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

Optimal. Leaf size=91 \[ \frac{a^3 c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^3 x \]

[Out]

(5*a^3*c^3*x)/16 + (5*a^3*c^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(24*
f) + (a^3*c^3*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)

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Rubi [A]  time = 0.0793481, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2635, 8} \[ \frac{a^3 c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a^3*c^3*x)/16 + (5*a^3*c^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(24*
f) + (a^3*c^3*Cos[e + f*x]^5*Sin[e + f*x])/(6*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 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))^3 (c-c \sin (e+f x))^3 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac{a^3 c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{6} \left (5 a^3 c^3\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{5 a^3 c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{8} \left (5 a^3 c^3\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{5 a^3 c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac{5 a^3 c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{16} \left (5 a^3 c^3\right ) \int 1 \, dx\\ &=\frac{5}{16} a^3 c^3 x+\frac{5 a^3 c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac{5 a^3 c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}\\ \end{align*}

Mathematica [A]  time = 0.0512375, size = 49, normalized size = 0.54 \[ \frac{a^3 c^3 (45 \sin (2 (e+f x))+9 \sin (4 (e+f x))+\sin (6 (e+f x))+60 e+60 f x)}{192 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*c^3*(60*e + 60*f*x + 45*Sin[2*(e + f*x)] + 9*Sin[4*(e + f*x)] + Sin[6*(e + f*x)]))/(192*f)

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Maple [A]  time = 0.013, size = 140, normalized size = 1.5 \begin{align*}{\frac{1}{f} \left ( -{c}^{3}{a}^{3} \left ( -{\frac{\cos \left ( fx+e \right ) }{6} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( fx+e \right ) }{8}} \right ) }+{\frac{5\,fx}{16}}+{\frac{5\,e}{16}} \right ) +3\,{c}^{3}{a}^{3} \left ( -1/4\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+3/2\,\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) +3/8\,fx+3/8\,e \right ) -3\,{c}^{3}{a}^{3} \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) +{c}^{3}{a}^{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))^3*(c-c*sin(f*x+e))^3,x)

[Out]

1/f*(-c^3*a^3*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)+3*c^3*a^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)-3*c^3*a^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)
+c^3*a^3*(f*x+e))

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Maxima [A]  time = 1.22467, size = 178, normalized size = 1.96 \begin{align*} -\frac{{\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{3} - 18 \,{\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^{3} c^{3} + 144 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{3} - 192 \,{\left (f x + e\right )} a^{3} c^{3}}{192 \, f} \end{align*}

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))^3,x, algorithm="maxima")

[Out]

-1/192*((4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*a^3*c^3 - 18*(12*f*x
+ 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^3*c^3 + 144*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c^3 - 192*
(f*x + e)*a^3*c^3)/f

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Fricas [A]  time = 1.40205, size = 163, normalized size = 1.79 \begin{align*} \frac{15 \, a^{3} c^{3} f x +{\left (8 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} + 10 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} + 15 \, a^{3} c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f} \end{align*}

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))^3,x, algorithm="fricas")

[Out]

1/48*(15*a^3*c^3*f*x + (8*a^3*c^3*cos(f*x + e)^5 + 10*a^3*c^3*cos(f*x + e)^3 + 15*a^3*c^3*cos(f*x + e))*sin(f*
x + e))/f

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Sympy [A]  time = 10.1909, size = 398, normalized size = 4.37 \begin{align*} \begin{cases} - \frac{5 a^{3} c^{3} x \sin ^{6}{\left (e + f x \right )}}{16} - \frac{15 a^{3} c^{3} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac{9 a^{3} c^{3} x \sin ^{4}{\left (e + f x \right )}}{8} - \frac{15 a^{3} c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac{9 a^{3} c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac{3 a^{3} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac{5 a^{3} c^{3} x \cos ^{6}{\left (e + f x \right )}}{16} + \frac{9 a^{3} c^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac{3 a^{3} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{3} c^{3} x + \frac{11 a^{3} c^{3} \sin ^{5}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{16 f} + \frac{5 a^{3} c^{3} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac{15 a^{3} c^{3} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} + \frac{5 a^{3} c^{3} \sin{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac{9 a^{3} c^{3} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac{3 a^{3} c^{3} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} & \text{for}\: f \neq 0 \\x \left (a \sin{\left (e \right )} + a\right )^{3} \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))**3*(c-c*sin(f*x+e))**3,x)

[Out]

Piecewise((-5*a**3*c**3*x*sin(e + f*x)**6/16 - 15*a**3*c**3*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*a**3*c**3
*x*sin(e + f*x)**4/8 - 15*a**3*c**3*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*a**3*c**3*x*sin(e + f*x)**2*cos(e
+ f*x)**2/4 - 3*a**3*c**3*x*sin(e + f*x)**2/2 - 5*a**3*c**3*x*cos(e + f*x)**6/16 + 9*a**3*c**3*x*cos(e + f*x)
**4/8 - 3*a**3*c**3*x*cos(e + f*x)**2/2 + a**3*c**3*x + 11*a**3*c**3*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 5*a
**3*c**3*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 15*a**3*c**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) + 5*a**3*c**3
*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 9*a**3*c**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 3*a**3*c**3*sin(e + f*
x)*cos(e + f*x)/(2*f), Ne(f, 0)), (x*(a*sin(e) + a)**3*(-c*sin(e) + c)**3, True))

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Giac [A]  time = 1.77461, size = 99, normalized size = 1.09 \begin{align*} \frac{5}{16} \, a^{3} c^{3} x + \frac{a^{3} c^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{3 \, a^{3} c^{3} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac{15 \, a^{3} c^{3} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end{align*}

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))^3,x, algorithm="giac")

[Out]

5/16*a^3*c^3*x + 1/192*a^3*c^3*sin(6*f*x + 6*e)/f + 3/64*a^3*c^3*sin(4*f*x + 4*e)/f + 15/64*a^3*c^3*sin(2*f*x
+ 2*e)/f