3.249 \(\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx\)

Optimal. Leaf size=112 \[ \frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{a^3 c^4 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^4 x \]

[Out]

(5*a^3*c^4*x)/16 + (a^3*c^4*Cos[e + f*x]^7)/(7*f) + (5*a^3*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^4*
Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^3*c^4*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)

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Rubi [A]  time = 0.10777, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2736, 2669, 2635, 8} \[ \frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{a^3 c^4 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^4 x \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4,x]

[Out]

(5*a^3*c^4*x)/16 + (a^3*c^4*Cos[e + f*x]^7)/(7*f) + (5*a^3*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^4*
Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^3*c^4*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 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))^3 (c-c \sin (e+f x))^4 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\left (a^3 c^4\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{6} \left (5 a^3 c^4\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{8} \left (5 a^3 c^4\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{5 a^3 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac{5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{16} \left (5 a^3 c^4\right ) \int 1 \, dx\\ &=\frac{5}{16} a^3 c^4 x+\frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{5 a^3 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac{5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}\\ \end{align*}

Mathematica [A]  time = 1.05507, size = 89, normalized size = 0.79 \[ \frac{a^3 c^4 (315 \sin (2 (e+f x))+63 \sin (4 (e+f x))+7 \sin (6 (e+f x))+105 \cos (e+f x)+63 \cos (3 (e+f x))+21 \cos (5 (e+f x))+3 \cos (7 (e+f x))+420 e+420 f x)}{1344 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4,x]

[Out]

(a^3*c^4*(420*e + 420*f*x + 105*Cos[e + f*x] + 63*Cos[3*(e + f*x)] + 21*Cos[5*(e + f*x)] + 3*Cos[7*(e + f*x)]
+ 315*Sin[2*(e + f*x)] + 63*Sin[4*(e + f*x)] + 7*Sin[6*(e + f*x)]))/(1344*f)

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Maple [B]  time = 0.016, size = 255, normalized size = 2.3 \begin{align*}{\frac{1}{f} \left ( -{\frac{{c}^{4}{a}^{3}\cos \left ( fx+e \right ) }{7} \left ({\frac{16}{5}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{6}+{\frac{6\, \left ( \sin \left ( fx+e \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{5}} \right ) }-{c}^{4}{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 ) +{\frac{3\,{c}^{4}{a}^{3}\cos \left ( fx+e \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) }+3\,{c}^{4}{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 ) -{c}^{4}{a}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) -3\,{c}^{4}{a}^{3} \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) +{c}^{4}{a}^{3}\cos \left ( fx+e \right ) +{c}^{4}{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))^4,x)

[Out]

1/f*(-1/7*c^4*a^3*(16/5+sin(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(f*x+e)^2)*cos(f*x+e)-c^4*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)+3/5*c^4*a^3*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)
*cos(f*x+e)+3*c^4*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)-c^4*a^3*(2+sin(f*x+e)^2)*c
os(f*x+e)-3*c^4*a^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+c^4*a^3*cos(f*x+e)+c^4*a^3*(f*x+e))

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Maxima [B]  time = 1.19201, size = 346, normalized size = 3.09 \begin{align*} \frac{192 \,{\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} + 1344 \,{\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} + 6720 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} - 35 \,{\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^{4} + 630 \,{\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^{4} - 5040 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{4} + 6720 \,{\left (f x + e\right )} a^{3} c^{4} + 6720 \, a^{3} c^{4} \cos \left (f x + e\right )}{6720 \, 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))^4,x, algorithm="maxima")

[Out]

1/6720*(192*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*a^3*c^4 + 1344*(3*cos
(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3*c^4 + 6720*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*c^4 -
35*(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^4 + 630*(12*f*x + 1
2*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^3*c^4 - 5040*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c^4 + 6720*(f
*x + e)*a^3*c^4 + 6720*a^3*c^4*cos(f*x + e))/f

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Fricas [A]  time = 1.38752, size = 207, normalized size = 1.85 \begin{align*} \frac{48 \, a^{3} c^{4} \cos \left (f x + e\right )^{7} + 105 \, a^{3} c^{4} f x + 7 \,{\left (8 \, a^{3} c^{4} \cos \left (f x + e\right )^{5} + 10 \, a^{3} c^{4} \cos \left (f x + e\right )^{3} + 15 \, a^{3} c^{4} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{336 \, 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))^4,x, algorithm="fricas")

[Out]

1/336*(48*a^3*c^4*cos(f*x + e)^7 + 105*a^3*c^4*f*x + 7*(8*a^3*c^4*cos(f*x + e)^5 + 10*a^3*c^4*cos(f*x + e)^3 +
15*a^3*c^4*cos(f*x + e))*sin(f*x + e))/f

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Sympy [A]  time = 18.6363, size = 631, normalized size = 5.63 \begin{align*} \text{result too large to display} \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))**4,x)

[Out]

Piecewise((-5*a**3*c**4*x*sin(e + f*x)**6/16 - 15*a**3*c**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*a**3*c**4
*x*sin(e + f*x)**4/8 - 15*a**3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*a**3*c**4*x*sin(e + f*x)**2*cos(e
+ f*x)**2/4 - 3*a**3*c**4*x*sin(e + f*x)**2/2 - 5*a**3*c**4*x*cos(e + f*x)**6/16 + 9*a**3*c**4*x*cos(e + f*x)
**4/8 - 3*a**3*c**4*x*cos(e + f*x)**2/2 + a**3*c**4*x - a**3*c**4*sin(e + f*x)**6*cos(e + f*x)/f + 11*a**3*c**
4*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 2*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)**3/f + 3*a**3*c**4*sin(e + f*
x)**4*cos(e + f*x)/f + 5*a**3*c**4*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 15*a**3*c**4*sin(e + f*x)**3*cos(e
+ f*x)/(8*f) - 8*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 4*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)**3
/f - 3*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)/f + 5*a**3*c**4*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 9*a**3*c**
4*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 3*a**3*c**4*sin(e + f*x)*cos(e + f*x)/(2*f) - 16*a**3*c**4*cos(e + f*x)
**7/(35*f) + 8*a**3*c**4*cos(e + f*x)**5/(5*f) - 2*a**3*c**4*cos(e + f*x)**3/f + a**3*c**4*cos(e + f*x)/f, Ne(
f, 0)), (x*(a*sin(e) + a)**3*(-c*sin(e) + c)**4, True))

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Giac [A]  time = 1.72489, size = 208, normalized size = 1.86 \begin{align*} \frac{5}{16} \, a^{3} c^{4} x + \frac{a^{3} c^{4} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac{a^{3} c^{4} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac{3 \, a^{3} c^{4} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac{5 \, a^{3} c^{4} \cos \left (f x + e\right )}{64 \, f} + \frac{a^{3} c^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{3 \, a^{3} c^{4} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac{15 \, a^{3} c^{4} \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))^4,x, algorithm="giac")

[Out]

5/16*a^3*c^4*x + 1/448*a^3*c^4*cos(7*f*x + 7*e)/f + 1/64*a^3*c^4*cos(5*f*x + 5*e)/f + 3/64*a^3*c^4*cos(3*f*x +
3*e)/f + 5/64*a^3*c^4*cos(f*x + e)/f + 1/192*a^3*c^4*sin(6*f*x + 6*e)/f + 3/64*a^3*c^4*sin(4*f*x + 4*e)/f + 1
5/64*a^3*c^4*sin(2*f*x + 2*e)/f