3.156 \(\int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=181 \[ \frac{9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac{d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac{9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac{d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}-\frac{9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac{d^3 \sin (6 a+6 b x)}{6912 b^4}-\frac{3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^3 \cos (6 a+6 b x)}{192 b} \]

[Out]

(9*d^2*(c + d*x)*Cos[2*a + 2*b*x])/(128*b^3) - (3*(c + d*x)^3*Cos[2*a + 2*b*x])/(64*b) - (d^2*(c + d*x)*Cos[6*
a + 6*b*x])/(1152*b^3) + ((c + d*x)^3*Cos[6*a + 6*b*x])/(192*b) - (9*d^3*Sin[2*a + 2*b*x])/(256*b^4) + (9*d*(c
+ d*x)^2*Sin[2*a + 2*b*x])/(128*b^2) + (d^3*Sin[6*a + 6*b*x])/(6912*b^4) - (d*(c + d*x)^2*Sin[6*a + 6*b*x])/(
384*b^2)

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Rubi [A]  time = 0.218944, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4406, 3296, 2637} \[ \frac{9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac{d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac{9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac{d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}-\frac{9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac{d^3 \sin (6 a+6 b x)}{6912 b^4}-\frac{3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^3 \cos (6 a+6 b x)}{192 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(9*d^2*(c + d*x)*Cos[2*a + 2*b*x])/(128*b^3) - (3*(c + d*x)^3*Cos[2*a + 2*b*x])/(64*b) - (d^2*(c + d*x)*Cos[6*
a + 6*b*x])/(1152*b^3) + ((c + d*x)^3*Cos[6*a + 6*b*x])/(192*b) - (9*d^3*Sin[2*a + 2*b*x])/(256*b^4) + (9*d*(c
+ d*x)^2*Sin[2*a + 2*b*x])/(128*b^2) + (d^3*Sin[6*a + 6*b*x])/(6912*b^4) - (d*(c + d*x)^2*Sin[6*a + 6*b*x])/(
384*b^2)

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac{3}{32} (c+d x)^3 \sin (2 a+2 b x)-\frac{1}{32} (c+d x)^3 \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac{1}{32} \int (c+d x)^3 \sin (6 a+6 b x) \, dx\right )+\frac{3}{32} \int (c+d x)^3 \sin (2 a+2 b x) \, dx\\ &=-\frac{3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^3 \cos (6 a+6 b x)}{192 b}-\frac{d \int (c+d x)^2 \cos (6 a+6 b x) \, dx}{64 b}+\frac{(9 d) \int (c+d x)^2 \cos (2 a+2 b x) \, dx}{64 b}\\ &=-\frac{3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^3 \cos (6 a+6 b x)}{192 b}+\frac{9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac{d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}+\frac{d^2 \int (c+d x) \sin (6 a+6 b x) \, dx}{192 b^2}-\frac{\left (9 d^2\right ) \int (c+d x) \sin (2 a+2 b x) \, dx}{64 b^2}\\ &=\frac{9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac{3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}-\frac{d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac{(c+d x)^3 \cos (6 a+6 b x)}{192 b}+\frac{9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac{d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}+\frac{d^3 \int \cos (6 a+6 b x) \, dx}{1152 b^3}-\frac{\left (9 d^3\right ) \int \cos (2 a+2 b x) \, dx}{128 b^3}\\ &=\frac{9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac{3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}-\frac{d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac{(c+d x)^3 \cos (6 a+6 b x)}{192 b}-\frac{9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac{9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}+\frac{d^3 \sin (6 a+6 b x)}{6912 b^4}-\frac{d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}\\ \end{align*}

Mathematica [A]  time = 2.38983, size = 132, normalized size = 0.73 \[ \frac{-324 b (c+d x) \cos (2 (a+b x)) \left (2 b^2 (c+d x)^2-3 d^2\right )+12 b (c+d x) \cos (6 (a+b x)) \left (6 b^2 (c+d x)^2-d^2\right )-4 d \sin (2 (a+b x)) \left (\cos (4 (a+b x)) \left (18 b^2 (c+d x)^2-d^2\right )-234 b^2 (c+d x)^2+121 d^2\right )}{13824 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(-324*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 12*b*(c + d*x)*(-d^2 + 6*b^2*(c + d*x)^2)*Co
s[6*(a + b*x)] - 4*d*(121*d^2 - 234*b^2*(c + d*x)^2 + (-d^2 + 18*b^2*(c + d*x)^2)*Cos[4*(a + b*x)])*Sin[2*(a +
b*x)])/(13824*b^4)

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Maple [B]  time = 0.027, size = 1100, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*cos(b*x+a)^3*sin(b*x+a)^3,x)

[Out]

1/b*(1/b^3*d^3*(1/4*(b*x+a)^3*sin(b*x+a)^4-3/4*(b*x+a)^2*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*
x+3/8*a)-1/24*(b*x+a)*sin(b*x+a)^4-1/96*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-1/18*b*x-1/18*a+1/8*(b*x+a)*c
os(b*x+a)^2-1/16*cos(b*x+a)*sin(b*x+a)+1/12*(b*x+a)^3-1/6*(b*x+a)^3*sin(b*x+a)^6+1/2*(b*x+a)^2*(-1/6*(sin(b*x+
a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a)+5/16*b*x+5/16*a)+1/36*(b*x+a)*sin(b*x+a)^6+1/216*(sin(b*x+a)
^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a))-3/b^3*a*d^3*(1/4*(b*x+a)^2*sin(b*x+a)^4-1/2*(b*x+a)*(-1/4*(si
n(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)+1/24*(b*x+a)^2-1/72*sin(b*x+a)^4-1/24*sin(b*x+a)^2-1/6*(b
*x+a)^2*sin(b*x+a)^6+1/3*(b*x+a)*(-1/6*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a)+5/16*b*x+5/1
6*a)+1/108*sin(b*x+a)^6)+3/b^2*c*d^2*(1/4*(b*x+a)^2*sin(b*x+a)^4-1/2*(b*x+a)*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a
))*cos(b*x+a)+3/8*b*x+3/8*a)+1/24*(b*x+a)^2-1/72*sin(b*x+a)^4-1/24*sin(b*x+a)^2-1/6*(b*x+a)^2*sin(b*x+a)^6+1/3
*(b*x+a)*(-1/6*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a)+5/16*b*x+5/16*a)+1/108*sin(b*x+a)^6)
+3/b^3*a^2*d^3*(1/4*(b*x+a)*sin(b*x+a)^4+1/16*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-1/24*b*x-1/24*a-1/6*(b*
x+a)*sin(b*x+a)^6-1/36*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a))-6/b^2*a*c*d^2*(1/4*(b*x+a)*
sin(b*x+a)^4+1/16*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-1/24*b*x-1/24*a-1/6*(b*x+a)*sin(b*x+a)^6-1/36*(sin(
b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a))+3/b*c^2*d*(1/4*(b*x+a)*sin(b*x+a)^4+1/16*(sin(b*x+a)^3+
3/2*sin(b*x+a))*cos(b*x+a)-1/24*b*x-1/24*a-1/6*(b*x+a)*sin(b*x+a)^6-1/36*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*s
in(b*x+a))*cos(b*x+a))-1/b^3*a^3*d^3*(-1/6*sin(b*x+a)^2*cos(b*x+a)^4-1/12*cos(b*x+a)^4)+3/b^2*a^2*c*d^2*(-1/6*
sin(b*x+a)^2*cos(b*x+a)^4-1/12*cos(b*x+a)^4)-3/b*a*c^2*d*(-1/6*sin(b*x+a)^2*cos(b*x+a)^4-1/12*cos(b*x+a)^4)+c^
3*(-1/6*sin(b*x+a)^2*cos(b*x+a)^4-1/12*cos(b*x+a)^4))

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Maxima [B]  time = 1.33376, size = 813, normalized size = 4.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/6912*(576*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*c^3 - 1728*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a*c^2*d/b
+ 1728*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^2*c*d^2/b^2 - 576*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^3*d^3
/b^3 - 18*(6*(b*x + a)*cos(6*b*x + 6*a) - 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) + 27*sin(2*b*x + 2*
a))*c^2*d/b + 36*(6*(b*x + a)*cos(6*b*x + 6*a) - 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) + 27*sin(2*b
*x + 2*a))*a*c*d^2/b^2 - 18*(6*(b*x + a)*cos(6*b*x + 6*a) - 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) +
27*sin(2*b*x + 2*a))*a^2*d^3/b^3 - 6*((18*(b*x + a)^2 - 1)*cos(6*b*x + 6*a) - 81*(2*(b*x + a)^2 - 1)*cos(2*b*
x + 2*a) - 6*(b*x + a)*sin(6*b*x + 6*a) + 162*(b*x + a)*sin(2*b*x + 2*a))*c*d^2/b^2 + 6*((18*(b*x + a)^2 - 1)*
cos(6*b*x + 6*a) - 81*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 6*(b*x + a)*sin(6*b*x + 6*a) + 162*(b*x + a)*sin(
2*b*x + 2*a))*a*d^3/b^3 - (6*(6*(b*x + a)^3 - b*x - a)*cos(6*b*x + 6*a) - 162*(2*(b*x + a)^3 - 3*b*x - 3*a)*co
s(2*b*x + 2*a) - (18*(b*x + a)^2 - 1)*sin(6*b*x + 6*a) + 243*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*d^3/b^3)/b

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Fricas [B]  time = 0.533301, size = 747, normalized size = 4.13 \begin{align*} \frac{9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 6 \,{\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{3} - b c d^{2} +{\left (18 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{6} - 9 \,{\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{3} - b c d^{2} +{\left (18 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 27 \,{\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2} + 3 \,{\left (9 \, b^{3} c^{2} d - 5 \, b d^{3}\right )} x -{\left ({\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{5} -{\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{3} - 3 \,{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 5 \, d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{216 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

1/216*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 6*(6*b^3*d^3*x^3 + 18*b^3*c*d^2*x^2 + 6*b^3*c^3 - b*c*d^2 + (18*b^3*
c^2*d - b*d^3)*x)*cos(b*x + a)^6 - 9*(6*b^3*d^3*x^3 + 18*b^3*c*d^2*x^2 + 6*b^3*c^3 - b*c*d^2 + (18*b^3*c^2*d -
b*d^3)*x)*cos(b*x + a)^4 + 27*(b*d^3*x + b*c*d^2)*cos(b*x + a)^2 + 3*(9*b^3*c^2*d - 5*b*d^3)*x - ((18*b^2*d^3
*x^2 + 36*b^2*c*d^2*x + 18*b^2*c^2*d - d^3)*cos(b*x + a)^5 - (18*b^2*d^3*x^2 + 36*b^2*c*d^2*x + 18*b^2*c^2*d -
d^3)*cos(b*x + a)^3 - 3*(9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d - 5*d^3)*cos(b*x + a))*sin(b*x + a))/b^
4

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Sympy [A]  time = 60.5033, size = 867, normalized size = 4.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*cos(b*x+a)**3*sin(b*x+a)**3,x)

[Out]

Piecewise((c**3*sin(a + b*x)**6/(12*b) + c**3*sin(a + b*x)**4*cos(a + b*x)**2/(4*b) + c**2*d*x*sin(a + b*x)**6
/(8*b) + 3*c**2*d*x*sin(a + b*x)**4*cos(a + b*x)**2/(8*b) - 3*c**2*d*x*sin(a + b*x)**2*cos(a + b*x)**4/(8*b) -
c**2*d*x*cos(a + b*x)**6/(8*b) + c*d**2*x**2*sin(a + b*x)**6/(8*b) + 3*c*d**2*x**2*sin(a + b*x)**4*cos(a + b*
x)**2/(8*b) - 3*c*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)**4/(8*b) - c*d**2*x**2*cos(a + b*x)**6/(8*b) + d**3*x
**3*sin(a + b*x)**6/(24*b) + d**3*x**3*sin(a + b*x)**4*cos(a + b*x)**2/(8*b) - d**3*x**3*sin(a + b*x)**2*cos(a
+ b*x)**4/(8*b) - d**3*x**3*cos(a + b*x)**6/(24*b) + c**2*d*sin(a + b*x)**5*cos(a + b*x)/(8*b**2) + c**2*d*si
n(a + b*x)**3*cos(a + b*x)**3/(3*b**2) + c**2*d*sin(a + b*x)*cos(a + b*x)**5/(8*b**2) + c*d**2*x*sin(a + b*x)*
*5*cos(a + b*x)/(4*b**2) + 2*c*d**2*x*sin(a + b*x)**3*cos(a + b*x)**3/(3*b**2) + c*d**2*x*sin(a + b*x)*cos(a +
b*x)**5/(4*b**2) + d**3*x**2*sin(a + b*x)**5*cos(a + b*x)/(8*b**2) + d**3*x**2*sin(a + b*x)**3*cos(a + b*x)**
3/(3*b**2) + d**3*x**2*sin(a + b*x)*cos(a + b*x)**5/(8*b**2) - 5*c*d**2*sin(a + b*x)**6/(36*b**3) - 7*c*d**2*s
in(a + b*x)**4*cos(a + b*x)**2/(24*b**3) - c*d**2*sin(a + b*x)**2*cos(a + b*x)**4/(8*b**3) - 5*d**3*x*sin(a +
b*x)**6/(72*b**3) - d**3*x*sin(a + b*x)**4*cos(a + b*x)**2/(12*b**3) + d**3*x*sin(a + b*x)**2*cos(a + b*x)**4/
(12*b**3) + 5*d**3*x*cos(a + b*x)**6/(72*b**3) - 5*d**3*sin(a + b*x)**5*cos(a + b*x)/(72*b**4) - 31*d**3*sin(a
+ b*x)**3*cos(a + b*x)**3/(216*b**4) - 5*d**3*sin(a + b*x)*cos(a + b*x)**5/(72*b**4), Ne(b, 0)), ((c**3*x + 3
*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4)*sin(a)**3*cos(a)**3, True))

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Giac [A]  time = 1.13246, size = 325, normalized size = 1.8 \begin{align*} \frac{{\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 18 \, b^{3} c^{2} d x + 6 \, b^{3} c^{3} - b d^{3} x - b c d^{2}\right )} \cos \left (6 \, b x + 6 \, a\right )}{1152 \, b^{4}} - \frac{3 \,{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{4}} - \frac{{\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (6 \, b x + 6 \, a\right )}{6912 \, b^{4}} + \frac{9 \,{\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{256 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/1152*(6*b^3*d^3*x^3 + 18*b^3*c*d^2*x^2 + 18*b^3*c^2*d*x + 6*b^3*c^3 - b*d^3*x - b*c*d^2)*cos(6*b*x + 6*a)/b^
4 - 3/128*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 - 3*b*d^3*x - 3*b*c*d^2)*cos(2*b*x + 2*
a)/b^4 - 1/6912*(18*b^2*d^3*x^2 + 36*b^2*c*d^2*x + 18*b^2*c^2*d - d^3)*sin(6*b*x + 6*a)/b^4 + 9/256*(2*b^2*d^3
*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d - d^3)*sin(2*b*x + 2*a)/b^4