3.248 \(\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx\)

Optimal. Leaf size=145 \[ \frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac{3 a^3 c^5 \sin (e+f x) \cos ^5(e+f x)}{16 f}+\frac{15 a^3 c^5 \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac{45 a^3 c^5 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{45}{128} a^3 c^5 x \]

[Out]

(45*a^3*c^5*x)/128 + (9*a^3*c^5*Cos[e + f*x]^7)/(56*f) + (45*a^3*c^5*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (15*
a^3*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(64*f) + (3*a^3*c^5*Cos[e + f*x]^5*Sin[e + f*x])/(16*f) + (a^3*Cos[e + f*
x]^7*(c^5 - c^5*Sin[e + f*x]))/(8*f)

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Rubi [A]  time = 0.157084, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2736, 2678, 2669, 2635, 8} \[ \frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac{3 a^3 c^5 \sin (e+f x) \cos ^5(e+f x)}{16 f}+\frac{15 a^3 c^5 \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac{45 a^3 c^5 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{45}{128} a^3 c^5 x \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5,x]

[Out]

(45*a^3*c^5*x)/128 + (9*a^3*c^5*Cos[e + f*x]^7)/(56*f) + (45*a^3*c^5*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (15*
a^3*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(64*f) + (3*a^3*c^5*Cos[e + f*x]^5*Sin[e + f*x])/(16*f) + (a^3*Cos[e + f*
x]^7*(c^5 - c^5*Sin[e + f*x]))/(8*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))^3 (c-c \sin (e+f x))^5 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=\frac{a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac{1}{8} \left (9 a^3 c^4\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac{1}{8} \left (9 a^3 c^5\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac{a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac{1}{16} \left (15 a^3 c^5\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{15 a^3 c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac{3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac{a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac{1}{64} \left (45 a^3 c^5\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{45 a^3 c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{15 a^3 c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac{3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac{a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac{1}{128} \left (45 a^3 c^5\right ) \int 1 \, dx\\ &=\frac{45}{128} a^3 c^5 x+\frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{45 a^3 c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{15 a^3 c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac{3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac{a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}\\ \end{align*}

Mathematica [A]  time = 1.23014, size = 89, normalized size = 0.61 \[ \frac{a^3 c^5 (1792 \sin (2 (e+f x))+280 \sin (4 (e+f x))-7 \sin (8 (e+f x))+1120 \cos (e+f x)+672 \cos (3 (e+f x))+224 \cos (5 (e+f x))+32 \cos (7 (e+f x))+2520 e+2520 f x)}{7168 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5,x]

[Out]

(a^3*c^5*(2520*e + 2520*f*x + 1120*Cos[e + f*x] + 672*Cos[3*(e + f*x)] + 224*Cos[5*(e + f*x)] + 32*Cos[7*(e +
f*x)] + 1792*Sin[2*(e + f*x)] + 280*Sin[4*(e + f*x)] - 7*Sin[8*(e + f*x)]))/(7168*f)

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Maple [B]  time = 0.02, size = 276, normalized size = 1.9 \begin{align*}{\frac{1}{f} \left ( -{c}^{5}{a}^{3} \left ( -{\frac{\cos \left ( fx+e \right ) }{8} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( fx+e \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( fx+e \right ) }{16}} \right ) }+{\frac{35\,fx}{128}}+{\frac{35\,e}{128}} \right ) -{\frac{2\,{c}^{5}{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 ) }+2\,{c}^{5}{a}^{3} \left ( -1/6\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{5}+5/4\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{15\,\sin \left ( fx+e \right ) }{8}} \right ) \cos \left ( fx+e \right ) +{\frac{5\,fx}{16}}+{\frac{5\,e}{16}} \right ) +{\frac{6\,{c}^{5}{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 ) }-2\,{c}^{5}{a}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) -2\,{c}^{5}{a}^{3} \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) +2\,{c}^{5}{a}^{3}\cos \left ( fx+e \right ) +{c}^{5}{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))^5,x)

[Out]

1/f*(-c^5*a^3*(-1/8*(sin(f*x+e)^7+7/6*sin(f*x+e)^5+35/24*sin(f*x+e)^3+35/16*sin(f*x+e))*cos(f*x+e)+35/128*f*x+
35/128*e)-2/7*c^5*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)+2*c^5*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)+6/5*c^5*a^3*(8/3+sin(f*x+e)^4+4/3*sin(f*x
+e)^2)*cos(f*x+e)-2*c^5*a^3*(2+sin(f*x+e)^2)*cos(f*x+e)-2*c^5*a^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+2
*c^5*a^3*cos(f*x+e)+c^5*a^3*(f*x+e))

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Maxima [B]  time = 1.21141, size = 381, normalized size = 2.63 \begin{align*} \frac{6144 \,{\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^{5} + 43008 \,{\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^{5} + 215040 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{5} - 35 \,{\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{5} + 1120 \,{\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^{5} - 53760 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{5} + 107520 \,{\left (f x + e\right )} a^{3} c^{5} + 215040 \, a^{3} c^{5} \cos \left (f x + e\right )}{107520 \, 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))^5,x, algorithm="maxima")

[Out]

1/107520*(6144*(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^5 + 43008*(3
*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3*c^5 + 215040*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*
c^5 - 35*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e) + 168*sin(4*f*x + 4*e) - 768*sin(2*f*x
+ 2*e))*a^3*c^5 + 1120*(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^5 - 53760*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c^5 + 107520*(f*x + e)*a^3*c^5 + 215040*a^3*c^5*cos(f*x + e))
/f

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Fricas [A]  time = 1.35618, size = 247, normalized size = 1.7 \begin{align*} \frac{256 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} + 315 \, a^{3} c^{5} f x - 7 \,{\left (16 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} - 24 \, a^{3} c^{5} \cos \left (f x + e\right )^{5} - 30 \, a^{3} c^{5} \cos \left (f x + e\right )^{3} - 45 \, a^{3} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{896 \, 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))^5,x, algorithm="fricas")

[Out]

1/896*(256*a^3*c^5*cos(f*x + e)^7 + 315*a^3*c^5*f*x - 7*(16*a^3*c^5*cos(f*x + e)^7 - 24*a^3*c^5*cos(f*x + e)^5
- 30*a^3*c^5*cos(f*x + e)^3 - 45*a^3*c^5*cos(f*x + e))*sin(f*x + e))/f

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Sympy [A]  time = 29.9373, size = 740, normalized size = 5.1 \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))**5,x)

[Out]

Piecewise((-35*a**3*c**5*x*sin(e + f*x)**8/128 - 35*a**3*c**5*x*sin(e + f*x)**6*cos(e + f*x)**2/32 + 5*a**3*c*
*5*x*sin(e + f*x)**6/8 - 105*a**3*c**5*x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 15*a**3*c**5*x*sin(e + f*x)**4*c
os(e + f*x)**2/8 - 35*a**3*c**5*x*sin(e + f*x)**2*cos(e + f*x)**6/32 + 15*a**3*c**5*x*sin(e + f*x)**2*cos(e +
f*x)**4/8 - a**3*c**5*x*sin(e + f*x)**2 - 35*a**3*c**5*x*cos(e + f*x)**8/128 + 5*a**3*c**5*x*cos(e + f*x)**6/8
- a**3*c**5*x*cos(e + f*x)**2 + a**3*c**5*x + 93*a**3*c**5*sin(e + f*x)**7*cos(e + f*x)/(128*f) - 2*a**3*c**5
*sin(e + f*x)**6*cos(e + f*x)/f + 511*a**3*c**5*sin(e + f*x)**5*cos(e + f*x)**3/(384*f) - 11*a**3*c**5*sin(e +
f*x)**5*cos(e + f*x)/(8*f) - 4*a**3*c**5*sin(e + f*x)**4*cos(e + f*x)**3/f + 6*a**3*c**5*sin(e + f*x)**4*cos(
e + f*x)/f + 385*a**3*c**5*sin(e + f*x)**3*cos(e + f*x)**5/(384*f) - 5*a**3*c**5*sin(e + f*x)**3*cos(e + f*x)*
*3/(3*f) - 16*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 8*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)**3/f
- 6*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)/f + 35*a**3*c**5*sin(e + f*x)*cos(e + f*x)**7/(128*f) - 5*a**3*c**5
*sin(e + f*x)*cos(e + f*x)**5/(8*f) + a**3*c**5*sin(e + f*x)*cos(e + f*x)/f - 32*a**3*c**5*cos(e + f*x)**7/(35
*f) + 16*a**3*c**5*cos(e + f*x)**5/(5*f) - 4*a**3*c**5*cos(e + f*x)**3/f + 2*a**3*c**5*cos(e + f*x)/f, Ne(f, 0
)), (x*(a*sin(e) + a)**3*(-c*sin(e) + c)**5, True))

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Giac [A]  time = 1.69469, size = 208, normalized size = 1.43 \begin{align*} \frac{45}{128} \, a^{3} c^{5} x + \frac{a^{3} c^{5} \cos \left (7 \, f x + 7 \, e\right )}{224 \, f} + \frac{a^{3} c^{5} \cos \left (5 \, f x + 5 \, e\right )}{32 \, f} + \frac{3 \, a^{3} c^{5} \cos \left (3 \, f x + 3 \, e\right )}{32 \, f} + \frac{5 \, a^{3} c^{5} \cos \left (f x + e\right )}{32 \, f} - \frac{a^{3} c^{5} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac{5 \, a^{3} c^{5} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac{a^{3} c^{5} \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))^3*(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

45/128*a^3*c^5*x + 1/224*a^3*c^5*cos(7*f*x + 7*e)/f + 1/32*a^3*c^5*cos(5*f*x + 5*e)/f + 3/32*a^3*c^5*cos(3*f*x
+ 3*e)/f + 5/32*a^3*c^5*cos(f*x + e)/f - 1/1024*a^3*c^5*sin(8*f*x + 8*e)/f + 5/128*a^3*c^5*sin(4*f*x + 4*e)/f
+ 1/4*a^3*c^5*sin(2*f*x + 2*e)/f