3.81 \(\int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx\)

Optimal. Leaf size=105 \[ \frac{3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac{3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}+\frac{3 d^3 \cos (4 a+4 b x)}{1024 b^4}-\frac{(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^4}{32 d} \]

[Out]

(c + d*x)^4/(32*d) + (3*d^3*Cos[4*a + 4*b*x])/(1024*b^4) - (3*d*(c + d*x)^2*Cos[4*a + 4*b*x])/(128*b^2) + (3*d
^2*(c + d*x)*Sin[4*a + 4*b*x])/(256*b^3) - ((c + d*x)^3*Sin[4*a + 4*b*x])/(32*b)

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Rubi [A]  time = 0.130323, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4406, 3296, 2638} \[ \frac{3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac{3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}+\frac{3 d^3 \cos (4 a+4 b x)}{1024 b^4}-\frac{(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^4}{32 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x]^2,x]

[Out]

(c + d*x)^4/(32*d) + (3*d^3*Cos[4*a + 4*b*x])/(1024*b^4) - (3*d*(c + d*x)^2*Cos[4*a + 4*b*x])/(128*b^2) + (3*d
^2*(c + d*x)*Sin[4*a + 4*b*x])/(256*b^3) - ((c + d*x)^3*Sin[4*a + 4*b*x])/(32*b)

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 2638

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

Rubi steps

\begin{align*} \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac{1}{8} (c+d x)^3-\frac{1}{8} (c+d x)^3 \cos (4 a+4 b x)\right ) \, dx\\ &=\frac{(c+d x)^4}{32 d}-\frac{1}{8} \int (c+d x)^3 \cos (4 a+4 b x) \, dx\\ &=\frac{(c+d x)^4}{32 d}-\frac{(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac{(3 d) \int (c+d x)^2 \sin (4 a+4 b x) \, dx}{32 b}\\ &=\frac{(c+d x)^4}{32 d}-\frac{3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}-\frac{(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac{\left (3 d^2\right ) \int (c+d x) \cos (4 a+4 b x) \, dx}{64 b^2}\\ &=\frac{(c+d x)^4}{32 d}-\frac{3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}+\frac{3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac{(c+d x)^3 \sin (4 a+4 b x)}{32 b}-\frac{\left (3 d^3\right ) \int \sin (4 a+4 b x) \, dx}{256 b^3}\\ &=\frac{(c+d x)^4}{32 d}+\frac{3 d^3 \cos (4 a+4 b x)}{1024 b^4}-\frac{3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}+\frac{3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac{(c+d x)^3 \sin (4 a+4 b x)}{32 b}\\ \end{align*}

Mathematica [A]  time = 0.685076, size = 106, normalized size = 1.01 \[ \frac{-4 b (c+d x) \sin (4 (a+b x)) \left (8 b^2 (c+d x)^2-3 d^2\right )-3 d \cos (4 (a+b x)) \left (8 b^2 (c+d x)^2-d^2\right )+32 b^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )}{1024 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x]^2,x]

[Out]

(32*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 3*d*(-d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] - 4*b*
(c + d*x)*(-3*d^2 + 8*b^2*(c + d*x)^2)*Sin[4*(a + b*x)])/(1024*b^4)

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Maple [B]  time = 0.022, size = 1074, normalized size = 10.2 \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)^2*sin(b*x+a)^2,x)

[Out]

1/b*(1/b^3*d^3*((b*x+a)^3*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-3/16*(b*x+a)^2*cos(b*x+a)^2+3/8*(b*x+a)*(
1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-21/128*(b*x+a)^2-3/128*sin(b*x+a)^2-3/32*(b*x+a)^4-(b*x+a)^3*(-1/4*(s
in(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-3/16*(b*x+a)^2*sin(b*x+a)^4+3/8*(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)+3/128*sin(b*x+a)^4)-3/b^3*a*d^3*((b*x+a)^2*(-1/2*cos(b*x+a)*sin(
b*x+a)+1/2*b*x+1/2*a)-1/8*(b*x+a)*cos(b*x+a)^2+1/16*cos(b*x+a)*sin(b*x+a)+7/64*b*x+7/64*a-1/12*(b*x+a)^3-(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)-1/8*(b*x+a)*sin(b*x+a)^4-1/32*(sin(b*x+a)^3
+3/2*sin(b*x+a))*cos(b*x+a))+3/b^2*c*d^2*((b*x+a)^2*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/8*(b*x+a)*cos
(b*x+a)^2+1/16*cos(b*x+a)*sin(b*x+a)+7/64*b*x+7/64*a-1/12*(b*x+a)^3-(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)-1/8*(b*x+a)*sin(b*x+a)^4-1/32*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a))+3/b^3*a^
2*d^3*((b*x+a)*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/16*(b*x+a)^2+1/16*sin(b*x+a)^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)-1/16*sin(b*x+a)^4)-6/b^2*a*c*d^2*((b*x+a)*(-1/2*cos(b*x+a)*
sin(b*x+a)+1/2*b*x+1/2*a)-1/16*(b*x+a)^2+1/16*sin(b*x+a)^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)-1/16*sin(b*x+a)^4)+3/b*c^2*d*((b*x+a)*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/16*(b*x+
a)^2+1/16*sin(b*x+a)^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)-1/16*sin(b*x+a)^4
)-1/b^3*a^3*d^3*(-1/4*sin(b*x+a)*cos(b*x+a)^3+1/8*cos(b*x+a)*sin(b*x+a)+1/8*b*x+1/8*a)+3/b^2*a^2*c*d^2*(-1/4*s
in(b*x+a)*cos(b*x+a)^3+1/8*cos(b*x+a)*sin(b*x+a)+1/8*b*x+1/8*a)-3/b*a*c^2*d*(-1/4*sin(b*x+a)*cos(b*x+a)^3+1/8*
cos(b*x+a)*sin(b*x+a)+1/8*b*x+1/8*a)+c^3*(-1/4*sin(b*x+a)*cos(b*x+a)^3+1/8*cos(b*x+a)*sin(b*x+a)+1/8*b*x+1/8*a
))

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Maxima [B]  time = 1.30413, size = 597, normalized size = 5.69 \begin{align*} \frac{32 \,{\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} c^{3} - \frac{96 \,{\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a c^{2} d}{b} + \frac{96 \,{\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{2} c d^{2}}{b^{2}} - \frac{32 \,{\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{3} d^{3}}{b^{3}} + \frac{24 \,{\left (8 \,{\left (b x + a\right )}^{2} - 4 \,{\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} c^{2} d}{b} - \frac{48 \,{\left (8 \,{\left (b x + a\right )}^{2} - 4 \,{\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac{24 \,{\left (8 \,{\left (b x + a\right )}^{2} - 4 \,{\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac{4 \,{\left (32 \,{\left (b x + a\right )}^{3} - 12 \,{\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \,{\left (8 \,{\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} c d^{2}}{b^{2}} - \frac{4 \,{\left (32 \,{\left (b x + a\right )}^{3} - 12 \,{\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \,{\left (8 \,{\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} a d^{3}}{b^{3}} + \frac{{\left (32 \,{\left (b x + a\right )}^{4} - 3 \,{\left (8 \,{\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - 4 \,{\left (8 \,{\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} d^{3}}{b^{3}}}{1024 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^2*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

1/1024*(32*(4*b*x + 4*a - sin(4*b*x + 4*a))*c^3 - 96*(4*b*x + 4*a - sin(4*b*x + 4*a))*a*c^2*d/b + 96*(4*b*x +
4*a - sin(4*b*x + 4*a))*a^2*c*d^2/b^2 - 32*(4*b*x + 4*a - sin(4*b*x + 4*a))*a^3*d^3/b^3 + 24*(8*(b*x + a)^2 -
4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*c^2*d/b - 48*(8*(b*x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) -
cos(4*b*x + 4*a))*a*c*d^2/b^2 + 24*(8*(b*x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*a^2*d^3/b
^3 + 4*(32*(b*x + a)^3 - 12*(b*x + a)*cos(4*b*x + 4*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a))*c*d^2/b^2 - 4
*(32*(b*x + a)^3 - 12*(b*x + a)*cos(4*b*x + 4*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a))*a*d^3/b^3 + (32*(b*
x + a)^4 - 3*(8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a) - 4*(8*(b*x + a)^3 - 3*b*x - 3*a)*sin(4*b*x + 4*a))*d^3/b^3)
/b

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Fricas [B]  time = 0.513568, size = 647, normalized size = 6.16 \begin{align*} \frac{4 \, b^{4} d^{3} x^{4} + 16 \, b^{4} c d^{2} x^{3} - 3 \,{\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{4} + 3 \,{\left (8 \, b^{4} c^{2} d - b^{2} d^{3}\right )} x^{2} + 3 \,{\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} + 2 \,{\left (8 \, b^{4} c^{3} - 3 \, b^{2} c d^{2}\right )} x - 2 \,{\left (2 \,{\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \,{\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} -{\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \,{\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^2*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

1/128*(4*b^4*d^3*x^4 + 16*b^4*c*d^2*x^3 - 3*(8*b^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - d^3)*cos(b*x + a)^
4 + 3*(8*b^4*c^2*d - b^2*d^3)*x^2 + 3*(8*b^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - d^3)*cos(b*x + a)^2 + 2*
(8*b^4*c^3 - 3*b^2*c*d^2)*x - 2*(2*(8*b^3*d^3*x^3 + 24*b^3*c*d^2*x^2 + 8*b^3*c^3 - 3*b*c*d^2 + 3*(8*b^3*c^2*d
- b*d^3)*x)*cos(b*x + a)^3 - (8*b^3*d^3*x^3 + 24*b^3*c*d^2*x^2 + 8*b^3*c^3 - 3*b*c*d^2 + 3*(8*b^3*c^2*d - b*d^
3)*x)*cos(b*x + a))*sin(b*x + a))/b^4

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Sympy [A]  time = 9.97599, size = 813, normalized size = 7.74 \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)**2*sin(b*x+a)**2,x)

[Out]

Piecewise((c**3*x*sin(a + b*x)**4/8 + c**3*x*sin(a + b*x)**2*cos(a + b*x)**2/4 + c**3*x*cos(a + b*x)**4/8 + 3*
c**2*d*x**2*sin(a + b*x)**4/16 + 3*c**2*d*x**2*sin(a + b*x)**2*cos(a + b*x)**2/8 + 3*c**2*d*x**2*cos(a + b*x)*
*4/16 + c*d**2*x**3*sin(a + b*x)**4/8 + c*d**2*x**3*sin(a + b*x)**2*cos(a + b*x)**2/4 + c*d**2*x**3*cos(a + b*
x)**4/8 + d**3*x**4*sin(a + b*x)**4/32 + d**3*x**4*sin(a + b*x)**2*cos(a + b*x)**2/16 + d**3*x**4*cos(a + b*x)
**4/32 + c**3*sin(a + b*x)**3*cos(a + b*x)/(8*b) - c**3*sin(a + b*x)*cos(a + b*x)**3/(8*b) + 3*c**2*d*x*sin(a
+ b*x)**3*cos(a + b*x)/(8*b) - 3*c**2*d*x*sin(a + b*x)*cos(a + b*x)**3/(8*b) + 3*c*d**2*x**2*sin(a + b*x)**3*c
os(a + b*x)/(8*b) - 3*c*d**2*x**2*sin(a + b*x)*cos(a + b*x)**3/(8*b) + d**3*x**3*sin(a + b*x)**3*cos(a + b*x)/
(8*b) - d**3*x**3*sin(a + b*x)*cos(a + b*x)**3/(8*b) + 3*c**2*d*sin(a + b*x)**2*cos(a + b*x)**2/(16*b**2) - 3*
c*d**2*x*sin(a + b*x)**4/(64*b**2) + 9*c*d**2*x*sin(a + b*x)**2*cos(a + b*x)**2/(32*b**2) - 3*c*d**2*x*cos(a +
b*x)**4/(64*b**2) - 3*d**3*x**2*sin(a + b*x)**4/(128*b**2) + 9*d**3*x**2*sin(a + b*x)**2*cos(a + b*x)**2/(64*
b**2) - 3*d**3*x**2*cos(a + b*x)**4/(128*b**2) - 3*c*d**2*sin(a + b*x)**3*cos(a + b*x)/(64*b**3) + 3*c*d**2*si
n(a + b*x)*cos(a + b*x)**3/(64*b**3) - 3*d**3*x*sin(a + b*x)**3*cos(a + b*x)/(64*b**3) + 3*d**3*x*sin(a + b*x)
*cos(a + b*x)**3/(64*b**3) - 3*d**3*sin(a + b*x)**2*cos(a + b*x)**2/(128*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)**2*cos(a)**2, True))

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Giac [A]  time = 1.14039, size = 207, normalized size = 1.97 \begin{align*} \frac{1}{32} \, d^{3} x^{4} + \frac{1}{8} \, c d^{2} x^{3} + \frac{3}{16} \, c^{2} d x^{2} + \frac{1}{8} \, c^{3} x - \frac{3 \,{\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{4}} - \frac{{\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 24 \, b^{3} c^{2} d x + 8 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \sin \left (4 \, b x + 4 \, 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)^2*sin(b*x+a)^2,x, algorithm="giac")

[Out]

1/32*d^3*x^4 + 1/8*c*d^2*x^3 + 3/16*c^2*d*x^2 + 1/8*c^3*x - 3/1024*(8*b^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2
*d - d^3)*cos(4*b*x + 4*a)/b^4 - 1/256*(8*b^3*d^3*x^3 + 24*b^3*c*d^2*x^2 + 24*b^3*c^2*d*x + 8*b^3*c^3 - 3*b*d^
3*x - 3*b*c*d^2)*sin(4*b*x + 4*a)/b^4