3.228 \(\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx\)

Optimal. Leaf size=83 \[ \frac{5 a c^3 \cos ^3(e+f x)}{12 f}+\frac{a \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{4 f}+\frac{5 a c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a c^3 x \]

[Out]

(5*a*c^3*x)/8 + (5*a*c^3*Cos[e + f*x]^3)/(12*f) + (5*a*c^3*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^
3*(c^3 - c^3*Sin[e + f*x]))/(4*f)

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Rubi [A]  time = 0.112533, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2736, 2678, 2669, 2635, 8} \[ \frac{5 a c^3 \cos ^3(e+f x)}{12 f}+\frac{a \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{4 f}+\frac{5 a c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a c^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*c^3*x)/8 + (5*a*c^3*Cos[e + f*x]^3)/(12*f) + (5*a*c^3*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^
3*(c^3 - c^3*Sin[e + f*x]))/(4*f)

Rule 2736

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 0]))

Rule 2678

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

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 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 8

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

Rubi steps

\begin{align*} \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx &=(a c) \int \cos ^2(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=\frac{a \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{4 f}+\frac{1}{4} \left (5 a c^2\right ) \int \cos ^2(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{5 a c^3 \cos ^3(e+f x)}{12 f}+\frac{a \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{4 f}+\frac{1}{4} \left (5 a c^3\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{5 a c^3 \cos ^3(e+f x)}{12 f}+\frac{5 a c^3 \cos (e+f x) \sin (e+f x)}{8 f}+\frac{a \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{4 f}+\frac{1}{8} \left (5 a c^3\right ) \int 1 \, dx\\ &=\frac{5}{8} a c^3 x+\frac{5 a c^3 \cos ^3(e+f x)}{12 f}+\frac{5 a c^3 \cos (e+f x) \sin (e+f x)}{8 f}+\frac{a \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{4 f}\\ \end{align*}

Mathematica [A]  time = 0.371495, size = 54, normalized size = 0.65 \[ \frac{a c^3 (24 \sin (2 (e+f x))-3 \sin (4 (e+f x))+48 \cos (e+f x)+16 \cos (3 (e+f x))+60 f x)}{96 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c^3*(60*f*x + 48*Cos[e + f*x] + 16*Cos[3*(e + f*x)] + 24*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/(96*f)

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Maple [A]  time = 0.017, size = 89, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ( -a{c}^{3} \left ( -{\frac{\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) -{\frac{2\,a{c}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+2\,a{c}^{3}\cos \left ( fx+e \right ) +a{c}^{3} \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.21324, size = 116, normalized size = 1.4 \begin{align*} \frac{64 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a c^{3} - 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 )} a c^{3} + 96 \,{\left (f x + e\right )} a c^{3} + 192 \, a c^{3} \cos \left (f x + e\right )}{96 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

1/96*(64*(cos(f*x + e)^3 - 3*cos(f*x + e))*a*c^3 - 3*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a
*c^3 + 96*(f*x + e)*a*c^3 + 192*a*c^3*cos(f*x + e))/f

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Fricas [A]  time = 1.50477, size = 154, normalized size = 1.86 \begin{align*} \frac{16 \, a c^{3} \cos \left (f x + e\right )^{3} + 15 \, a c^{3} f x - 3 \,{\left (2 \, a c^{3} \cos \left (f x + e\right )^{3} - 5 \, a c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

1/24*(16*a*c^3*cos(f*x + e)^3 + 15*a*c^3*f*x - 3*(2*a*c^3*cos(f*x + e)^3 - 5*a*c^3*cos(f*x + e))*sin(f*x + e))
/f

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Sympy [A]  time = 2.18679, size = 196, normalized size = 2.36 \begin{align*} \begin{cases} - \frac{3 a c^{3} x \sin ^{4}{\left (e + f x \right )}}{8} - \frac{3 a c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac{3 a c^{3} x \cos ^{4}{\left (e + f x \right )}}{8} + a c^{3} x + \frac{5 a c^{3} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} - \frac{2 a c^{3} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{3 a c^{3} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac{4 a c^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac{2 a c^{3} \cos{\left (e + f x \right )}}{f} & \text{for}\: f \neq 0 \\x \left (a \sin{\left (e \right )} + a\right ) \left (- c \sin{\left (e \right )} + c\right )^{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))**3,x)

[Out]

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

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Giac [A]  time = 1.90293, size = 109, normalized size = 1.31 \begin{align*} \frac{5}{8} \, a c^{3} x + \frac{a c^{3} \cos \left (3 \, f x + 3 \, e\right )}{6 \, f} + \frac{a c^{3} \cos \left (f x + e\right )}{2 \, f} - \frac{a c^{3} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac{a c^{3} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^3,x, algorithm="giac")

[Out]

5/8*a*c^3*x + 1/6*a*c^3*cos(3*f*x + 3*e)/f + 1/2*a*c^3*cos(f*x + e)/f - 1/32*a*c^3*sin(4*f*x + 4*e)/f + 1/4*a*
c^3*sin(2*f*x + 2*e)/f