c^{2} f x + 3 \, {\left (2 \, B a c^{2} \cos \left (f x + e\right )^{3} + {\left (4 \, A - B\right )} a c^{2} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

1/24*(8*(A - B)*a*c^2*cos(f*x + e)^3 + 3*(4*A - B)*a*c^2*f*x + 3*(2*B*a*c^2*cos(f*x + e)^3 + (4*A - B)*a*c^2*c
os(f*x + e))*sin(f*x + e))/f

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giac [A]  time = 0.14, size = 114, normalized size = 1.18 \[ \frac {B a c^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {A a c^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, A a c^{2} - B a c^{2}\right )} x + \frac {{\left (A a c^{2} - B a c^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac {{\left (A a c^{2} - B a c^{2}\right )} \cos \left (f x + e\right )}{4 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^2,x, algorithm="giac")

[Out]

1/32*B*a*c^2*sin(4*f*x + 4*e)/f + 1/4*A*a*c^2*sin(2*f*x + 2*e)/f + 1/8*(4*A*a*c^2 - B*a*c^2)*x + 1/12*(A*a*c^2
- B*a*c^2)*cos(3*f*x + 3*e)/f + 1/4*(A*a*c^2 - B*a*c^2)*cos(f*x + e)/f

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maple [B]  time = 0.39, size = 185, normalized size = 1.91 \[ \frac {-\frac {A \,c^{2} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \,c^{2} a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,c^{2} a \cos \left (f x +e \right )+B \,c^{2} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {B \,c^{2} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \,c^{2} a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,c^{2} a \left (f x +e \right )-B \,c^{2} a \cos \left (f x +e \right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^2,x)

[Out]

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

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maxima [B]  time = 0.33, size = 179, normalized size = 1.85 \[ \frac {32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a c^{2} - 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c^{2} + 96 \, {\left (f x + e\right )} A a c^{2} - 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c^{2} + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{2} - 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{2} + 96 \, A a c^{2} \cos \left (f x + e\right ) - 96 \, B a c^{2} \cos \left (f x + e\right )}{96 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

1/96*(32*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a*c^2 - 24*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a*c^2 + 96*(f*x + e
)*A*a*c^2 - 32*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a*c^2 + 3*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x +
2*e))*B*a*c^2 - 24*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a*c^2 + 96*A*a*c^2*cos(f*x + e) - 96*B*a*c^2*cos(f*x +
e))/f

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mupad [B]  time = 13.38, size = 345, normalized size = 3.56 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a\,c^2+\frac {B\,a\,c^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,a\,c^2-2\,B\,a\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,A\,a\,c^2-2\,B\,a\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {2\,A\,a\,c^2}{3}-\frac {2\,B\,a\,c^2}{3}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (A\,a\,c^2+\frac {B\,a\,c^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (A\,a\,c^2-\frac {7\,B\,a\,c^2}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (A\,a\,c^2-\frac {7\,B\,a\,c^2}{4}\right )+\frac {2\,A\,a\,c^2}{3}-\frac {2\,B\,a\,c^2}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a\,c^2\,\mathrm {atan}\left (\frac {a\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,A-B\right )}{4\,\left (A\,a\,c^2-\frac {B\,a\,c^2}{4}\right )}\right )\,\left (4\,A-B\right )}{4\,f}-\frac {a\,c^2\,\left (4\,A-B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^2,x)

[Out]

(tan(e/2 + (f*x)/2)*(A*a*c^2 + (B*a*c^2)/4) + tan(e/2 + (f*x)/2)^4*(2*A*a*c^2 - 2*B*a*c^2) + tan(e/2 + (f*x)/2
)^6*(2*A*a*c^2 - 2*B*a*c^2) + tan(e/2 + (f*x)/2)^2*((2*A*a*c^2)/3 - (2*B*a*c^2)/3) - tan(e/2 + (f*x)/2)^7*(A*a
*c^2 + (B*a*c^2)/4) + tan(e/2 + (f*x)/2)^3*(A*a*c^2 - (7*B*a*c^2)/4) - tan(e/2 + (f*x)/2)^5*(A*a*c^2 - (7*B*a*
c^2)/4) + (2*A*a*c^2)/3 - (2*B*a*c^2)/3)/(f*(4*tan(e/2 + (f*x)/2)^2 + 6*tan(e/2 + (f*x)/2)^4 + 4*tan(e/2 + (f*
x)/2)^6 + tan(e/2 + (f*x)/2)^8 + 1)) + (a*c^2*atan((a*c^2*tan(e/2 + (f*x)/2)*(4*A - B))/(4*(A*a*c^2 - (B*a*c^2
)/4)))*(4*A - B))/(4*f) - (a*c^2*(4*A - B)*(atan(tan(e/2 + (f*x)/2)) - (f*x)/2))/(4*f)

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sympy [A]  time = 2.13, size = 396, normalized size = 4.08 \[ \begin {cases} - \frac {A a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {A a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + A a c^{2} x - \frac {A a c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {A a c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 A a c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {A a c^{2} \cos {\left (e + f x \right )}}{f} + \frac {3 B a c^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 B a c^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {B a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 B a c^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {B a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {5 B a c^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {B a c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 B a c^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {B a c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 B a c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a c^{2} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right ) \left (- c \sin {\relax (e )} + c\right )^{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**2,x)

[Out]

Piecewise((-A*a*c**2*x*sin(e + f*x)**2/2 - A*a*c**2*x*cos(e + f*x)**2/2 + A*a*c**2*x - A*a*c**2*sin(e + f*x)**
2*cos(e + f*x)/f + A*a*c**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*A*a*c**2*cos(e + f*x)**3/(3*f) + A*a*c**2*cos(
e + f*x)/f + 3*B*a*c**2*x*sin(e + f*x)**4/8 + 3*B*a*c**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 - B*a*c**2*x*sin(
e + f*x)**2/2 + 3*B*a*c**2*x*cos(e + f*x)**4/8 - B*a*c**2*x*cos(e + f*x)**2/2 - 5*B*a*c**2*sin(e + f*x)**3*cos
(e + f*x)/(8*f) + B*a*c**2*sin(e + f*x)**2*cos(e + f*x)/f - 3*B*a*c**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) + B*
a*c**2*sin(e + f*x)*cos(e + f*x)/(2*f) + 2*B*a*c**2*cos(e + f*x)**3/(3*f) - B*a*c**2*cos(e + f*x)/f, Ne(f, 0))
, (x*(A + B*sin(e))*(a*sin(e) + a)*(-c*sin(e) + c)**2, True))

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